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M be a continuous function and let / € H(D) be such that Re f'(z] + C is a continuous function on [0,oo) such that (p(t) ^ 0, t > 0, \ • C" is called a Bloch mapping if \\g\\ = sup{\\D(g o y,)(0)|| : a e Aut(B) be given by (6.1.9). Then II JMI
z E D.
Show that / is univalent on D. (Janiec, 1989 [Jan].) 2.4.5. Show that if / : U —> C is normalized locally univalent and satisfies
then / is close-to-convex. 2.4.6. Let 7 : [0,2vr] —> R be a nonconstant function such that 7 is nondecreasing on [0,7r] and nonincreasing on [7r,27r]. Let
Show that / is close-to-convex. (Kaplan, 1952 [Kap].) 2.4.7. Prove Theorem 2.4.7. 2.4.8. Prove Theorem 2.4.8. 2.4.9. Show that the a-spiral Koebe function is not close-to-convex when c*G(-7r/2,7r/2)\{0}. (Duren, 1983 [Dur].) 2.4.10. Show that f ( z ) = —-- —r^-, cos(p ^ 0, is close-to-convex, but is not spirallike. (Duren, 1983 [Dur].) Hint. The image of the unit disc under this function is the complement of a half-line which is not collinear with the origin.
2.4. Close-to-convexity, spirallikeness and ^-likeness
85
2.4.11. Show that if / is close-to-convex with respect to the convex function /i, then
Hint. Use similar reasoning as in the proofs of Lemmas 2.4.1 and 2.4.2. 2.4.12. Prove Theorem 2.4.9. 2.4.13. Let A,/i e C with |A| = |p| = 1 and
Show that / is close-to-convex. 2.4.14. Use Theorem 2.1.8 to obtain another proof of necessity of the condition in Theorem 2.4.10. Hint. Let g(z,t) = (1 - te~ia)f(z) and p = 1 in Theorem 2.1.8.
Chapter 3
The Loewner theory 3.1
3.1.1
Loewner chains and the Loewner differential equation Kernel convergence
One of the most powerful techniques in the theory of univalent functions is Loewner's theory [L62], developed in the early 1920's. This theory enables us to obtain many far-reaching results which are not accessible via the elementary methods we have considered up to this point. Loewner was influenced by the work of S. Lie on semigroups associated to differential equations. His idea was to express an arbitrary function in a dense subclass of S as the limit of a time-dependent flow starting from the identity function. The flow satisfies a differential equation - the Loewner differential equation. Prom the outset, one of the main applications of Loewner's method was to the study of the Bieberbach conjecture. Not surprisingly, the method plays a central role in de Brange's proof of this famous conjecture [DeB]. However Loewner's method also yields geometric results of considerable interest, such as the radius of starlikeness for the class S and the rotation theorem for the class S. For this reason, we are going to present some of the basic results of Loewner's theory. For further details, the reader may consult [Ahl2], [Becl-2], [Con], [Gol4], [Gon5], [Hay], [Hen], [Kufl-4], [L62], [Pom3], [Ros-Rov], and especially 87
88
The Loewner theory
[Pom5, Chapter 6] and [Dur, Chapter 3]. We begin this section with some basic ideas about kernel convergence. Definition 3.1.1. Let {Gn}neN be a sequence of domains in C such that oo
0 € Gn, for all n € N. If 0 is an interior point of [ j G n , we define the kernel G n=l
of {Gn}n6N to be the largest domain containing 0 such that if K is a compact subset of G, then there exists a positive integer no such that K C Gn for oo n > riQ. If 0 is not an interior point of M G n , we define the kernel to be {0}. n=l
We first prove that the kernel is well defined. We may assume that G ^ {0}. Let G denote the set of domains D such that 0 € D and every compact subset K of D is contained in all but a finite number of sets Gn. We claim that G= D belongs to Q. Certainly G is a domain which contains 0. By the definitions of Q and G, we know that for each z 6 G there is a domain Dz 6 G such that z & Dz, and there is a disc U(z, pz) centered at z and of radius pz whose closure is contained in Dz. If K is a compact subset of G, we may cover K by a finite number of discs U ( z j , p Z j ) , j = l , . . . , f c , such that U(ZJ,PZJ) C Dj for some Dj e GAgain using the definition of G, we have for j = 1, . . . , fc, U(zj, pZj) C G n , for all but a finite number of integers n. Therefore, K C Gn for all but a finite number of sets Gn. We have thus proved that G € G, and it is evident that no larger domain can belong to GWe say that {Gn}ngN converges (kernel converges) to G if each subsequence of {Gn}n€N has the same kernel G. We denote this by Gn —> G. The importance of this concept arises from the Caratheodory convergence theorem [Cara2]. Theorem 3.1.2. Let {Gn}n^ be a sequence of simply connected domains with 0 € Gn and Gn ^ C, for all n € N. Let fn be the conformal mapping of U onto Gn such that fn(0) = 0 and fn(Q) > 0. Let G be the kernel of {Gn}neNThen fn^f locally uniformly on U if and only if Gn —»• G =^ C. In the case of convergence, either G = {0} and f = Q, or else G ^ {0} and then G is a simply connected domain, f gives a conformal map of the unit disc U onto G, and f~* -> f~l locally uniformly on G.
3.1. Loewner chains and the Loewner differential equation
89
Proof. Necessity. First assume that /„ -> / locally uniformly on U. Then / € H(U) and it is either constant or a univalent function, by Hurwitz's theorem. Case (a): If / is constant then it is obvious that / = 0 since /(O) = 0. To prove that the kernel of {Gn}neN is {0}, we argue by contradiction. If there is a domain H such that 0 6 H and each compact subset of H lies in Gn for sufficiently large n, then there exists e > 0 such that U€ C Gn for all n. Let gn = f~l, n € N. Then gn is a holomorphic function on C/€, <7n(0) = 0 and |<7n(w)| < 1 for w 6 Ue. By the Schwarz lemma, one deduces that | e, for sufficiently large n. However, this is a contradiction to the fact that fn —>• 0 locally uniformly on U as n —> oo. Therefore, the kernel of {Gn}ngN is {0}. Similarly, every subsequence of {Gn}neN has kernel {0}, and hence Gn -» {0}. Case (b): We now assume / ^ 0. Let ft = f ( U ) . Then ft is a simply connected domain in C. We have to prove that G = ft and Gn —>• G in the sense of kernel convergence. This is carried out in the following three steps: First step. First, we prove that ft C G. To this end, we have to prove that if K is a. compact subset of ft, then K C Gn for sufficiently large n. To do so we enclose K within a smooth Jordan curve F lying in ft \ K. Let 77 be the distance between F and K, and let 7 = /-1(F) (see Figure 3.1). If VQ € K then \f(z)-vQ\ > n , z €7. Now since fn —t f uniformly on 7, there exists no € N such that
Consequently if n > no and 2 6 7 , then
By Rouche's theorem, one deduces that both equations f ( z ) — VQ = 0 and fn(z) — VQ = 0 have the same number of zeros inside 7 for n > riQ. Since the function f ( z ) — VQ has precisely one zero inside 7, we see that VQ € fn(U) = Gn for all n > no- Noting that no is independent of VQ e K, we conclude that K C Gn for sufficiently large n, and hence ft C G as claimed.
The Loewner theory
90
Figure 3.1: A compact subset K of ft = f(U)
Second step. There exists a subsequence {njt}fceN such that f lim /~ locally uniformly on fi. fc-»00
—
*
Indeed, the inverse functions gn = f~l are well denned on any fixed compact subset of fi for n sufficiently large, and also |sw(^)| < 1 for such n. By Montel's theorem, there exists a subsequence {<7nfc}fc€N of {<7n}neN which converges locally uniformly on 17 to a holomorphic function g on H. This function satisfies 0(0) = 0 and 0, since 0<
1 .. = lim /'(O)
1
= limp' (0) =
Thus g is not constant, and therefore by Hurwitz's theorem g is univalent. We next prove that g = f~l. To this end, let ZQ € U and WQ = /(ZQ). Let 6 > 0 be small enough that the closed disc U(ZQ, 6) is contained in U. Also let ^ = dU(zo, 6) and FI = f ( j i ) . If 771 denotes the distance between w0 and FI, then clearly \f(z) — WQ\ >rji for z € 71, and since /n —> / locally uniformly on £7, there exists a positive integer n'0 such that \fn(z) — f(z)\ < r)i for n > n'0 and 2 € 71. An argument using Rouche's theorem again implies that both equations fn(z] -WQ = Q and /(2) - WQ — 0 have the same number of zeros in 17(20, £) for n > n(), that is one. We then conclude that for each n > n0, there exists a point zn € U(zQ,6) such that fn(zn) = WQ, or zn = ^n(^o)- Thus for sufficiently large ft, we have - g(w0)\
3.1. Loewner chains and the Loewner differential equation
91
Consequently, we obtain that \g(wo) - ZQ\ < \g(wQ) - 9nk(w0)\ + \znk - zQ\ < 26. Since 6 is arbitrary, we must have g(wo) = ZQ, and since ZQ is arbitrary, one deduces that g = f~l. Third step. /-1 = lim f ~ l locally uniformly on ft and ft = G. n—Kx> The argument in the second step implies that each subsequence of {#n}neN contains a further subsequence which converges locally uniformly on ft to f~l. But this implies that the whole sequence {<7n}neN converges to f~l. On the other hand, by Definition 3.1.1 each compact subset of the kernel G of {Gn}n£N is contained in all but a finite number of sets Gn. In view of the first step we know that ft C G. Since f~l = lim f~* exists locally n—¥oo
uniformly on ft, an application of Vitali's theorem shows that 6 = lim f~l n-Kx>
exists locally uniformly on G. Furthermore 0 is univalent on G and |n = f~1. Now if ft ^ G, there exists a point w\ E G \ ft and we must have (iui) € U. However since f~l is a one-to-one map of ft onto U, 0(u>i) = f~1(w2) = (w>2) for some w% € ft. But this is a contradiction with the univalence of
,
V1
\z\)
2,
and this implies that /„ —t 0 locally uniformly on U.
92
The Loewner theory
Case (b): If G ^ {0} and G ^ C, we first prove that {/n(0)}neN is a bounded sequence. Otherwise there exists a subsequence {nfc}jt€N such that /n fc (°) > fc for fc > 1. By the Koebe 1/4-theorem, Gnk D Uk/4. It follows that the sequence {Gnfc}jtgN has the kernel C. However, this is a contradiction with the hypothesis, and hence there exists M > 0 such that /^(O) < M for all n e N. By the growth theorem for the class S, we have
l/n^l < M^-tL,
zeU.
Therefore {/n}neN is a locally uniformly bounded sequence, and hence by Montel's theorem {/n}n€N has a subsequence {/nfc}fceN which converges locally uniformly on U to a function /. By the first part of the proof applied to this subsequence, / is a univalent function on U which maps U onto the kernel of {Gnk}keN- This kernel must be G since Gn ->• G. Since /(O) = 0 and /'(O) > 0, / is uniquely determined by the uniqueness assertion of the Riemann mapping theorem. The same reasoning may be applied to subsequences of {/n}neN) and shows that any subsequence contains a further subsequence which converges locally uniformly on U to the same function /. Hence {/n}neN converges locally uniformly on U to /, as desired. This completes the proof. We remark that Theorem 3.1.2 does not hold for conformal mappings of C. Indeed, if / n (z) = z/n, z € C, then Gn = /n(C) -» C, but /„ -> 0 which is not a univalent function. We recall that a function / € S is called a slit mapping if / maps the unit disc conformally onto the complex plane minus a set of Jordan arcs. These arcs must tend to oo because f ( U ) is a simply connected domain. The function / is called a single-slit mapping if the complement of f ( U ) is a single Jordan arc. Caratheodory's convergence theorem enables us to prove the important result that with respect to uniform convergence on compact sets, the set of single-slit mappings is dense in S. Theorem 3.1.3. Let f G S. Then there exists a sequence of single-slit mappings fn € S such that f n — * f locally uniformly on U. Proof. By considering dilations of /, i.e. fr(z) = f(rz)/r, 0 < r < 1, we may reduce to the case in which / is univalent on U. In this case f ( U ) is a domain bounded by an analytic Jordan curve F: [a, 6] —» C. Let WQ —
3.1. Loewner chains and the Loewner differential equation
93
Figure 3.2: Density of single-slit maps
F(a) = F(6), and let {wn}neN be a sequence of points on F corresponding to an increasing sequence of parameter values {in}neN C [a, 6], such that tn -» b as n —> oo. Also let Tn be a Jordan arc consisting of an arc running from oo to WQ in the exterior of F, followed by an arc which moves from WQ along F to wn (see Figure 3.2). Let fin be the complement of Fn. Also let /„ be the univalent function that maps U conformally onto ftn such that /n(0) = 0 and /^(O) > 0. From this construction it is geometrically clear that ft = f ( U ) is the kernel of {ftn}n€N and that ftn -» ft in the sense of kernel convergence. Using the result of Theorem 3.1.2, we conclude that fn —> / locally uniformly on U. In addition, from the Weierstrass convergence theorem, we deduce that /4(0) —>• /'(O) = 1. Finally, if gn(z) = /n(z)//4(0), then {pn}neN is a sequence of single-slit mappings in 5 such that gn —> / locally uniformly on U. This completes the proof. The foregoing result has the following very important consequence which is used frequently in the Loewner theory. Let \P : 5 —> R be a real-valued continuous functional on 5, in the sense that if {/n}neN C S and fn —> / locally uniformly on U, then ^(/ n ) —>• ^(/)- If we wish to study the problem of finding the number
94
The Loewner theory
then in view of Theorem 3.1.3 it suffices to determine the number sup
where the supremum of \I> is taken over the subset S' of S consisting of singleslit mappings in S. However, single slit mappings do not exist in higher dimensions, and partly for this reason we shall give some applications of the Loewner method in one variable which avoid the reduction to single-slit mappings. Notes. See [Dur]. See also [Gol4], [Pom5], [Con], [Hen] and [Ros-Rov].
3.1.2
Subordination chains and kernel convergence
In this section we shall study properties of Loewner chains (univalent subordination chains), and in the next we shall show how they may be described by the Loewner differential equation. Much of this section is based on [Pom5, Chapter 6]. For further discussion about this subject see [Dur], [Gol4], [Ahl2], [Con], [Hay], [Hen], [Ros-Rov]. In the theory of Loewner chains, if h is a function which depends holomorphically on z G U and is also a function of other (real) variables, it is customary to write h'(z, •) instead of ||(z, •). Definition 3.1.4. The function / : Ux [0, oo) —> C is called a subordination chain if /(•,£) is holomorphic on U, /'(0,t) is a continuous function of t > 0 with /'(0,t) ^ 0 for t > 0, |/'(0,i)l is strictly increasing, /'(0,t) -)• oo as t —> oo, and (3.1.1) /(z, s) -< f ( z , t), zeU, 0 < s < t < oo. Hence, if f ( z , t ] is a subordination chain then there exists a family of Schwarz functions v = v(z, s, £), called the transition functions, such that /Q 1 O~\ ^O.l.^
\ — f ( n^c/^Z, i ( - r eo, +\ Jf(~ ^Z, eo^ — J l/^, +\ t/y,
7 Z c. t Jl (_/,
nU< 2i ^ c* < T_^ «" ^^-' rv-i (-*•'•
The subordination chain f ( z , t ) is called a Loewner chain (or a univalent subordination chain) if f ( - , t ) is univalent on U for all t > 0. There is no loss of generality in assuming that the functions in a subordination chain satisfy the following normalization:
/(0,t) = 0, f'(Q,t) = e*
t>Q.
3.1. Loewner chains and the Loewner differential equation
95
Indeed, if /(z,i) = a0 + a\(t)z + ... is a general subordination chain, and
''>-••] becomes a normalized subordination chain. Prom now on we shall assume that all Loewner chains under discussion are normalized. For a Loewner chain f ( z , t ) let ft denote the univalent function defined by ft(z) — f(z>t). We may think of this Loewner chain as a parameterized family of univalent functions {ft}te[Q,<x>) on ^> indexed by time, /o being the first element, and as t increases to oo the images of the functions expand to fill out the complex plane. If f ( z , t ) is a Loewner chain, then there is a unique family of Schwarz functions v(z, s, i) such that f ( z , s ) = f ( v ( z , s, £),£), z £ U, 0 < s < t < oo. The normalization implies that t/(0, s, t) = e8~t for 0 < s < t. Moreover, since the functions ft and fa are univalent on U, u(-,s,£) is likewise univalent on U. In addition, from (3.1.2) we deduce that the functions v(z,s,t) have the following semigroup property: (3.1.3)
v(z, s, u) = v(v(z, s,t),t,u),
z£U,
0<s
To verify this property, it suffices to observe that for w = v(z, s,t), we have f(v(w,t,u),u) = f(w,t) = f ( z , s ) = f(v(z,s,u),u), hence (3.1.3) follows by the univalence o Further, using the semigroup property (3.1.3) and the fact that \v(z, s,t)\ < \z\, we deduce that |v(2,s,w)| < \v(z,s, t)\,
z£U,
0<s
and thus |v(z, s, t)| is a decreasing function of t G [s, oo). We now prove that between the notions of a Loewner chain and kernel convergence there is a duality. For this purpose, let {G(t)}t>o be a family of simply connected domains such that (3.1.4)
0 € G(s) g G(t) C
0<s<*
96
(3 15)
The Loewner theory f G(tn} -> G(to) G(tn] ->• C
if tn -> t0 < oo if tn -+ oo.
The convergence in question is kernel convergence. We prove the following result (see [Con], [Pom5]): Theorem 3.1.5. (i) Let {G(t)}t>o be a family of simply connected domains which satisfy the conditions (3.1.4) and (3.1.5). For each t > 0 let gt(z) = g(z,t) be the conformal map of U onto G(t) such that
ztU.
Using the relation (3.1.4), we have (3.1.6)
g(z, s) -< g(z, t),
0 < s < t < oo,
and therefore there is a family of Schwarz functions v = v (z, s, t} such that g(z, s) = g(v(z, s, t), t), z e U, 0 < s < t < oo. Hence
a(8) = C as tn —>• oo, we must have a(t) —>• oo as t —> oo. On the other hand, from Theorem 3.1.2 we know that g±n -> gt locally uniformly on U as tn -> t < oo, so the function a is continuous. These arguments prove (a). We next prove the assertion (b). To this end, it suffices to observe that a : [0,oo) -» [ao,oo) is strictly increasing and continuous, hence one-to-one.
3.1. Loewner chains and the Loewner differential equation
97
Consequently (3 is also a strictly increasing function from [0,oo) onto [0, oo). Using the relation (3.1.6) and the above argument, we deduce that f ( z , t ) = aQlg(z,ft~1(t)) is a univalent subordination chain. Moreover, if r = P~l(t] then t = /3(r) and e* = a(r)/ao. Consequently, we obtain = e*.
We conclude that f ( z , t ) is normalized, and moreover f(U,i) = aQ1G(/3~l(t)). This completes the proof of part (i). We now prove part (ii). Clearly G(s) C G(t) for 0 < s < t < oo, and also G(s) 7^ G(t) for s ^ t, by the uniqueness assertion of the Riemann mapping theorem. This implies (3.1.4). The first part of (3.1.5) follows from Theorem 3.1.2 and the estimate (3.1.9) below. On the other hand, since for each t > 0, e~*/('» t) is a function in the class 5, the Koebe 1/4-theorem gives e~*/(£A*) 3 t^i/4- From this we can conclude that G(tn) = f(U,tn) -> C as tn —> oo, and this completes the proof of (ii). The following estimates are very useful in many situations. Among other things they imply that a (normalized) Loewner chain is locally Lipschitz continuous in t, locally uniformly in z. Lemma 3.1.6. If f ( z , t ) is a Loewner chain then
> 0,
(3.1.9) \f(z,t)-f(z,t)\<
. I1
|v,(e'-e'), z € 17, 0 < s < t< oo. \Z\)
Proof. Obviously the relations (3.1.7) and (3.1.8) are simple consequences of the growth and distortion theorems for the class 5, because e~*/(-,£) € S for each t > 0. In order to prove (3.1.9), let v(z, s, t) be the family of transition functions defined by f ( z , t ) , i.e. f ( z , s) = f ( v ( z , s,t),t),
zeU,
Q<s
98
The Loewner theory
Also let P(z,S,t}
= l±g±. *-*(*'*'*), 8 l
l —e ~
z + v(z,s,t)
zeU
Q<S
This function is holomorphic on U and has positive real part. Moreover, p(0, s,t) = 1 and hence p(-,s,t) € P. Therefore, from (2.1.4) we deduce that "s~t
z-v(z,s,t)
and from this it follows that I
v
-
'
/
l
' ' 1
~
v
1 — \Z\
Using (3.1.8) and the above relation we obtain
\f(z, t) - f ( z , s)\ = \f(z, t) - f(v(z, 5,*), t)\ = f /'(C, ^v(z,s,t)
proving (3.1.9). This completes the proof. Example 3.1.7. Let 7: [0, oo) —>• C be a Jordan arc that does not contain the origin, such that i(t} —> oo as t —»• oo, and consider the domains G(t) = C \7([t, oo)), 0 < t < oo. The family {G(t)}t>o satisfies the conditions (3.1.4) and (3.1.5) and hence, using part (b) of Theorem 3.1.5, we obtain an example of a Loewner chain. Example 3.1.7 shows that any single-slit mapping is the initial element of a Loewner chain. For example, f ( z , t ) =
etz
^ is a Loewner chain whose (!- z) initial element is the Koebe function. (It is clear geometrically that /(z, t) = etf(z) is a Loewner chain iff / is starlike.) In fact, we have the following deeper result, which asserts that any function in 5 can be embedded in a Loewner chain. Theorem 3.1.8. For each function f in S there is a Loewner chain f ( z , t ) such that f ( z , 0 ) — j ( z ] on U. Proof. We first assume that / is holomorphic on U. Then the image of the unit circle is a closed Jordan curve C. Let G denote the inner domain of C and
3.1. Loewner chains and the Loewner differential equation
99
let H be the outer domain of C. Let C be the extended complex plane and g be a conformal map of C \ U onto H such that g(oo) = oo. For t > 0 consider the closed Jordan curve Ct = {g(etete) : 0 < 6 < 27r} and its inner domain G(t). Then (3(0) = G and the family {G(t)}t>o satisfies (3.1.4) and (3.1.5). If h(z,t) is the conformal map of U onto G(t) with h(0,t) = 0, h'(0,£) = a(t) > 0, then the uniqueness assertion of the Riemann mapping theorem implies that h(z,0) = f ( z } . Hence letting (3(t) = log[a(t)], we deduce from Theorem 3.1.5 (i) (b) that f ( z , t ) = h(z,/3~l(t)) is a Loewner chain satisfying /(z,0) = /(z), In the general case let / € S and let fn(z) = /(r n z)/r n , where rn = 1 — 1/n and n € N\{1}. Then /„ 6 S and is analytic on U. Using the above arguments we conclude that /„ can be embedded as the initial element of a Loewner chain fn(z,i). The result is now a consequence of the following theorem: Theorem 3.1.9. Any sequence of Loewner chains {/n(z>i)}neN contains a subsequence which converges locally uniformly on U for each fixed t > 0 to a Loewner chain. Proof. Using the relation (3.1.9), we deduce that for each r 6 (0,1), \fn(z,t) - fn(z, s)\ <
8T
(1 - r)*
(et - es),
n € N, |z| < r, 0 < s < t < oo.
The upper bound in (3.1.8) implies that \fn(z,t)-fn(w,t)\<\z-w\ \-\-T
< e*—
rg \z - w\,
I |/'((1 - r)z + TW,t)\dr
Jo
\z\ < r, H < r, t > 0, n € N.
The above estimates imply that the functions fn(z,t), n 6 N, are equicontinuous on {(z,t) : \z\ < 1 — l//c,0 < t < k}, k = 2,3,— Applying the Arzela-Ascoli theorem on the compact space E/i_i/fc x [0,fejx(seee.g. [Roy, p. 167-169]), we conclude that for k fixed, there is a subsequence {/np(z5£)}peN which converges pointwise on {(z,t) :. \z\ < 1 — l/fc,0 < t < k} to a limit /fc(z,t), and furthermore the convergence is uniform. A diagonal sequence argument then shows that there exists a subsequence, which we denote again by {/np(^>*)}p6N5 which converges pointwise in {\z\ < 1,0 < t < oo}, and
100
The Loewner theory
furthermore the convergence is locally uniform on this space. In particular, the convergence is locally uniform on U for each fixed t > 0. Since the limit function f ( z , t ] satisfies the conditions /(0,£) = 0 and /'(O, t) = ef, it follows that f(z,i) cannot be identically constant, and hence is univalent on U, for each t > 0. Also since / np (z, s) -< fnp(z, t) for p = 1,2,..., 0 < s < t < oo, we deduce that fnp(z,s} = fnp(vnp(z,S,t),t),
p = 1,2,...,
where vnp(z, s, t) are univalent Schwarz functions. Since the set {vnp(-, s, £)}PGN is locally uniformly bounded on U, there is a subsequence of {rop}peN> which we denote again by {np}p£N, such that {vnp(z,s,i)}p(=N converges locally uniformly on U to an analytic function v(z, s, t) which is easily seen to be a Schwarz function. Moreover, taking limits for this subsequence, we deduce that /(z,s) = f(v(z,s,t),t}, that is f ( z , s ) -< f(z,t). We have therefore proved that f(z,t) is a Loewner chain. This completes the proof.
3.1.3
Loewner's differential equation
We now derive one of the basic results in Loewner's theory, the Loewner differential equation. There are two versions of the equation, one for the Loewner chain itself and one for the transition functions. A (time-dependent) function of positive real part appears in both of them, and this corresponds to the fact that a Loewner chain represents an expanding flow. There is a more specialized version of the differential equation for single-slit mappings, first considered by Loewner [L62] (see (3.1.29)) and later by Kufarev [Kufl-4]. However, we shall work mainly with the general version, which was first studied by Kufarev [Kufl,3]. (Loewner did, however, write down equation (3.1.10).) Important contributions were made by Pommerenke [Pom3] and also Becker [Bee 1-3]. Our treatment is similar in spirit to that of [Pom5, Chapter 6]. Theorem 3.1.10. Assume p(z,t) belongs to P for each t > 0 and is a measurable function of t € [0, oo) for each z € U. Then for each z £ U and s > 0, the initial value problem (3.1.10)
dv — = -vp(v,t),
a.e. t>s,
v(z,s,s) = z,
3.1. Loewner chains and the Loewner differential equation
101
has a unique locally absolutely continuous solution v(t) = v(z,s,t) = es~tz + — Furthermore, for fixed s > 0 and z € U, u(z, s, t) is Lipschitz continuous in t > s locally uniformly with respect to z. The functions v(-,s,t) are univalent Schwarz functions for 0 < 5 < t < oo, and for each s > 0, the limit (3.1.11)
f(z,s)= lim
exists locally uniformly on U. Moreover /(•, s) is univalent and f(v(z, s, £), t) = f(z,s) for all z £ U and s < t < oo. Thus the function f : U x [0,oo) —>• C given by (3.1.11) is a Loewner chain, and in addition it satisfies the differential equation (3.1.12)
( z , t ) = zp(z,t)f'(z,t)
a.e. t > 0,
for all z € U. (The exceptional set of measure 0 is independent of z.) Proof. The proof combines the classical method of successive approximations with properties of the set P, which allow us to conclude that global rather than local solutions (for a.e. t > s) are obtained. Fix s > 0. Also fix r e (0, 1) and let z € Ur. We shall prove that (3.1.10) has a unique solution on each interval s < t < t\ with t\ > s. We define the successive approximations vn(t) = vn(z,t) such that vo(t) = 0 and
r\- Ir* pK(r),r)dri ,
(3.1.13)
L
J8
J
for t > s. These approximations are well defined, since |vn(£)| < r by an inductive argument using the fact that p(z,t) € P. For each t > s, vn(z,t) is holomorphic in z, vn(0, t) = 0, v'n(Q,i) = es~l, and vn(z,s) = z, for n = 1,2,.... Now, the fact that Re p(z,t) > 0 implies by (2.1.6) that \p'(z,t)\<
( 1 _j* r ) 2 .
N
t>0.
Since for a, 6 € C7, we have /"b p(6, t) — p(o, i) = / p(C*)dC> ^o
102
The Loewner theory
where the integration is over the straight line segment [a,b], we easily deduce that (3.1.14)
lp(a^.p(b^\<^^,
for |a| < r, \b\
(3.1.15)
for all zi, Z2 € C with Re z\ > 0, Re z^ > 0, we deduce that vn(i}\
-p(vn-i(T),r)\dT
Js
<
\vn(T)
(l-r)'/ From this we obtain by induction that
Vn ]
Since
the above estimates now imply that the limit v(t) = v(z,s,t) = lim vn(t) = lim n—too
n—>oo
j=l
exists uniformly on \z\ < r, s < t < t\, for each t\. This limit is a continuous function of t, and is analytic as a function of z. Moreover, using (3.1.13), we have \v(z,s,t}\ < \z\. Taking into account the fact that ^(0, t) = es~*, n = 1, 2, . . ., we deduce that v'(0,s,t) = eS ~*Using Lebesgue's dominated convergence theorem, we conclude that v(z,s,t] satisfies the integral equation v(z,s,t) - zexp T - //"* p(v(z,s,T),T)dr\ 1 . L
Js
J
3.1. Loewner chains and the Loewner differential equation
103
Hence v(z, s, t) is a locally absolutely continuous function of t € [s, oo) and satisfies the differential equation
dv — = —vp(v,t),
a.e. t>s,
v(s) = z.
C7C
Moreover, v(z, s, t) is a Lipschitz continuous function of t > s locally uniformly with respect to z. Indeed, using the relations (3.1.15) and (2.1.4), we obtain
= z
exp [ - /
p(v(z, s, r), r)dr] - exp [ - /
p(v(z, s, T), r)dr]
for \z\ < r and s < t\ < t-z < oo. We next prove that the solution of (3.1.10) is unique. To this end, suppose u(t) is another locally absolutely continuous solution of (3.1.10) such that u(s) = z. Using the relations (3.1.14) and (3.1.15), we obtain \u(t)-v(t)\
if*
i
r /**
11
z exp - / p(u(r},r}dr\ -zexp - / p(v(r),r)dr L Js J L Js JI
T)— Ml*, for t > s and \z\ < r. Consider a subinterval [s,£i] of [s, oo), and let M > 0 be such that \u(t) — v(t)\ < M on [s,ti]. Then for t e [ s , t i ] , we obtain the estimate
Again using the method of mathematical induction, we conclude that
!«(«) - v(t)\ < for all n e N. Therefore w(t) = v(t) on [s,ti] and this assures us that the solution of (3.1.10) is unique, as claimed.
The Loewner theory
104
The semigroup property (3.1.16)
V(*,S,*) = V(V
,T,*),
0<s
also follows from the uniqueness of solutions to initial value problems of the form (3.1.10). Indeed, both sides of (3.1.16) satisfy the differential equation and have the same value when t = r. We next prove that the solution v(z,s,t) of (3.1.10) is univalent for each t > s. Indeed, suppose that to > s and z\, z% € U are such that v(zi,s,to] — v(zz, s, to)- Also let Vj(t) = v ( z j , s, t) for j = 1, 2, and let w(t) — vi(i) — v?.(t}. Then w(t0) = 0. Since Re p(z,t) > 0 and p(0,t) = 1, the Herglotz integral formula (2.1.3) gives the estimate + N 1 + \Z2\ , , j—r • --j— r Zi - Z 2 |, - \Zi\ 1 - \Z2\
for zi, 22 € U and t > 0 (see Problem 3.1.3). Consequently, we obtain ,t) <
^+_N . 1+_N,
V2(t)p(v2(t},t)\
((),
8
Now choose K > 0 such that \w(t)\ < K for s < t < to- Then Tr(r)dT
N
-(t 0 -t). f\
Substituting back into the estimate for — (t) gives C'C
r° dw_ dw, to
Jt
dr (to -
3.1. Loewner chains and the Loewner differential equation
105
By induction, we may obtain a sequence of estimates which lead to the conclusion that w(t] = 0 on [s,to]. Then the relation w(s) = 0 and the fact that v(zj, s, 5) = Zj for j = 1,2, imply that z\ = z-i, as claimed. Now, since the function e*~ s u(-,s,£), t > s, belongs to 5, the growth theorem gives (3.1.17)
\v(z,s,t)\<
|Z|
e8-*,
zeU, t>a. v \ \) Further, using (3.1.14) and (3.1.17), we obtain for \z\ < r and t > s that 1
z
Combining this relation with the equality et~sv(z,s,t) = zexp / (1 -p(v(z,s,r),r))dr\ , \.Ja
J
t s
we conclude that the family {e ~ v(z,s,i)}t>a is Cauchy (locally uniformly in 2), and hence the limit /(z, s) = lim etv(z, s, t) t—^oo exists locally uniformly in z. Prom (3.1.16) we deduce that /(-z, s) = lim etv(z, s, t) t—too
= lim e*v(v(z, s, T),r,t) = f ( v ( z , s,r),r), t—too
EU and 0 < s < r < oo. Thus f ( z , i ) is a subordination chain. Also ),£) = 0 and /'((),£) = e* for t > 0. Then by Hurwitz's theorem, /(•,<) is univalent on U for each t > 0. Therefore f ( z , i ) is a Loewner chain. The proof that equation (3.1.12) is satisfied is contained in Theorem 3.1.12. We are now able to prove the following characterization of Loewner chains, which is one of the main results of the theory. There is also a version of this theorem for non-normalized Loewner chains [Pom3, FolgerungS] (see Problem 3.1.6). For further discussion of equation (3.1.19), the reader may consult [Becl] and [Pom3].
106
The Loewner theory
Theorem 3.1.11. The function f : U x [0,oo) ->• C with /(0,t) = 0, /'(0,£) = et , t > 0, is a Loewner chain if and only if the following conditions hold: (i) There exist r e (0, 1) and a constant M > 0 such that f ( z , t ) is holomorphic on Ur for each t > 0, locally absolutely continuous in t > 0 locally uniformly with respect to z € Ur, and (3.1.18)
|/(z,t)|<Me',
|z|
t > 0.
fz'z^ There exists a function p(z,t) such that p ( - , t ) € P for each t > 0, p(z, •) is measurable on [0, oo) for each z € U, and for all z € Ur, (3.1.19)
(M) = */'(*, «)POM),
a.e. t > 0.
(mj .For each t > 0, f ( - , t ) is the analytic continuation of /(-,£) |t/r to Z7, and furthermore this analytic continuation exists under the assumptions (i) and (ii). Proof. First assume that f ( z , t ) is a Loewner chain. We shall show that the conditions (i) and (ii) hold for each r € (0, 1). Since e "*/(-,£) is a function in S for each t > 0, the growth theorem implies that for each r 6 (0, 1) there is a positive number M = M(r) such that \ f ( z , t \ < Me* for \z\
=
[f(z, t + h)- f(v(z, t,t + h),t +
= A(z, t, h] (- \z - v(z, t, t + /i)l V
z e t/, i > 0, h > 0,
where A(z,t, /i) is a real-linear operator which tends to the complex-linear operator f'(z, t) as h —> 0 + . Now (3.1.9) implies that /(z, t) is locally Lipschitz
3.1. Loewner chains and the Loewner differential equation
107
in t, locally uniformly in z. Hence from Vitali's theorem, except for a set of measure 0 in £, the left-hand side of (3.1.21) has a limit as h —* 0 for all z. For such t, the one-sided limit as h —>• 0+ of the right-hand side must also exist, and we obtain the function p(-,i) in (3.1.19) using Theorem 2.1.7 and the normalization of the transition functions. For t in the exceptional set of measure 0, we define p(z, t) = 1, z € U. The measurability of p(z, •) on [0, oo) follows from the measurability of -$[(z, •) and f'(z, •) on [0, oo). (From the Cauchy integral formula and the local Lipschitz continuity of f ( z , •), it follows that f ' ( z , - ) is also locally Lipschitz continuous on [0, oo).) This completes the proof of (ii). We now prove the converse statement. For this purpose, let r € (0,1), M > 0, f(z,i) and p(z,t) satisfy the assumptions (i) and (ii). We show that f(z,i) is locally Lipschitz continuous in t locally uniformly with respect to z € Ur. To accomplish this, let p 6 (0,r) and let T > 0. Using the Cauchy integral formula and (3.1.18), we deduce that there exists an L = L(p,T) > 0 such that (3.1.22)
|/'(z, t)|
\z\
t€[0,T\.
Moreover, in view of (3.1.19), (3.1.22), and the fact that
one deduces that there is an N = N(p, T) > 0 such that
—( z it) ^ ^(PJ^OJ \z\ ^ PI °-e- ^£ [O'-^1]Further, since £ f -, 4. Z
\
•( (~ Z
4-
\
J\ 1 ll) ~ }\ 1 ^lj
=:
/**2 df*
/
/ ./t!
/ A/
4> \ sJ-l:
~Q~V' Z > ^l^l ^*
f\ *** J-
^4*
^ T1
" S ^1 < ^2 S -^ J
we deduce that (3.1.23)
\ f ( z , ti) - f ( z , t2)\ < N(p,T)(t2 - ti)
for |z| < p and 0 < t\ < ti < T. Since p € (0,r) and T > 0 were arbitrarily chosen, the conclusion follows.
108
The Loewner theory
Taking into account Theorem 3.1.10, we conclude that the initial value problem dv — = -Vp(v, t), a.e. t>s, v(z, s, s) = z, has a unique locally absolutely continuous solution v(t) = v(z, s,t). Moreover, for fixed s and t, u(-, s, t) is a univalent Schwarz function. For z 6 Ur, s > 0 and t > s, let f ( z , s , t ) = f ( v ( z , s , t ) , t ) . Since v = v(z,s,t) is Lipschitz continuous in t G [s, oo) locally uniformly with respect to z G U by the proof of Theorem 3.1.10, and /(z, t) is also locally Lipschitz in t, we easily deduce that /(z, s, t) is locally Lipschitz continuous in t for t € [s, oo) locally uniformly with respect to z e C/,. as well. Indeed, from (3.1.22) we obtain I/CM) -/(«;,«)!< / |/'((1 -T)z + TtM)|dT
S, ti), ti) - /(«(«, S, t i ) , t 2 )| + !/(«(«, S, tl), t 2 ) - /(U(^, 5> *2), *2)|
< N(p,T)(t2 - ti) + R(p,T)(t2 - *i), for |z| < p and s < ti < i2 < T. Q
It follows that for all z e t/r, —f(z,s,t) at moreover
exists for almost all t > s, and
/(Z, S , t) = f(V(z, S, «), * ) ( « , «. *) +
(W(Z, 5, *), t)
= 0,
a.e. t > s.
Because v(z, s,s) = z and /(z, 5, t) is a locally absolutely continuous function of t, we deduce that /(z, s, t) is identically constant as a function of t, i.e. /(z, s, i) = /(z, s, s), and hence (3.1.24)
f ( v ( z , s, t ) , t ) = /(z, s),
|z| < r,
0 < s < t< oo.
3.1. Loewner chains and the Loewner differential equation
109
We next extend the function f ( z , t) univalently to the whole disc U. Using (3.1.18), we have |e~*/(2, t) - z\ < 1 + M, and since e~tf(z11) — z = a2(t)z2 -\ obtain
\z\ < r,
t > 0,
, we can apply the Schwarz lemma to
r2
—
—
t s
Also since e ~ v(-,s,t) belongs to S, the growth theorem gives \v(z, s, t)| <
'*'
h2e
8
(.1 ~ FU
"*,
zeU,
0<s
Hence from (3.1.24) we obtain I f( y
a\
o 11 (v
a +\\ — t> \o
|/v*>*y — 6 " v > > y l —
I"
f(ii(y
a +}
#^
ti( v
c ^M
/ V \ ' " » " / » " / — *'^*> "> "/|
Prom this we see that (3.1.25)
e*v(z,8tt) ->• f ( z , s ) as t -* oo,
uniformly on \z\
110
The Loewner theory
Moreover, for each s > 0 and z e U, v — v(z,s,t) is the unique locally absolutely continuous solution of the initial value problem (3.1.27)
dv — = — vp(v,t), c/c
and the limit (3.1.28)
a.e. t > s,
v(z,s,s) = z,
lim e*v(z, s, t) = /(z, s)
t—>oo
exists locally uniformly on U. Proof. The existence of the function p(z, t) such that (3.1.26) holds follows from the first part of the proof of Theorem 3.1.11. Let u = u(z, s, t) be the locally absolutely continuous solution of the initial value problem r\
— = — up(u, t), or
a.e. t>s, u(z,s,s) = z,
for each fixed s > 0 and z € U. Then u(-,s,t) is a univalent Schwarz function for t > s. Since f ( z , t ) is a locally absolutely continuous function of t, it is differentiate a.e on [0,oo), and with similar reasoning as in the proof of Theorem 3.1.11 we deduce that f(u(z,s,i),t) is also a locally absolutely continuous function of t and hence differentiable a.e. on [s,oo). Therefore in view of (3.1.26) we obtain that n -f(u(z,s,t),t) fjtll
= f'(u,t)—+uf'(u,t)p(u,t) ot
= Q,
a.e. t > s.
Hence
f(u(z, s, t),t) = f(u(z, s, s),s) = f ( z , s), and thus f ( v ( z , s, t), t} = f ( u ( z , s,t),t),
zeU,
t>s.
Since /(•, t) is univalent on U, we must have w(z, s, t) = v(z, s, i) for all z £ U and 0 < s < t < oo. Consequently, v(z, s,t) satisfies the initial value problem (3.1.27). Moreover, from Theorem 3.1.10, (3.1.28) follows. This completes the proof.
3.1. Loewner chains and the Loewner differential equation
111
We remark that both equations (3.1.26) and (3.1.27) are called the Loewner differential equations. Also we remark that Theorems 3.1.10, 3.1.11 and 3.1.12 imply the following uniqueness result for Loewner chains which satisfy the differential equation (3.1.26) (compare with [Bec3-4]). The corresponding uniqueness result in several variables is false, as we shall see in Chapter 8. Theorem 3.1.13. Letp : U x [0, oo) ->• C be a function such thatp(-,t) € P, t > 0, and p ( z , - ) is measurable on [0, oo), z € U. Then there exists a unique Loewner chain f ( z , t ) which satisfies the Loewner differential equation (3.1.26). Proof. Existence. In view of Theorem 3.1.10, the initial value problem dv — = -vp(v,t), ot
a.e. t>s, v(s) = z,
has a unique locally absolutely continuous solution v(t) = v(z,s,t) for all z € U and s > 0, and the limit lim e*v(z, s, t) = f ( z , s) exists and gives a Loewner chain which satisfies (3.1.26). Uniqueness. Let g(z,t) be a Loewner chain which satisfies (3.1.26). Also let w(z,s,t) be the transition functions associated to g(z,t). Prom Theorem 3.1.12, g(z, s) = lim etw(z:s,t) t—KX>
locally uniformly on U for each s > 0, and w(t) = w(z, s, t) satisfies the initial value problem (3.1.27). Thus g ( z , s ) = f ( z , s ) for all z e U and s > 0. This completes the proof. Loewner realized that the single-slit mappings in S could play a special role in his theory because of Theorem 3.1.3. In this case the associated Loewner chains are of the type discussed in Example 3.1.7, and the Loewner differential equation takes the form
&->-*•<••*>£$;•
112
The Loewner theory
where k is a continuous function from [0,oo) into C such that \k(t)\ = 1 for 0 < t < oo. The basic result for single-slit mappings is the following [L62] (cf. [Ahl2], [Con], [Dur], [Gol4], [Hay], [Hen]): Theorem 3.1.14. (i) Letc : [0, oo) —> C be a continuous function such that \c(t)\ = I, t 6 [0, oo). Then for each z e U , there exists a unique function w = w(z,t) such that w(z,-) is of class C1 on [0, oo) and satisfies the differential equation
Moreover, w(-,t) is a univalent Schwarz function and w'(Q,t) = e~* for each t > 0. (ii) If f € S is a single-slit mapping, then there exists a continuous function k : [0,oo) ->• C with \k(t)\ = 1, t > 0, such that lim etv(z,t) = f ( z ) locally t—HX> uniformly on U, where v = v(z, t) is the solution of the differential equation dv
1 + k(i)v
n
, ^
for all z € U. (Hi) Further, in case (ii) there exists a Loewner chain f ( z , t ) such that f ( z ] = f(z, 0), zeU, f ( z , •) € ^([O, oo)) for each zeU, and
The equation (3.1.29) was also studied in detail by Kufarev in the 1940's [Kufl-4]. In particular he observed that given a continuous function c : [0, oo) —>• C with \c(t)\ = 1, t > 0, the solution w = w(z, t) of the Loewner differential equation (3.1.29) need not give rise to a single-slit mapping [Kuf2].
3.1.4
Remarks on Bieberbach's conjecture
Perhaps the most famous problem associated with the class 5 is the Bieberbach conjecture, formulated in 1916 [Biel] and solved by de Branges [DeB] in 1985. This conjecture and two related conjectures due to Robertson and Milin
3.1. Loewner chains and the Loewner differential equation
113
influenced the development of univalent function theory for more than sixty years. The Loewner method played an important role in the solution, but there are other ideas involved which do not have generalizations to several variables, and we shall therefore not give a proof of de Branges' theorem here. As well as the paper of de Branges, the reader may consult the references [Fit-Pom], [Con], [Gon5], [Hay], [Hen], [Ros-Rov] for the complete proof. The Bieberbach conjecture is the following: Conjecture 3.1.15. If / 6 S has the power series expansion f ( z ) = z + a,2Z2 + . . . + anzn + . . . , then (3.1.30)
\0n\ < n,
n > 2.
Moreover, if there is an integer n > 2 such that equality holds in (3.1.30), then / is a rotation of the Koebe function. Robertson [Robe2] observed in 1936 that the following statement about odd functions in S implies the Bieberbach conjecture: Conjecture 3.1.16. For each odd function h € S, h(z) — z + b$z3 + b5Z5 + • •• + b2n-iz2n~l + . . ., the inequality (3.1.31)
l + |&3|2 + ... + l&2n-i| 2
holds for each n > 2. If there is an integer n > 2 such that equality holds in (3.1.31), then [h(z)f = /(z 2 ), z € U, and / is a rotation of the Koebe function. Next we mention Milin's conjecture (see [Mili2]) which implies Robertson's conjecture, and thus the Bieberbach conjecture. oo
Conjecture 3.1.17. Let / € S and g(z] = \og[z~lf(z}] = ^ckzk, z£U, k=l
where the branch of the logarithm is chosen such that g(0) = 0. Then the inequality
(3.1.32)
E£
l=i fc=i holds for each n > 1. If there is an integer n such that equality holds in (3.1.32), then / is a rotation of the Koebe function. In a remarkable paper in 1985, de Branges [DeB] proved
114
The Loewner theory
Theorem 3.1.18. The Milin conjecture is true. De Branges' proof of the Milin conjecture uses the reduction to singleslit mappings and some ingenious ideas about systems of special functions. A simplified proof was given by FitzGerald and Pommerenke [Fit-Pom]. Other versions of the proof are presented in the books cited above. Loewner in his original paper [L62] applied the differential equation (3.1.29) to prove the Bieberbach conjecture in the case n = 3. Here we show how the general form of the Loewner differential equation (3.1.26) can be used to prove the same result (see also [Pom5]). Theorem 3.1.19. /// € S is given by f ( z ) = z + a2z2+asz3 + . . ., z € U, then \a,3\ < 3. This bound is sharp. Proof. Since / G 5, it follows from Theorem 3.1.8 that there is a Loewner chain f ( z , t ) such that /(z) = /(z,0), z € U. Let f ( z , t) = e*z + a2(t)z2 + ...+ an(t)zn + . . . , z 6 U, t > 0, where the coefficients an(t) are locally absolutely continuous functions of t e [0, oo) (see Problem 3.1.2) and an = an(Q) for n = 2, 3, — Now by Theorem 3.1.12, there is a function p(z,t) such that p ( - , t ) € P, p(z,t) is measurable in t € [0, oo) for all z G 17, and for almost all t > 0, (3.1.33)
(z,t) = z/'(;M)p(z,t),
Vz € U.
Let p(z,f) = l+pi(t)z+p2(t)z2 + ...+pn(t)zn + ..., z<EU,t>0. Identifying the coefficients in both sides of (3.1.33), we deduce that n l d ~~ — an(t] = ^ja,j(t)pn-j(t) j=i
+ non(t),
a.e. t > 0,
for n = 2, 3, ---- Multiplying both sides of this equality by e~nt and integrating, we obtain for 0 < s < t < oo that T ar ^-(e-nr an(r}]dr = ]T j f ** e~nr a^pn-
Js
- =1 Js
3.1. Loewner chains and the Loewner differential equation
115
Letting t —>• oo in the above, we obtain (3.1.34)
an(s) = v
1*—1
&
forn = 2,3,..., and s > 0. Here we have used the fact that lim e~ntan(t) = 0, t—>oo
n = 2,3,..., since e~*/(-,£) € S, t > 0 (compare with Chapter 1). Consequently, for n = 2 in (3.1.34), one deduces that a2(s) = -e23 f
(3.1.35)
e-TPi(r)dr.
Js
Moreover, for n = 3 and s = 0 in (3.1.34), we obtain by an elementary computation using (3.1.35) the equality 03 = - /f°° e i p2(r)dr-2 If°° e~3Ta2(T)pi(r)dr Jo
Jo
/•OO
POO
e~2Tp2(r)dr + 2 /
=Jo
.
Jo oo
/
f<X>
e~Tpi(r}(
^
e^pi^dt^dr \Jr
'
/ /-oo
\2
\Jo
J
It suffices to show that Re 03 < 3, since there is a rotation of / whose third coefficient is |os|. We have (3.1.36)
Re 03 = - f
e~2TRe p2(r)dr + ( f
U
oo
e
T
e~TRe pi(r)dr J \2
Impi(r)dr J .
Using Problem 2.1.9, we have z
&
\ r > 0.
Taking the negative real part on the left-hand side, we obtain 2 + ReP2(r) > ^Re pf (r) + i|pi(r)|2 = [Re pi(r)]2, r > 0.
116
The Loewner theory
We use this estimate in the first integral on the right side of (3.1.36). We apply the Schwarz inequality to the second integral, breaking the integrand into the factors e~ r / 2 and e~ T / 2 Re pi(r), and we neglect the third term. This gives Re as < / [2e-2T + (e~r - e~ JT )(Re pi(r)Y]dT < 3, Jo using the fact that |Re Pi(r)| < 2, r > 0 (see Theorem 2.1.5). Equality |as | = 3 occurs for the Koebe function and its rotations. This completes the proof. Notes. The results of Section 3.1 concerning Loewner chains and the Loewner differential equation may be found along with further material in [Pom3], [Pom5], [Dur], [Ros-Rov], [Ahl2], [Con], [Gol4], [Hay], [Hen], [Gon5], [Becl-4], [Bran], [L62], [Kufl-5], [Biel-Lew]. See also [Bae-Dra-Dur-Mar]. Problems
3.1.1. (i) The transition functions v(z,s,t) of a Loewner chain satisfy estimates which correspond to (3.1.9), namely \v(z,
5i, t) — v(z, 52, t)\ < —
2 (1
— e S l ~ S 2 ),
0 < 5i < 52 < t < OO,
and
1*1 — el \v(z,s,ti) — v(z,s,£2)l < 2|z| 1 + —(\. *
"1 1
I -y I \Z\
2
),
0 < s
Prove these estimates. (ii) Prove that on any interval (0, £],
dv
— (z,s,t) = zp(z,s)v'(z,s,t),
a.e. s e ( 0 , t ] ,
v(z,t, t) = z,
where p(z,s) is the function given by Theorem 3.1.12. 3.1.2. Show that the Taylor series coefficients of the functions ft in a Loewner chain /(z, t) are locally absolutely continuous functions of t. 3.1.3. Show that if p G P and z\, z2 € C/, then
3.2. Applications
117
3.1.4. If/ is the Koebe function and f ( z , t) is the Loewner chain associated to /, i.e. /(M)=
(T:V
2617
' *- 0 >
find the function k with \k(t)\ = 1 that appears in Loewner's equation
Do the same for a rotation of the Koebe function. 3.1.5. Prove Theorem 3.1.14. 3.1.6. Deduce the following from Theorem 3.1.11 by making a change of variable: Let r 6 (0, 1). The function f ( z , t ) = a\(t}z + • • • on Ur x [0, oo) is the restriction of a (non-normalized) Loewner chain denned on U x [0, oo) if the following conditions hold: (i) /(•,£) is holomorphic on Ur for each t > 0, and f ( z , - ) is locally absolutely continuous on [0,oo) locally uniformly with respect to z € Ur\ (ii) ai(-) e C^QO, oo)), a\(i} ^ 0, t > 0, |ai(-)| is strictly increasing on [0, oo), |ai(£)| —> oo as t —)• oo, and there exists a constant M > 0 such that |/OM)| < M|oi(t)|,
|z|
t>0.
(iii) There exists a function p(z,t) on U x [0, oo) such that p ( - , t ) is holomorphic and has positive real part on U (but is not normalized), p(z, •) is measurable on [0, oo) for each z € Z7, and for all z € Ur, t),
a.e. t>0.
(Pommerenke, 1965 [Pom3].)
3.2
Applications of Loewner's differential equation to the study of univalent functions
In this section we shall investigate certain applications of Loewner's method to geometric problems such as the radius of starlikeness and the rotation theorem for the class S. We shall also show that for certain subclasses of S there exist alternative characterizations in terms of Loewner chains.
The Loewner theory
118
3.2.1
The radius of starlikeness for the class S and the rotation theorem
We first apply Loewner's method to determine the radius of starlikeness of S. This result was obtained by Grunsky [Gru2]; Goluzin [Gol2] applied Loewner's theory to the related Theorems 3.2.1 and 3.2.4 and Corollary 3.2.2. We begin with the following bound for arg
for / € 5 (see [Grul]),
where the branch is chosen such that arg /(*)
Theorem 3.2.1. /// 6 S then arg
700
= 0. z=0
This bound is sharp for each z G U.
Proof. Step 1. Using Theorem 3.1.3, we may suppose that / is a single-slit map. Then from Theorem 3.1.14 there exists a continuous function k : [0, oo) —>• C with \k(t)\ = 1, t > 0, such that (3.2.1) locally uniformly on U, where v — v(z,t) is the unique solution of the initial value problem (3.2.2)
dv _ _ 1_+ k(t)v
>0,
u(z,0) = z.
For fixed z € U \ {0}, log v(z, t) can be defined as a continuous function of t beginning with a particular value of logz when t — 0, and (3.2.2) yields
d
l + k(t)v(z,t)
Taking real and imaginary parts, we obtain C\ 9 Q^ \3-4-3)
and (3.2.4)
v z f\\ — —\u(z ^.\\ii(y \ ^)\— \V\Z1
l-\v(z,t)\
3.2. Applications
119
Hence from (3.2.3) we deduce that \v(z, t)\ decreases monotonically from \z\ to 0 as t increases from 0 to oo, and therefore there is a one-to-one correspondence between t and |u|, which enables us to consider \v\ as the independent variable. Moreover, from (3.2.3) we obtain dt= -•dt\v\
(3.2.5)
|1-H2
Taking into account the relations (3.2.3), (3.2.4) and (3.2.5) we easily deduce that \dtargv(z,t)\ <
2dt\v(z,t)\
2\v(z,t)\dt \l-k(t)v(z,t)\*
On the other hand, since arg
~v(z,t)
d o or
we obtain arg
v(z,t) z V(Z,T)\
Letting t —> oo in this inequality and using (3.2.1), we obtain the desired bound. Step 2. To prove the statement about sharpness, it is enough to show that for each z 6 U \ {0}, we can find a continuous function k with \k(t)\ = 1 on [0, oo), and for which the function v = v(z, t) given by (3.2.2) satisfies the relation (3.2.6) Im [k(t)v(z,t]\ = -\v(z,t)\, t € [0,oo). Indeed, if the relation (3.2.6) holds with \k(t)\ = 1, then Re [k(t)v(z, t)] = 0,
t € [0, oo),
and in view of (3.2.3) and (3.2.4) we deduce that f1) 0 O 7\
V -^-'/
\1l( 9
t\\ — —\9i(y f^l U Z l
^.\U\Z^)\—
\ \ l )\-i
I
V
, i /
>
/I
,\|0
120
The Loewner theory
and (3.2.8) \
— argv(z.t) = *j i
/
O
\
/
/
-i
I
-r^-r-—rpr\ „,( -j
4.\\ s.
Separating variables in (3.2.7), integrating from 0 to £, and using the fact that u(z, 0) = z, we deduce that
Solving this equation, we obtain |i>(z,t)| as a function of t G [0, oo) which decreases monotonically from \z\ to 0 as t increases from 0 to oo. Next, we may use this function and the relation (3.2.8) to deduce that
dtargv(z,t) = —
2dt\v(z,t)\
2.
Hence, integrating both sides of this equality from 0 to t, we obtain
0.2.9)
ar
where again we have used the equality v(z,Q) = z. Therefore we have defined the function v = v(z,t), and taking into account (3.2.6) the function k is determined by the equality k(t)v(z,t) = -i\v(z,t)\,
t>0.
This function is continuous in t € [0, oo) and satisfies the condition \k(t)\ = 1, t > 0. Also it is clear that the Loewner differential equation (3.2.2), corresponding to the function fc, has exactly the solution v(z, t) determined above. Moreover, if we let f(z] — lim e*v(z,t), we deduce in view of (3.2.9) that f(z) arg -^ = log
This proves the sharpness of the upper bound in Theorem 3.2.1. In a similar manner, we can prove the sharpness of the lower bound. A direct consequence of the above result is the following bound for arg
when / G S (see [Grul]). The branch is chosen such that
3.2. Applications
arg
*/'(*) /(*)
121
= 0. This result can be used to obtain the radius of starlike2=0
ness of S. Corollary 3.2.2. // / € S then (3.2.10)
arg "*/'(*)
M
T/MS estimate is sharp for each z &U. Proof. Fix ZQ 6 U \ {0} and consider the Koebe transform
9(*) = Then g € S and in view of Theorem 3.2.1 we obtain
r*«i|< z \\ On the other hand, it is obvious that g(-zo) -zo
f(z0) z0f'(z0)
l-
and hence the above relations imply that
Jl as desired. The sharpness of (3.2.10) follows from the sharpness of the estimate in Theorem 3.2.1. This completes the proof. The following result, due to Grunsky [Gru2], establishes the radius of starlikeness of S. For comparison, we recall that the radius of convexity of 5 is 2 — v^ w 0.27 (see Theorem 2.2.22), while the radius of close-to-convexity is given in Theorem 3.2.5. 7T Corollary 3.2.3. The radius of starlikeness of S is tanh— . i.e. 4
, TT 6 2 — 1 = tanh- = — 4 e z 1
0.66.
122
The Loewner theory
Proof. Taking into account Theorem 2.2.2 and Corollary 3.2.2, we conclude that / € S*(Ur), where r is determined by solving the equation log
1 +rl TT 1-r ~ 2 '
7T
that is, r = tanh —. Thus 4
,
W
and this relation may fail for \z\ > r, in view of the sharpness of the estimate (3.2.10). Hence r*(5) = tanh . There is a related estimate for log —T^T when / € 5, where the branch /(*) */'(*) is chosen such that log = 0. This result was also obtained by
/(*)
z=0
Grunsky [Grul]. We leave the proof for the reader, since it suffices to use similar arguments as in the proof of Theorem 3.2.1. Theorem 3.2.4. // / e S then
108
[TC?
This estimate is sharp for each z e U. In 1962 Krzyz [Krz2] obtained the radius of close-to-convexity for the class 5. This is the largest number rcc such that every function in the class S is close-to-convex on Urcc- We omit the proof of this result; however we mention that it is of interest for the reader. Theorem 3.2.5. For r e (r*(S),l), let
and let XQ(r) be the unique root of the polynomial
P(X) = X3 - AX2 + A2X -A = Q. Then the radius of close-to-convexity of the class S is the unique real root rc of the equation 2arccot i X
0
( r ) + log(l + X 0 2 (r)) - 2log
3.2. Applications
123
contained in the interval (r*(S), 1). In particular, 0.80 < r^ < 0.81. Another application of Loewner's theory is the following estimate for | arg f ' ( z } \ due to Goluzin [Gol2]. (In fact the sharpness of the bound for \z\ > l/\/2 was first established by Bazilevich [Bazl].) Because of the geometric interpretation of arg/'(z) as the local rotation factor under the mapping / € S, Theorem 3.2.6 is called the rotation theorem for the class S. Again the branch is chosen such that arg f'(z)\z=Q — 0. We note that with an elementary proof, based on inequality (1.1.9), it is possible to show that if / e 5, then |arg/ / (z)|<21ogfitM], L 1 PlJ
z G U.
However, this estimate is not sharp when z ^ 0. Theorem 3.2.6. Let f € S. Then 4arcsin|;z|,
\z\ < —=.
I arg f ' ( z ) | <
These estimates are sharp for each z G U.
Proof. Step 1. Again using Theorem 3.1.3 we may suppose that / is a single-slit mapping. We may also assume that z ^ 0. Then using Theorem 3.1.14 we may express / as the limit
f(z] - lim etv(z,t) t —K3O
locally uniformly on U, where v = v(z,t) is the solution of the initial value problem
and k : [0, oo) —> C is a continuous function such that \k(t)\ = 1, t > 0. By Weierstrass' theorem, we conclude that
124
The Loewner theory
Differentiating both sides of equation (3.2.11) with respect to z (this is possible since v(z, •) is of class Cl on [0, oo) for each z E U), one deduces that d_ fdv\ = cV = _ ,l + 2kv-k2v2 dz(dt) ~ dt ~ V (I- kv}2 and hence
Q — logv' = 1-
(1-kv)*'
Taking imaginary parts in this equality, we have
On the other hand, since 9 , , , Idvll -logH = Re 1^—-J, we obtain from (3.2.11) the equality (3-2.13)
— log Id = -,
~ [ ,0.
|1 — kv\*
dt
This equation shows that |v(z, t)\ decreases from |z| to 0 as t increases from 0 to oo. Hence there is a one-to-one correspondence between \v\ and t, and we can regard \v\ as the independent variable. Eliminating t between (3.2.12) and (3.2.13), one concludes that (3.2.14)
- ImU-
Since \k(t)\ = 1 we see that (3.2.15)
J-^p—
2 ' = |sin{2arg(l - kv}}\
sin(2arcsin|t>|) = 2|v|
,
, , . 1 v2
<
i,
H>
.
3.2. Applications
125
Now, because |u(z,£)| decreases as t increases we have dt\v\ < 0, and from (3.2.14) we obtain 4dt\v\
<
\dtaxgv'(z,t)\ <
I
M(i-M') , H>
Integrating the above inequality from t = 0 to t = oo, and noting that axgv'(z,t) —t arg/'(z) as t —> oo, we obtain |arg/'(z)l<
0
4dx = 4arcsm|z|, V1 - x2
I \z\<~7= V2
and 1/V2
|arg/'(z)|<
|z| /"1*1
4dx
T^x*
+
2dx
JiN/2
This is the desired relation. Step 2. In order to prove that these estimates are sharp for each z € U we show that we can find a continuous function k with \k(t)\ = 1 on [0, oo), such that if v(z, t) is the solution of the differential equation (3.2.11), then for this function equality holds for all t in (3.2.15). Fix z 6 U \ {0} and consider the resulting equation (3.2.15). An equivalent form of this equation is the following: |sin{arg(l- kv}}\ =
The above equation enables us to determine k(t)v(zj i) as a function of |v(2,£)|. (The function k(t)v(z,i) is uniquely determined for \v\ < 1/2, while for \v\ > 1/2 there are two possible choices, and we can make a continuous choice.) If we substitute this function kv into (3.2.13), we may compute \v(z, t)\ as a function of t.
126
The Loewner theory
On the other hand, from (3.2.11) we have d
l + k(t)v(z,t)
Hence, taking real and imaginary parts we obtain (3.2.13) and also ^ 91^ (3.2.16)
d
*
Therefore, from (3.2.16) we may determine argi> (z,t) in terms of t. Thus we have sufficient information to determine v(z,i) and k(t) in such a way that the Loewner equation (3.2.11) and (3.2.15) are satisfied for all t. This proves the sharpness of Theorem 3.2.6. Remark 3.2.7. There are refinements of the rotation theorem for certain subclasses of S: for normalized close-to-convex functions, we have | arg f'(z)\ < 4arcsin|,z| (Theorem 2.4.9), while for normalized convex functions we have | arg f ( z ) \ < 2arcsin|z| (see Problem 2.2.8).
3.2.2
Applications of the method of Loewner chains to characterize some subclasses of S
We begin by using Theorem 3.1.11 to give an alternative characterization of spirallike functions of type a with |a| < ir/2. At the same time, we give another proof of Theorem 2.4.10 by using the method of subordination chains (see Spacek [Spa], Robertson [RobeS], [Pom5, Theorem 6.6]). Theorem 3.2.8. Let f : U —> C be a normalized holomorphic function and let a € E with \a\ < vr/2. Then f is spirallike of type a if and only if Re em-J-y > 0 on U. I f(z) J Proof. First assume the condition (3.2.17) holds. Let /: U x [0, oo) ->• C be defined by
(3.2.17)
(3.2.18)
f ( z , t ) = e'iaf(eiatz),
z e U, t>0,
where a = tana. Then f ( z , t ) is holomorphic on \z\ < 1, /(0,t) = 0 and /;(0,t) = e* for t > 0. For r 6 (0,1) let M = max|/(z)|. Then \f(z,t)\ < Me1 \z\
for \z\ < r and t > 0, and hence the condition (3.1.18) is satisfied.
3.2. Applications
127
A simple computation yields that
where /o o or\\ (o.Z.Z\J)
*-./., j.\ „*« i /i *«\ » V J 2v( Z i * } — ^Oi ~r v(1 — %CL] —: tat
:
/
.
' >e zf'(etatz) Obviously p(z,t) is holomorphic on |z| < 1, p(0,£) = 1 and from (3.2.17), Rep(z,£) > 0, |z| < 1, t > 0. The measurability condition from Theorem 3.1.11 is also satisfied, and hence /(z, t) is a Loewner chain. In particular / is univalent on U and f ( z ) -< f ( z , t ] -< e - / C z ) , for z € U and t > 0. This implies that exp (-e-*°-L-\ f ( z ) € f ( U ) , \ cos OL j
zeU,
t > 0,
and we conclude that / is a spirallike function of type a. Conversely, assume that / is spirallike of type a. Then / is univalent on U and f ( z ) -< e^-ia^f(z) for z 6 tf and t > 0. Let /
p-ia
\
(^(-^T^-^ }f( \ cos a /
\ eiaa
^ /'
for z € U and 0 < s < t < oo. This function is well defined on U since / is spirallike. Also v(Q,s,t) = 0 and |v(z,s,t)| < 1 on If, hence v is a Schwarz function. Moreover, v(-,s,t) is univalent on U for 0 < s < t < oo. Now, if f ( z , t ) is the map defined by (3.2.18) then it is easy to see that f ( z , s) = f ( v ( z , s,t),t),
zeU,
Q<s
and therefore f ( z , t ) is a subordination chain. Moreover, since / is univalent on U it follows that /(•,£) is also univalent on U, and f ( z , t ) is normalized, so that f ( z , t ) is a Loewner chain. Using the relations (3.2.19), (3.2.20) and Theorem 3.1.12, we must have Re p(z,t) > 0 for \z\ < 1 and t > 0. Setting t = 0 in (3.2.20), we deduce that
128
The Loewner theory
This completes the proof. Using the above arguments we obtain the following (cf. [Pom5]): Corollary 3.2.9. Let f be a normalized holomorphic function on U and a € M. with \a\ < 7T/2. Also let a = tana. Then f is spirallike of type a if and only if
is a Loewner chain. In particular, f is starlike if and only if f ( z , t) = e t f ( z ) is a Loewner chain. With similar reasoning as in the proof of Theorem 3.2.8 we may prove the following alternative characterization of close-to-convexity (see [Biel-Lew]): Theorem 3.2.10. Let f : U —» C be a normalized holomorphic function and let g be starlike on U. Then f is close-to- convex with respect to the convex function h(z) = / —^dt if and only if Jo t f ( z , t} = f ( z ) + (e* - l ) g ( z ) ,
ztU,
t > 0,
is a Loewner chain. (The Loewner chain f ( z , t ) is not normalized if g is not normalized.) Proof. First we suppose that / is close-to-convex with respect to h. Then /(•,£) is also close-to-convex with respect to h for any fixed t > 0, and hence univalent. It is also easily checked that (3.2.21)
Re [|£CM)/WCM)}] > 0,
t > 0, z & U.
This implies that f ( z , t) is a subordination chain. Conversely, the condition (3.2.21) when t — 0 implies that / is close-toconvex with respect to h. This completes the proof. We remark that the above characterization of close-to-convexity in terms of Loewner chains may be used to prove the linear accessibility property of image domains for close-to-convex functions. Corollary 3.2.11. Let f be a normalized close-to-convex function on U. Then for each r, 0 < r < I, the complement of f ( U r ) is a union of nonintersecting rays.
3.2. Applications
129
Proof. Since / is close-to-convex on U there exists a starlike function g such that f ( z , t ) = f ( z ) + (e1 — l)g(z) is a Loewner chain. Hence L(z,r) = [f(z) + tg(z) : t > 0, z fixed, \z\ = r} are disjoint rays which fill up the complement of f ( U r ) . It is also possible to give a characterization of convexity in terms of Loewner chains, using Theorems 2.2.6 and 3.2.10. Theorem 3.2.12. Let f : U —t C be a normalized holomorphic function. Then f € K if and only if (3.2.22)
f ( z , t) = f ( z ) + (e* - l ) z f ' ( z ) ,
z£U,
* > 0,
is a Loewner chain. Proof. First assume / € K. Then g(z) = z f ( z ) is normalized starlike on U, by Theorem 2.2.6. Using Theorem 3.2.10, it is obvious that f(z,i) given by (3.2.22) is a Loewner chain. Conversely, assume f ( z , t ) given by (3.2.22) is a Loewner chain. In view of Theorem 3.1.12 we obtain Re
'•'(7 +\ >o, zee/, t>o. v z >v j
A straightforward computation yields Re 1 + (1 - e'^f,, Y > °>
-^ef/,
t>0.
Letting t —)• oo in the above, we deduce that
as desired. This completes the proof. Notes. In this section we have proved some of the most well known and basic applications of Loewner's differential equation, apart from coefficient estimates. For most of the results in this section, we have used the following references: [Dur], [Gol4], [Hay], [Pom3], [Pom5]. For other applications of Loewner's differential equation the reader may consult [Ahl2], [Avk-Aks2], [Baz3], [Becl-4], [Bran], [Con], [Gon5], [Hen], [Leu], [Lewi,2], [Ros], [Ros-Rov], [Rot], [Rov].
130
The Loewner theory
Problems
3.2.1. Prove Theorem 3.2.4. 3.2.2. Prove Theorem 3.2.5. 3.2.3. Complete the details in the proof of Theorem 3.2.10.
3.3 3.3.1
Univalence criteria Becker's univalence criteria
We finish this chapter with some applications of the method of Loewner chains to univalence criteria involving higher derivatives of the function. First we prove the following result due to Becker, 1972 [Becl] (an improvement of an earlier result of Duren, Shapiro, and Shields [Dur-Sh-Sh]): Theorem 3.3.1. Let f : U -> C be a holomorphic function such that /'(O) + 0. // zf"(z) (3.3.1) (1 < i, /'(*) then f is univalent on U. Proof. Clearly we may suppose that / is normalized. Using the relation (3.3.1), one deduces that f(z) / 0 for z e U. Let
Then /(-,*) is holomorphic on U, /(0,t) = 0, /'(0,t) = e* for t > 0, and f ( z , •) e C°°([0, oo)) for each z eU.lt is clear that as t -^ oo,
locally uniformly in 2, and hence lim e~*f(z,t} — z
t—too
locally uniformly in z. Consequently {e~tf(z, t)}t>o is a normal family, and for each r € (0,1) there is a constant M = M(r] > 0 such that |/(z, t)| < Me*
3.3. Univalence criteria
131
for \z\ < r and t > 0. Hence the condition (i) from Theorem 3.1.11 is satisfied. To check that condition (ii) is satisfied, we first note that
and
zf(z,t) = e*zf(e-*z) + (1 - e^V/ =
etzf'(e-tz)(l-E(z,t)),
where
/;
Using the relation (3.3.1) and the fact that 1 — e~2t < 1 — le""*^!2, z € ?7, we easily deduce that
and hence f'(z, t) ^ 0 for z € U and t > 0. If we define p(z, t) by
then straightforward computation yields that
and also p(0, t) = 1. The holomorphy and measurability conditions of Theorem 3.1.11 are clearly satisfied and we have Re p(z,i) > 0 for \z\ < 1 and t > 0. Since all conditions of Theorem 3.1.11 are satisfied, we conclude that /(z, t) is a Loewner chain. Hence /, as the first element of this chain, is univalent on U. This completes the proof. It is of interest to see whether there is any room for improvement of the constant on the right-hand side of (3.3.1). Consider the function f ( z ) = e^, z € U, where (3 > TT. Then it is clear that / is not univalent on U. On the other hand, since
132
The Loewner theory
and
= sup < (Jr(l - r 2 ) : 0 < r < 1 [ = Q "^A v. j y we see that the constant 1 in (3.3.1) cannot be replaced by any number strictly 2 ,greater than - V37T w 1.21. Pommerenke [Pom8] gave an example which shows that the constant 1 in (3.3.1) cannot be replaced by 1.121. This improves the above bound obtained by Becker [Becl]. We remark that an interesting extension of Becker's result was obtained by Ruscheweyh [Rusl]. With similar reasoning as in the proof of Theorem 3.3.1, one may prove the following criterion for univalence, due to Ahlfors [Ahl3] and Becker [Becl]:
Theorem 3.3.2. Lei f : U —>• C be a holomorphic function with /'(O) ^ 0. Let c € C with \c\ < 1, c ^ —I, and assume that < 1,
z € U.
Then f is univalent on U. In this case it suffices to prove that /OM) =/(e-V) +-4 is a (non-normalized) univalent subordination chain. We leave the details for the reader.
3.3.2
Univalence criteria involving the Schwarzian derivative
Another very important class of univalence criteria are those involving the Schwarzian derivative. The basic result was discovered by Nehari [Nehl] and is a sufficient condition for univalence. A more elementary necessary condition was found earlier by Kraus [Krau] and rediscovered by Nehari. The aim of this section is to present the results of Nehari and Kraus and a general criterion
3.3. Univalence criteria
133
for univalence due to Epstein. We shall see that there are some connections with the theory of Loewner chains. Recall that the Schwarzian derivative of a locally univalent function / is defined by
It is noteworthy because of its invariance properties under linear fractional transformations. If h = f o g is the composition of locally univalent functions, then
If T is a linear fractional transformation then {T; z} = 0, and hence we obtain {T o g-z} = {g; z} and {/ o T; z} = {/; T(z)}(T'(z))2. Theorem 3.3.3. (i) If f e S then
This result is sharp. (ii) Conversely, if f : U —>• C is a holomorphic function such that (3-3.2)
|{/;z
,
,
then f is univalent on U. Proof. First we prove the necessary condition for univalence given in part (i). To this end, fix z € U and consider the Koebe transform of /:
Then g G S and simple computations yield that
and
134
The Loewner theory
We now consider the function (p defined by 1
0:1
.
Since g € S we have
From Corollary 1.1.3 we have |ai| < 1, and thus r\
K/;4I< ( i _ N 2 ) 2 '
z u
^ -
The sharpness of the above estimate follows from the sharpness of the estimate |ai| < 1. In fact, equality occurs for the Koebe function. The sufficient condition for univalence in part (ii) is usually deduced from a comparison theorem for ordinary differential equations. A different proof, using the Loewner method, was given by Becker [Bed]. We shall present Becker's proof. It is known that if p is a holomorphic function on U, then any (meromorphic) function / on U such that {/; z} — p(z) has the form / = wi/w-2, where wi and W2 are linearly independent solutions of the differential equation (3.3.3)
w"(z] + ^-w(z) = 0. 2*
The converse is also true. (See [Nehl], [Dur].) It follows that if / and g are holomorphic functions on U and { f ; z } = {g;z}, then there exists a linear fractional transformation T such that g = T o f . Now, assume that / is holomorphic on U and satisfies (3.3.2). Then /'(z) ^ 0 for z <E U (Problem 3.3.1), and if p ( z ) = {/; z}, z e U, then p is also a holomorphic function on the unit disc. With this choice of p, let w\ and w-2 be the solutions of equation (3.3.3) such that (3.3.4)
wi(0) = 0, it;i(0) = 1,
(3.3.5)
w2(Q} = 1, w'2(Q) = 0.
3.3. Univalence criteria
135
The Wronskian of these solutions is constant, and taking into account the conditions (3.3.4) and (3.3.5), we see that (3.3.6)
w[ (z)w2(z) - wi(z)w'2(z) = 1, z e U.
Let u(z) = wi(z)/w2(z), for z € U. Then {u;z} = p(z), and using (3.3.3), (3.3.4), and (3.3.5), we obtain the expansion . ..,
,
O
As already noted, there exists a linear fractional transformation T such that u = T o /. Consequently, in order to prove that / is univalent, it suffices to show this property for the function u. To this end, let A
_ wi(e~V) + (e* - e-^zw'^e'tz) ' ^~ -* * - t ' - t
^
Then /(O, t) = 0 and /'(O, t) = e* for t > 0. Also f ( z , 0) = wi(z)/w2(z) = u(z). We shall show that f ( z , t) is a Loewner chain. Let
g(z,t) = w2(e~*z) + (e* - e'^zw'^e^z) = ti^e-'z) + (1 - e-2t)z* \W'2(e _~tz}] , z e 17, t > 0. l e z \ Then g ( - , t ) is holomorphic on J7 and using (3.3.3) and (3.3.5), we obtain (3.3.7)
lim g(z, t} = w2(Q) + w'2(0)z2 = I -
t—¥OO
Z
z\
locally uniformly on U. In view of (3.3.2), we have \p(Q)\ < 2, and thus for each r e (0, 1) we have g(z,t) ^ 0 for \z\ < r and t sufficiently large. Together with the fact that g(0, t) = w2(Q) = 1 for t > 0, this implies that there exists a number r £ (0, 1) such that g(z,t) ^ 0 for \z\ < r and t > 0. For this r, the function /(•, t) is holomorphic on Ur for each t > 0, and /(z, •) is of class C°° on [0, oo) for each z G Ur. Further, using the relations (3.3.4), (3.3.5) and (3.3.7), we deduce that lim e~tf(z,t) = -77-, *->«, p(0)
136
The Loewner theory
locally uniformly on Ur. Taking into account (3.3.3) and (3.3.6), by differentiation we easily obtain the relations
and df _ —ez dtv ' ' [w2(e-*z) + z(et - e-*)w;2(e-**)]2' for \z\ < r and t > 0. Let
^(M) + */'(*,*)
= -iz 2 (l-e- 2 ') 2 p(e-**),
|z|
*>0.
Then A(z, t) is holomorphic on \z\ < r for t > 0, and clearly we may extend this function holomorphically to the whole disc U. Moreover, using the assumption (3.3.2), one deduces that
,
for \z\ < 1 and t > 0. Hence, if h(z, t) = [l + \(z, t]\/[l - \(z, t)], then in view of the above inequality we conclude that /i(-,t) is a holomorphic function on U for t > 0, and Re h(z, t) > 0 for z 6 U and t > 0. Moreover, it is obvious that h(Q,t) = 1, h(z, •) is a measurable function on [0, oo) for each z € U, and ^(z,t) = zf'(z,t)h(z,t),
\z\
t>0.
Since all conditions of Theorem 3.1.11 are satisfied, we conclude that f ( z , t ) is a Loewner chain, and therefore /(z, 0) = u(z] is univalent on 17, as desired. This completes the proof.
3.3. Univalence criteria
137
Remark 3.3.4. The constant 2 in the estimate (3.3.2) is best possible. To see this, let
In this case r£
i
2 2(1 \ _ >v I iJ
It is easy to see that if 7 = i(3 with 0 e R and /? 7^ 0, then / is not univalent on U. In this case
This example is due to Hille [Hill]. In addition to the univalence condition (3.3.2), Nehari [Nehl] obtained another sufficient condition for univalence, as follows. We leave the details of the proof for the reader. Theorem 3.3.5. If f is a holomorphic function on U and
(3.3.8)
!{/;*>! < y , *eu,
then f is univalent on U. Again the constant 7T2/2 is best possible in (3.3.8), because if f ( z ) = e*z, A > TT, then / is not univalent on U, and in this case |{/; z}\ = A 2 /2. Nehari gave further univalence criteria in [Neh4]. Osgood and Stowe [OsStl-2] showed that the criteria of Nehari in Theorems 3.3.3 and 3.3.5 could be viewed as particular cases of a single theorem, in which the Schwarzian derivative is defined relative to conformal metrics on the domain and range of /.
Epstein [Epl,2] obtained a general criterion for univalence, which includes most previously known criteria. His proof uses hyperbolic geometry and is quite different from classical approaches. Pommerenke [Pom9] gave a very interesting proof of the analytic function case of Epstein's result, using the method of Loewner chains. He also showed that an additional assumption on g made by Epstein was not necessary. Here we shall present Pommerenke's proof. (In the proofs of Pommerenke and Epstein the function / is allowed to be meromorphic.)
138
The Loewner theory
Theorem 3.3.6. Let f and g be holomorphic and locally univalent functions on U. If -Zr.H ( ~\
< 1, z € U, then f is univalent on U. Proof. Suppose
f ( z ) — OQ + a\z + ...,
z € U,
and
g(z) = bo + biz + ..., z£U. Since / and g are locally univalent, it follows that ai ^ 0 and &i ^ 0. Also since the Schwarzian is invariant under linear fractional transformations, we may assume without loss of generality that / and g are normalized. We next introduce the functions
and
= f ( z ) w i ( z ) = z + dz2 + O(z3). These functions are holomorphic on U since /' and g' cannot have zeros on U. Now, let ft(z) — f ( z , t ) be the chain given by *--*^(e-^) 'Z ~ wi(e~'z) + (e* - e-t}zw((e -
G
^'
~
(Compare with the proof of Theorem 3.3.3.) Then f ( - , t ) is meromorphic on U. Since w\(z] = 1 + cz"2 + . . . it follows that the denominator in the expression of f ( z , t ) has the form 1 + O(z2) as z —> 0, uniformly in t. Hence there exist constants r € (0, 1) and M > 0 such that /(•,£) is holomorphic on Ur for each t > 0, and N
3.3. Univalence criteria
139
Moreover, it is easy to see that
ft(z) - e*z + O(z2)
as z -> 0,
since the numerator is etz + O(z2) as z —>• 0. Thus f ( z , t) is a normalized chain. Letting a(t) = 1 — e~2t, we obtain after elementary computations that
Af, fls CM) + */'(*,«)
e~taz(w'2wi — w"w2) + a2z2(wl2w'1 — w"w'2) where wi, W2, w{, w'2, w"t w'2 are computed at e^z. Using the definitions of iui and W2, and writing Sf(z) = {/; z} and 55(-2) = {
2Wl
- w'{w2 = f"w2 + 2/Xwi = f'wlg"/g',
Zw{ - w'{w'2 = f"w(wi - f'w'{Wl + 2f'w(2
Therefore, we obtain
for all z 6 Cf and t > 0. It can be seen that A(-, t) is zero for t = 0 and is holomorphic on U if t > 0. In view of the maximum modulus theorem, we obtain for t > 0 that
|A(z,t)| < max|A(tM)| = |A(iy 0 ,*)l = 1^0^(^0,01, |to|=l
for some WQ € C, |iyo| = 1. Next, using the inequality from the hypothesis for z = e~*WQ^ we have
|A(ti*>,t)l =
140
The Loewner theory
< 1.
This estimate together with the fact that A(0, t) = 0, t > 0, implies that if we define
then p ( - , t ) 6 P, t > 0. Also it is obvious that p(z, •) is measurable on [0, oo) for each z € U. Taking into account Theorem 3.1.11 we deduce that f ( z , t ) is a Loewner chain, and thus /(2,0) = w-2.(z) / w\(z) = f ( z ) is univalent on U. This completes the proof. Remark 3.3.7. If g(z) = z in Theorem 3.3.6, we obtain the sufficient condition for univalence due to Nehari (see Theorem 3.3.3). On the other hand, if / = g in Theorem 3.3.6, we obtain Becker's univalence result given in Theorem 3.3.1. Recently Schippers [Schip] derived a differential equation for the Loewner flow of the Schwarzian derivative of a univalent function on C7, and used this result to obtain sharp bounds for higher order analogs of the Schwarzian derivative. Introducing the operators cr n (/), defined by *3(/) = {/; z}
and an+1(f) = (*„(/))' - (n - l)^a n (/),
he obtained the following results via the Loewner method: Theorem 3.3.8. sup [\(z)]n-l\(Tn(f)(z)\ = 4 n ~ 3 (n - 2)16, feH*(U) where X(z) = 1/(1 — |z|2) is the hyperbolic metric on U. This estimate is sharp for a suitable rotation of the Koebe function. (Higher order Schwarzians have been considered by other authors, e.g. [Aha], [Harl-2], [Tarn].)
3.3.3
A generalization of Becker's and Nehari's univalence criteria
We finish this chapter with a result which generalizes Theorems 3.3.1, 3.3.2 and 3.3.3. This generalization was obtained by Pascu [Pas], using the method of Loewner chains.
3.3. Univalence criteria
141
Theorem 3.3.9. Let F = F(u,v) : U x C -* C satisfy the following assumptions: (i) The function L(z,t) = F(e~tz,etz) is holomorphic on U for all t € [0, oo) and is locally absolutely continuous in t G [0, oo) locally uniformly with respect to z € U. (U) The function —-^—/[zL'(z,t]] is holomorphic on U for t > 0 and is c/t holomorphic on U for t = 0. QF OF dF (Hi) -^-(0,0) ^ 0, —(0,0)/—(0,0) £ (-00, -1], and there exists ai : ov ou gv [0,ooj —> C such that \a\(t)\ is strictly increasing to oo as t —> oo, and dF dF al(t) = e't — (0,0) + ef — (0,0),
te[0,oo).
. , _, , .. ,, . t F(e~lz, etz)} . , , ., ,_ (iv) The family of junctions < r-r > is a normal family on U. I ai(t) J t>o Let dF dF G(u,v) = u—(u,v)/[v—(u,v)}. If (3.3.9)
\G(z,z)\
zeU,
and
(3.3.10)
\G(z,l/z)\
z<=U\{0},
then F(z, z) is a univalent function on U. Remark 3.3.10. Prom the assumption (iii) it follows that ai(t) ^ 0 for t e [0,00).
Proof of Theorem 3.3.9. Let Qr p(z,t) = —(z,t)/[zL'(z,t)],
zeU,
t 6 [0,oo).
From the condition (ii) one deduces that p ( - , t ) is holomorphic on U for each t > 0, and p(z, •) is a measurable function on [0, oo) for each z € U. We prove that Re p(z, t) > 0 for z € U and t > 0. To this end, let
142
The Loewner theory
After short computations, it is easy to obtain that w(z,t) = G(e~tz,etz) for z € U and t > 0. Hence w ( - , t ) is a holomorphic function of U for each t > 0. From the condition (3.3.9) it follows that |tw(2,0)| = \G(z,z}\ < 1, z e U. If t > 0 is fixed, then p(-, t) is holomorphic on U, by assumption (ii). Also, from (3.3.9) we have (3.3.11) HO,t)| Further, from (3.3.10) we obtain (3.3.12)
\w(z. t)\ ~ < max \w(z. t)\ = max \G(e^z, e*z)\
where A is a complex number with |A| = 1 and ZQ = e~*\ € U. Consequently, from (3.3.11), (3.3.12) and using the maximum principle for holomorphic functions, one obtains that |w;(;z,£)| < 1 for z € U and t > 0. Thus Re p(z,t) > 0, as claimed. Taking into account Theorem 3.1.11 and Problem 3.1.6, one concludes that L(z,t) is a univalent subordination chain. Therefore L(z,0) = F(z,z) is univalent on U. This completes the proof. Remark 3.3.11. Let / : U —> C be a normalized holomorphic function. Also let Fj : U x C —» C, j = 1, 2, 3, be the functions given by
where c e C with |c| < 1, c 7^ —1, and 171 f
\
£1. \ i
V
* 6\">, ") — J \«-i i
v-u
f"(u)'
l
~~^'J(uj
Setting successively F = Fj, j = 1,2,3, in Theorem 3.3.9 yields the results of Theorems 3.3.1, 3.3.2 and the sufficient condition in Theorem 3.3.3. On the other hand, if a(u] + (v - u)b(u) F(u,v} = c(u) + (v — u}d(u]
3.3. Univalence criteria
143
in Theorem 3.3.9, where a, 6, c, d are analytic functions on U such that a(z)d(z) — c(z)b(z) ^ 0 for z & U, then we obtain a sufficient condition of univalence due to Betker [Bet2]. Notes. We remark that in the literature there are many generalizations of Nehari's univalence criteria, given in Theorems 3.3.3 (ii) and 3.3.5. The reader may consult the following papers: [Ahl3], [Becl], [And-Hin], [Chul], [Epl,2], [Geh-Pom], [Lew-St], [Neh4], [Os-Stl], [Pfa4], [Schw]. Problems
3.3.1. Let / be a holomorphic function on U and suppose that f ' ( a ) = 0 for some point a 6 U. Show that {/; z} cannot have a removable singularity at a. 3.3.2. Prove Theorem 3.3.2. 3.3.3. Prove Theorem 3.3.5. 3.3.4. Let / : U —>• C be an analytic and locally univalent function such that
Show that / is univalent on U. (Pokornyi, 1951 [Pok], Nehari, 1979 [Neh4].) 3.3.5. Let / : U —> C be an analytic and locally univalent function such that 4z2 4 Show that / is univalent on U. (Chuaqui, 1995 [Chul].) 3.3.6. Complete the details in the proof of Theorem 3.3.6.
Chapter 4
Bloch functions and the Bloch constant In this chapter we shall discuss some of the basic properties of Bloch functions, and their role in the study of the Bloch constant problem and other problems of the same type. We also point out some other connections between Bloch functions and univalent functions, and between the related but larger class of normal functions and univalent functions. We show how the classical lower estimate of Ahlfors for the Bloch constant was obtained by Bonk [Bonl,2] as a consequence of a function theoretic distortion theorem. We also discuss analogous results of Liu and Minda [Liu-Min] for locally univalent Bloch functions. Finally we consider the analog of the Bloch constant problem for normalized convex functions. This problem can be solved exactly, as shown by Szego [Sze],
4.1
Preliminaries concerning Bloch functions Definition 4.1.1. Let / be a holomorphic function on U and let
(4.1.1)
11/11= sup (l-\z\*)\f(z)\. M
The function / is called a Bloch function if ||/|| < oo, and in this case ||/|| is called the Bloch seminorm of /. 145
146
Bloch functions and the Bloch constant
The Bloch seminorm is invariant under pre-composition with automorphisms of the unit disc and post-composition with (orientation preserving) Euclidean motions of the complex plane. Let B denote the set of Bloch functions on U. This set is a Banach space with respect to the norm
It is known that B is isomorphic to i°° (the space of bounded sequences with the supremum norm) (see [Shi-Wil]) and also B is isomorphic to the second dual of the subspace of B spanned by the polynomial functions (see [And-Cl-Pom], [Shi-Wil]). On the other hand, B is not separable (see [CamCim-Pfa]; [And-Cl-Pom]). An alternative characterization of Bloch functions is given in the following theorem (see [Pom4]). First we note that the group of holomorphic automorphisms of the unit disc U is given by Aut(C7) =
J.
|~
*
(Jt/Z
Now if / 6 H(U), we consider the family Tj defined by Ff = {d G H(U) : g(z} = /Mz)) - /MO)), y> € Aut(tf)}. Then we have Theorem 4.1.2. The function f G H(U) is a Bloch function if and only if the family Tj is a (finitely) normal family. Proof. If Ff is a normal family, then {(/ o v>)'(0) :
[Z /'MOV(CK, ^ e 17,
./O
where the integration is along the straight line segment from 0 to z. It is elementary to see that the integrand is bounded by
C
l-a2
C
4.1. Preliminaries
147
where C = ||/||, and hence
This shows that the family Tj is locally uniformly bounded on U and completes the proof. There is a variety of other useful characterizations of Bloch functions (see [And], [And-Cl-Pom], [Cim], [Pom4], [Timl]). We shall give some of them in this chapter (Theorem 4.1.7, Problem 4.1.4, Problem 4.2.2). The related notion of normal function was introduced by Lehto and Virtanen [Leh-Virl], Definition 4.1.3. Let / be a meromorphic function on U and let
f* is called the spherical derivative of /. Also let
!/(/)= SUP (i-MVCO.
N
Then we have [Leh-Virl] Theorem 4.1.4. The meromorphic function f on U is normal if and only if the family Fj is a normal family (viewed as a family of mappings into the Riemann sphere with the spherical metric --r^r)1 + \w\* It is clear that a Bloch function is normal, and that a bounded holomorphic function on U is Bloch. In fact, for bounded holomorphic functions on U we have the inequality [And-Cl-Pom] (4-1.2)
\\f\\B < 211/Hoo,
148
Bloch functions and the Bloch constant
where H/H^ = sup |/(z)|. N
where the branch of the logarithm is chosen so that log 1 = 0. Then / is an unbounded Bloch function and ||/|| = ||/||g = l. (ii) If a > 0 and
then / is not a Bloch function. In fact if / is a Bloch function of Bloch seminorm 1 such that /(O) = 0, it follows from the proof of Theorem 4.1.2 that
Hence the function /(z) = - log - , z e £/, is extremal for the growth
2
\\ — z\
of such functions. We now consider some connections between Bloch functions and univalent functions. First we have the following theorem of Pommerenke [Pom5]: Theorem 4.1.6. /// : U —> C is a univalent function then f(z) and f ' ( z ) are normal functions. Proof. Since / 6 HU(U), f may be written in the form
where g G S and a, (3 £ C, a ^ 0. Hence we have
r(l-N 2 )
9'(z)
Using the fact that g £ 5 and the estimate (1.1.7), we conclude that sup(l-|z| 2 )/*(z)
4.1. Preliminaries
149
Thus / is normal. Next we consider /'. In this case we obtain /"(*)
making use of the fact that the function g € S satisfies (see (1.1.9))
zeU.
— z
Therefore /' is also a normal function. The following beautiful result of Pommerenke [Pom4,5] (see also [Dur-ShSh]) exhibits a surprising connection between the set S and the set of Bloch functions in U. Theorem 4.1.7. Let f e H(U). Then f is a Bloch function if and only if there exist a function g & S and a complex number a such that (4.1.3)
U.
The branch of the logarithm is chosen so that logp'(O) = 0. Proof. First assume there exist a 6 C and g € S such that / can be written as in (4.1.3). Then <6|a|,
C/,
where again we have used (1.1.9). Thus / is a Bloch function and ||/|| < 6|a|. Conversely, suppose that / is a Bloch function. Without loss of generality, we may assume that ||/|| =^ 0 (otherwise, if ||/|| = 0, we can take a = 0 and g(z) = z) and let g : U —> C be given by 9(*) = f exp ./o
7(0-
z € U,
where the integration is along the straight line segment from 0 to 2. Then g e H(U), g(0) = 0 and g'(0) = 1. Also let a = \\f\\. Obviously (4.1.3) holds, and taking into account (4.1.1) we obtain
(i-N 2 )
1 ,.
,
150
Bloch functions and the Bloch constant
Hence, in view of the Becker univalence criterion (see [Becl] or Theorem 3.3.1), we conclude that g € S, as desired. Remark 4.1.8. More precisely, if BS denotes the set of functions
then we have ([And], [Becl]) {/ 6 B : II/U < 1 and /(O) = 0} § Bs § {/ 6 B : \\f\\ < 6 and / (0) = 0}.
The inclusions follow from arguments in the above proof. (Showing that the inclusions are strict is an exercise.) The constant 6 is sharp, as one can see by considering the Koebe function. An elementary result about the coefficients of Bloch functions is given in the following theorem (see [And-Cl-Pom] , [Pom4]). We leave the proof for the reader. Theorem 4.1.9. Let f : U —>• C be a Bloch function such that oo
f ( z ) = ^anzn, z€U. n=0
Then \an\ < 2||/||B, n = 0,l,.... In particular, this theorem states that the coefficients of a Bloch function are bounded. Of course, not every holomorphic function on U with bounded coefficients is a Bloch function. A counterexample is given by the function Other theorems about the coefficients of Bloch functions are proved in [And-Cl-Pom] . The Banach space structure of B is also studied in that paper; in addition see the survey papers [And] and [Cim]. There are no inclusions between the set of Bloch functions and any of the classical Hp spaces (p ^ oo) or the Nevanlinna class. In one direction, this is shown by Example 4.1.5 (ii); in the other, there exist Bloch functions constructed using gap series which do not belong to the Nevanlinna class (see the discussion following Theorem 2 in [And]). Coifman, Rochberg, and Weiss [Coi-Roc-Wei] showed that Bloch functions could be characterized by a bounded mean oscillation condition on U. Relations between univalent functions, analytic functions on U whose boundary
4.2. Bonk's distortion theorem
151
values have bounded mean oscillation, and Bloch functions have been studied by Baernstein [Bae] and Pommerenke [Pom6,7]. In particular, Pommerenke showed that a univalent Bloch function belongs to the space BMOA of analytic functions whose boundary values on the unit circle have bounded mean oscillation. For further information about the material in this section, see [And], [AndCl-Pom], [Cim], [Pom4,5], [Timl].
Problems 4.1.1. Prove the estimate (4.1.2). 4.1.2. Give examples which show that the inclusions in Remark 4.1.8 are proper. 4.1.3. Prove Theorem 4.1.9. 4.1.4. Show that if / is a Bloch function and dh is the hyperbolic distance on U, then
Hence / is a Bloch function if and only if there is a constant M > 0 such that \f(a)-f(b)\<Mdh(a,b), a,beU. It follows that any Bloch function is uniformly continuous with respect to the hyperbolic metric on U and the Euclidean metric on its range. 4.1.5. Let / : U —)• C be a holomorphic function and n > 2. Show that / € B if and only if (1 - \z\2}nf(n\z) is bounded in U. (K. Zhu, 1990 [Zhu].)
4.2
The Bloch constant problem and Bonk's distortion theorem
Bloch functions arose originally in the study of the Bloch constant problem. In this section we discuss this problem and related problems, and we
152
Bloch functions and the Bloch constant
show how the classical lower estimate of Ahlfors for the Bloch constant was obtained from a function-theoretic distortion theorem by Bonk [Bonl,2]. In 1924 Bloch [Blol,2] formulated a rather intricate covering theorem for non-univalent holomorphic functions on the unit disc, requiring only the condition /'(O) = 1. To state it, we need Definition 4.2.1. Let / € H(U) and a € U. (i) A schlicht disc of / centered at /(a) is a disc with center /(a) such that / maps conformally a subdomain of U containing a onto this disc. (ii) Let r(a, /) be the radius of the largest schlicht disc of / centered at
/(a). (iii) Let r(f) = sup |r(a, /) : a € U\. (iv) Let B = inf [r(f) : f 6 H(U), /'(O) = l}. B is called the Bloch constant. Bloch showed that B is positive. In 1929 Landau [Lan] gave numerical estimates for B and certain related constants, making use of an important reduction which accounts for the name "Bloch function". (However, this terminology was not introduced until much later by Pommerenke [Pom4].) To state it, we introduce a subclass B\ of B defined by (4.2.1)
B! = {/ e B : /(O) = 0, /'(O) = 1, ||/|| = l}.
Landau 's reduction is the following: Theorem 4.2.2. B = inf |r(/) : / e BI\. Proof. In computing B it is clear that we can take the infimum over functions which are holomorphic in U. For any such function /, the quantity (1 — |z|2)|/'(z)| has a maximum at an interior point ZQ of U. Suppose that the maximum value C is larger than 1. Then ZQ ^ 0. Let
- and let g(z) =
1 + ZQZ
2
U
[(/ o
where a = arg{(l — \ z o \ } f ' ( z o ) } . Then use of the relation (1 - \z\2)\(f o tf(z)\ = (1 - b(*) shows that g G B\ and r(g) < r(f). This implies that inf (r(/) : / e BI} < inf (r(/) : / € H(U), /'(O) = l}.
4.2. Bonk's distortion theorem
153
Since the opposite inequality is obvious, we axe done. oo
Remark 4.2.3. Functions / E B\, f(z) = z + \^anzn, satisfy the growth n=2
estimate
Together with the condition /'(O) = 1, this implies 02 = 0 and |as| < 1/3. Bonk [Bon2] proved that |a4| < 5/4 and Chen and Gauthier [Che-Gaul] improved this bound by showing that |a4J < 4.2/4 = 1.05. Landau also introduced two closely-related constants - the Landau constant L and the univalent Block constant A. To define L one omits the requirement that the discs in the range of / be schlicht, i.e. we set Lf = sup < r : f(U) contains a disc of radius r > and
For A one restricts to univalent functions, so that the distinction between schlicht discs and arbitrary discs in the range of / disappears. Thus we define
Then B < L < A. The locally univalent Bloch constant BO was introduced much later in the 1960's (Peschl [Pesl], Chern [Cher], Pommerenke [Pom4]). It is defined by (4.2.2)
Bo = inf |r(/) : / € H(U), f ( z ) ± 0, z € U, /'(O) = l}.
None of these constants is known precisely, but the classical upper bounds for B,Bo, and L, which are based on constructions of Ahlfors and Grunsky [Ahl-Gru] in the case of B and Rademacher [Rad] in the case of L (and BO) are conjectured to be sharp. Proving this is a long-standing open problem in geometric function theory. These bounds are
B < r(i/3)r(1i/i2) ^
154
Bloch functions and the Bloch constant
and B
°-L~
r(i/3)r(5/6) ^ 5433 -°' -
For further details see [Ahl-Gru], [Gol4], [Hil2], [Mini]. (The upper bound for L is incorrectly stated in [Gol4] and [Hil2].) There is no conjecture for the precise value of A, but the estimates 0.57088 < A < 0.6564155 are known ([Bel-Hum], [ZhaS], [Jen3]). It should also be mentioned that in 1923, prior to Bloch's work, Szego [Sze] had solved the univalent Bloch constant problem exactly for the case of convex univalent functions. We shall consider this case below. In a remarkable paper in 1938 demonstrating a connection between the Schwarz lemma and curvature, Ahlfors [Ahll] obtained the lower bound B > \/3/4, using a differential geometric argument and the maximum principle for subharmonic functions. Extending Ahlfors'methods, Heins [Hei] showed in 1962 that B > x/3/4. It was not until the 1988 thesis and a related publication in 1990 by Bonk [Bonl,2], that it was finally shown that Ahlfors' lower estimate for B could be obtained by function-theoretic methods. Bonk also obtained a small numerical /O
improvement in the lower bound for B: B > ——\- 10~14. Subsequently Chen v/3 and Gauthier [Che-Gaul] showed that B > ^— + 2 x 10~4. See also [CheH]. Here, however, we are mainly interested in Bonk's proof of the classical lower estimate of Ahlfors, since the methods are similar to those encountered in covering theorems in univalent function theory. We shall consider lower bounds for BO in the next section. The following distortion result is due to Bonk [Bon2]: Theorem 4.2.4. Suppose f € B\. Then
(4.2.3)
Re
4.2. Bonk's distortion theorem
155
Proof. It suffices to prove (4.2.3) for z e [0, l/-\/3], since we may introduce rotations of /. Let
-75 V3 and
1
w -3
9 / 1 \2 h(w} = -wl 1- -w\ .
An easy computation yields (4.2.4)
\h(w)\(l-\g(w)\2) = l,
H
We now define the function
We note that f'(g(w)) and h(w) are holomorphic on ?7, and q has removable singularities at 0 and I. Indeed, it is easy to see that w = 0 is a removable singularity, and w = 1 is also removable since /"(O) = 0 by Remark 4.2.3. Therefore q is holomorphic on U. From the fact that / 6 B\ and (4.2.4) we deduce that
h(w)
_ ~7H
I
/
\ m\ I * / \ " 7 — ••• i
(l-\g(w)\*)\h(w)\
I**'
*-•
W
We also note that in the expression for q(w), the factor V.J-
^ is the W)
Koebe function and has negative real part on dU \ {!}. These two observations imply that Re q(w) > 0 for w e dU \ {!}. Using the fact that q has a removable singularity at 1 and the minimum principle for harmonic functions, we therefore obtain Re q(w) > 0, \w\ < I. In particular for 0 < w < 1, where the Koebe function is positive, we deduce that
He I
h(w)
1 > 1.
156
Bloch functions and the Bloch constant
This is equivalent to (4.2.5)
Re f(g(w)) > h(w),
0 < w < 1,
since h(w) is positive for 0 < w < I. Setting z = g(w) in (4.2.5) and noting that z ranges from l/\/3 to 0 as 11; ranges from 0 to 1 gives the desired result. Corollary 4.2.5. B > v^/4. Proof. By Landau's reduction it suffices to show that r(f) > -s/3/4 for all / 6 Si. In fact we shall show that r(0, /) > \/3/4 for such /. If / € BI, then Bonk's distortion theorem implies that
Re f'(z) > 0, \z\ < -^. Using the Wolff-Noshiro-Warschawski theorem (see Lemma 2.4.1), we deduce that / is univalent on the disc U^,^. Thus / maps the disc U-^,^ conformally onto a simply connected domain D, and clearly 0 € D. Moreover, the boundary of D is the image of the circle \z\ = l/\/3 and if w = /(e^/\/3), 0 e [0,2?r], is a point on this image, then
•-I.
7S
1 - V3p
Prom this we conclude that the domain D contains the disc U^u and hence r(0, /) > \/3/4. Since / e BI is arbitrary, we have B > \/3/4, as desired. In order to understand fully the roles of the functions g and h in the proof of Bonk's distortion theorem, one needs to study the theory of extremal problems for Bloch functions, as developed by Cima and Wogen [Cim-Wog], Ruscheweyh and Wirths [Rus-Wirl,2], and Bonk himself in his thesis [Bonl]. Bonk's thesis contains another proof of his distortion theorem. A more geometric proof of Bonk's distortion theorem based on Julia's lemma was given by Minda [Min5j. Minda's proof is essentially the one-variable
4.3. Locally univalent Bloch functions
157
case of Theorem 9.1.6. Bonk [Bonl] and Minda [Min5] also determined the extremal functions. They are given by the function
and its rotations. This function is a two-sheeted branched analytic covering of U onto the disc U[——, -——). V 4 4 /
Problems 4.2.1. Show that inf (r(0,/) : / e H(U), /'(O) = l} = 0. (This is why one must allow the schlicht discs to have arbitrary centers in Bloch's theorem.) 4.2.2. Show that a holomorphic function / on U is Bloch if and only if there exists M > 0 such that r(a, /) < M for all a € U. 4.2.3. Let / be a holomorphic function on U such that /(O) = 0, /'(O) = a > 0 and \f(z)\ < M, z € U. Show that (i) / is univalent on the disc £7^, where _ a a P °~ M (ii) For any positive number p < po, f(Up) contains the disc UR, where
R=
M -ap (Landau, 1929 [Lan], Dieudonne, 1931, [Die2].) 4.2.4. Let T be the set of functions / which satisfy the conditions in Problem 4.2.3. Show that po is the radius of univalence of T (i.e. po is the largest value of p such that each function in T is univalent on Up). (Dieudonne, 1931 [Die2].)
4.3 4.3.1
Locally univalent Bloch functions Distortion results for locally univalent Bloch functions
It is also possible to obtain a distortion theorem for locally univalent Bloch functions using a version of Julia's lemma. This was carried out by Liu and
158
Bloch functions and the Bloch constant
Minda [Liu-Min] who deduced the known estimate BQ > 1/2. (Peschl [Pesl], Chern [Cher], and Pommerenke [Pom4] all showed that BQ > 1/2, and Pommerenke showed that the inequality is strict.) In this section we shall present their proofs. Part of the distortion theorem was obtained much earlier by Peschl [Pesl,2] using different methods. We also note that Yanagihara [Yan2] obtained a very small numerical improvement in the lower estimate for the locally univalent Bloch constant (Bo > 0.5 + 10~335). A further improvement B0 > 0.5 + 2 x 10~8 was announced by Chen in [CheH]. We introduce the subclass BQ of B denned by BQ = {/ 6 B : /'(*) + 0, z & U, H / l l = 1, /(O) = /'(O) -1 = 0}.
Because of Landau's reduction it suffices to consider the class BQ when estimating the locally univalent Bloch constant, i.e. (4.3.1)
BO = ii
For r > 0, let
,„. n-^i 2
z—
l+r
I +r
Then A(l,r) is a horodisc in [7, i.e. a disc in U that is internally tangent to dU at 1. Also let A(l,r) denote the closure of A(l,r) relative to U (which means that 1 ^ A(l,r)). We begin with Julia's lemma on the unit disc (in less than full generality, since we do not need the version which considers the angular derivative). Lemma 4.3.1. If f is a holomorphic function onUiJ {!}, f maps U into U and /(I) = 1, then /'(I) = a > 0 and for each r > 0, the function f maps the closed horodisc A(l,r) into the closed horodisc A(l,ar). Moreover, a point on the boundary o/A(l,r) (relative to U) is mapped into the boundary of A(l,ar) if and only if f is a conformal automorphism of the unit disc U such that /(I) = 1. Julia's lemma is proved in full generality in [Ahl2, Section 1-4]; the restricted version given above appears in [Poly-Sze, Problem 292]. By composing with a suitable linear fractional transformation, we obtain the following version of Julia's lemma for maps from the unit disc into the right half plane.
4.3. Locally univalent Bloch functions
159
Lemma 4.3.2. Suppose f is a holomorphic function on U\J{1}, f maps U into the right half plane II = {z S C : Re z > 0} and /(I) = 0. Then for each r > 0, / maps the closed horodisc A(l,r) into the closed disc U(0r,/3r), where (3 = —/'(I) > 0. Moreover, a point on the boundary of A(l,r) is mapped into the boundary of U((3r, (3r) if and only if f is a conformal mapping of U onto n such that /(I) = 0. A direct consequence of Lemma 4.3.2 is the following (see [Liu-Min]): Corollary 4.3.3. Suppose f is a holomorphic function on U U {1} satisfying the assumptions of Lemma 4-3.2. Then for any x 6 (— 1, 1), 1 —x Re f ( x ) < 2/3 - , 1+x
with equality for some x € (—1, 1) if and only if
Proof. Let x 6 (—1, 1) be fixed and r = (1 — x)/(l + x). Then x lies on the boundary of A(l, r), and in view of Lemma 4.3.2 we deduce that f ( x ) lies inside the circle which passes through 0 and 2/3(1 — x)/(l + x) and intersects the real axis orthogonally. The statement regarding equality is clear. This completes the proof. Before proceeding to the distortion theorem of Liu and Minda [Liu-Min] , we consider a certain function which is extremal for this theorem. Example 4.3.4. Let v(4.3.2)
'
l F(z) = ~exp{-^—-^ > +^ v ' 2 F \ 1-zJ 1
z e U.
It is easy to see that F is a universal covering projection of U onto the punctured disc {w : 0 < \w — 1/21 < e/2}. Moreover,
and equality holds if and only if z belongs to the circle |l-z| 2
160
Bloch functions and the Bloch constant
(To see this it suffices to note that for t > 0 the function te1"* assumes its maximum value 1 when t = 1.) Therefore F is a Bloch function and ||F|| < 1. It is easy to see that r(0, F) = 1/2 and r(F) = e/4. One may also verify (Problem 4.3.1) that (4.3.3)
|F'(z)| > F'(|2|) = - l e x p - L , z € V.
We now present the distortion result of Liu and Minda [Liu-Min]. We remark that the inequality |/'(2)| > F'(|z|), z € U, was first obtained by Peschl [Pesl,2], though Peschl did not determine all the extremal functions. Theorem 4.3.5. Suppose f E BQ and let F denote the function given by (4-3.2). Then the following inequalities hold: (i) For all z<=U,
and equality holds for some z = re10 ^ 0 if and only if f ( z ) = elQF(e some 6 e R. (it) For \z\ < 111, Re f ' ( z ) > F'(\z\) = „ *
2 exp{-
l9
z] for
- z
and equality holds for some z = re10 ^ 0 if and only if f ( z ) = el&F(e l°z) for some 0 &R. (in) Re f ' ( z ) > 0 for \z\ < ./ & 0.6633. In particular f is univalent V 4 + TT in this disc. Proof, (i) Let 6 be a real number and let
Clearly g is holomorphic on U with the possible exception of the point z — -1. We also observe that g(l) — 1 and g(z) ^ 0, z 6 U. Further, since /'(O) = 1 and H/ll = 1, we deduce from Remark 4.2.3 that /"(O) = 0, and
4.3. Locally univalent Bloch functions
161
from this it follows that g'(l) = 1. Using the fact that ||/|| = 1, we deduce that
Moreover, since g'(l) = 1, g cannot be constant and thus g maps U into U \ {0}. We can therefore find a holomorphic function h on U which maps U into II such that g(z) = exp(— h(z)), z € U, and h(l) = 0. A simple computation shows that h'(l) = —g'(l) = — 1, and in view of Corollary 4.3.3 we deduce that \g(x)\ = exp{-Re h(x)} > exp -2, I 1 + zJ
x € (-1, 1).
Furthermore, equality holds for some x e (—1,1) if and only if h(z) = 2——^, 1+z
z€U.
Replacing (1 — x)/2 by t in the above inequality, we conclude that
' * €[0 ' 1) Moreover, equality holds for some t € (0, 1) if and only if f ( z ) = et0F(e~iez) for some real 6. (ii) Keeping the same notation as in (i), we deduce that for all x € (—1, 1), h(x) lies in the disc centered at (1 — x)/(l + x) and of radius (1 — #)/(! + or). Since f 1 —x (4.3.4) min < Re exp(— w) : w — 1+x
n = exp y < -2
for all x € [0,1] (see Problem 4.3.2), we obtain Re g(x) = Re exp(-/i(x)) > exp \ -2-—- > , I 1 + zJ
x e [0,11.
The above relation is equivalent to Re f'(ei0t) >
±- exp -
, t € [0,1/2].
162
Bloch functions and the Bloch constant
The sharpness of this result follows as in (i).
(iii) Let p — \ -
7T
and let
Then p is holomorphic on U and, as in the proof of (i), p maps U into U \ {0}. Moreover, p(l) = 1 andp'(l) = Tr/2. (The value of p is chosen so that the latter condition is satisfied.) Hence we can find a function q which is holomorphic on [7, maps U into II, and satisfiesp(z) = exp(—q(z)), q(l) = 0 and q'(l) = —7T/2. Again applying Corollary 4.3.3, we conclude that q(x) lies in the disc with center TT(! - x)/[2(l + x)] and radius TT(! - x)/[2(l + x)]. On the other hand, since min < Re exp(—£) :
7T 1 — X
7T 1 — X
21+X
for all x e (0, 1], we conclude that Re p(x) > 0 for all x € (0, 1], and hence Re f'(ei0t] > 0 for all t & [0,pj. This completes the proof. (An improvement of the constant in part (iii) of Theorem 4.3.5 (to « 0.6654) was recently obtained by Chen [CheH].) As shown by Liu and Minda [Liu-Min], the lower bound BO > 1/2 follows directly from the distortion theorem, and strict inequality follows from a knowledge of the extremal functions for the distortion theorem. Corollary 4.3.6. Assume f G BQ. Then r(0, /) > 1/2 with equality if and only if f ( z ) = el0F(e~l9z) for some real 0, where F is given by (4-3.2). Moreover, BQ > 1/2. Proof. Let Jl C U be the domain containing 0 which is mapped conformally onto f/ r ( 0) /) by /. Since / is locally univalent, the boundary of ^r(o,/) must meet the boundary of the Riemann surface f ( U ) . Thus there is a line segment F in C^r(o,/) Joining 0 to a boundary point of f(U). Then 7 = (/|n)-1 ° F is an arc beginning at 0 which tends to the boundary of U. Using part (i) of Theorem 4.3.5, we deduce that
r(o, /) = I \dw\ = f |/'
4.3. Locally univalent Bloch functions
163
It is clear that equality occurs if and only if / is a rotation of F. We now prove that BO > 1/2. Landau's reduction together with a normal family argument shows that we can find a function / € BQ such that r(f) = BQ. The first part of the proof implies that r(0, /) > 1/2. If r(0, /) > 1/2 then we are done. On the other hand, if r(0, /) = 1/2 then / is a rotation of F, and from the discussion in Example 4.3.4 we have r(f) = r(F) = e/4 > 1/2. This completes the proof. 4.3.2
The case of convex functions
We end this chapter with a classical theorem of Szego [Sze] which solves the Bloch constant problem for the case of normalized convex functions. We also mention a related theorem of Graham and Varolin [Gra- Var2] . Theorem 4.3.7. // / € K then f(U) contains a disc of radius Tr/4 and this result is sharp. The extremal functions are given by
and its rotations. Proof. There are at least three different ways of proving this result. Szego's proof used a subordination argument to reduce to the case of conformal maps onto infinite strips or triangles. Computations show that a strip mapping is the extremal case. A differential geometric proof using Ahlfors' method was given by Zhang [ZhaM] and rediscovered by Minda [Min3]. The third proof is based on Landau's reduction, which shows that it is sufficient to consider the case of normalized convex functions with vanishing second coefficient (cf. Remark 4.2.3). The covering theorem for such functions (Corollary 2.2.14) states precisely that the image contains a disc of radius ?r/4 centered at 0. One can prove an analogous theorem for normalized convex functions with 0,2 = . . . = afc = 0- This condition holds in particular for convex functions with fc-fold symmetry. The following result was obtained by Graham and Varolin [Gra-Var2] and uses Theorem 2.2.12.
Bloch functions and the Bloch constant
164
Theorem 4.3.8. For the class of normalized convex functions of the form f ( z ) = z + ak+izk+l + ak+2Zk+2 + . . . , z 6 U, the Bloch constant coincides with the Koebe constant and has the value «i dt k>2.
= ./o/'
Extremal functions are given by 1.
2 log fk(z) =
r*
l
dt_
Jo U-* f c
Problems 4.3.1. Prove the inequality in (4.3.3). 4.3.2. Prove (4.3.4). Why is the restriction x 6 [0,1] needed ?
Chapter 5
Linear invariance in the unit disc 5.1
General ideas concerning linear-invariant families
In this chapter we shall investigate some growth and distortion results for locally univalent functions in the unit disc. The setting of this work is Pommerenke's theory of linear-invariant families of locally univalent functions
f(z) = z + a2z2 + o3z3 + . . . ,
z€U.
(See [Poml], [Pom2].) The basic notion in this theory is the order of a given linear-invariant family, introduced by Pommerenke [Poml]. We shall give a number of applications of this notion, including generalizations of some classical results for the set S which have been studied in the previous chapters. Let £S denote the set of normalized locally univalent functions on the unit disc U. If / € £S and 0 is an automorphism of U, we let A^(/) denote the Koebe transform of / with respect to 0, i.e. ( / < > * ) ( * ) - ( / 00(0)
(/o0'(0) 165
- rr
Linear invariance in the unit disc
166
We note that the Koebe transform has the following group property:
We recall that Aut(C7) denotes the set of holomorphic automorphisms of the unit disc. Definition 5.1.1. The family T is called a linear-invariant family (L.I.F.) if (i) f C £S, (ii) A^(/) e .F, for all / e F and 0 <E Aut(U). Definition 5.1.2. For a linear- invariant family F, let
r /"(O) /"(o) . . r f\
ord T = sup \
2 -
/ e >
J
denote the order of the L.I.F. T. Evidently, if T is a compact family, then ord.F < oo. The first result of this chapter is due to Pommerenke [Poml] and it is useful in many applications. Lemma 5.1.3. Let T be a linear-invariant family and a = ord^7. Then (5.1.1)
a = sup sup
'/'(C)
Proof. Fix ( € U. Given / € T, we consider the Koebe transform j(z; £) of / with respect to the disc automorphism (j)(z) = (z + C)/(l + C^)> i-e-
Letting z* —
—, it is clear that
(5.1.3) Hence
,,, a2/ ,, (5. .1.4)
^(z,C) = / (z,C) = -
2? /'(z») 1
»
i-KP /"(»•)
+ i
, « '
5.1. General ideas
167
so that
<<*, and thus (5.1.5)
sup sup
2 v-
,/"«)
i.i / / / ( c )
On the other hand, since f C CS we clearly have sup sup
2
V
m
./"(C) > sup
/"(O)
'/'(C)
and together with (5.1.5), this implies (5.1.1) as desired. We now give some examples of L.I.F.'s in the unit disc (for further examples see [Poml]). Example 5.1.4. The class S of normalized univalent functions on U is a L.I.F. Example 5.1.5. The class K of normalized convex functions on U is a L.I.F. The class C of normalized close-to-convex functions on U is also a L.I.F. However, the class S* of normalized starlike functions on U is not a L.I.F. Example 5.1.6. CS is a L.I.F. of infinite order. To see this, let k > 0 and /(z) = [ekz - l]/fc, z € U. Then it is obvious that / € CS and |/"(0)/2| = fc/2 —)• oo as k —> oo. The universal L.I.F. U(a), consisting of all L.I.F.'s contained in CS with order not greater than a, is also a L.I.F. Example 5.1.7. Let Q be a non-empty subset of CS and let
denote the L.I.F. generated by Q. Clearly Q is a L.I.F. if and only if A[<7] = Q. (This construction gives a number of interesting examples of L.I.F.'s on U.} We now consider a basic distortion and growth theorem for a L.I.F. of order a which generalizes many classical results from the theory of univalent functions on the unit disc. This result was obtained in 1964 by Pommerenke, together with other estimates for L.I.F.'s [Poml, Satz 1.1]. We give only an upper bound in the growth result (5.1.7), since the general L.I.F. may include functions that have zeros in 0 < \z\ < I.
168
Linear invariance in the unit disc
Theorem 5.1.8. Let T be a L.I.F. with a — ordT" < oo. Also let f e and \z\ = r < 1. Then (5.1.6)
1-r
and
Equality in (5.1.6) and (5.1.7) is achieved by the function
Remark 5.1.9. Taking real parts in (5.1.6) and exponentiating, we deduce the following distortion theorem for the L.I.F. T with ord^" = a < oo:
Also, taking imaginary parts in (5.1.6), we obtain
We mention that Campbell [Cam] showed that equality in (5.1.8) holds if and only if / is a rotation of the generalized Koebe function, i.e. (5.1.9)
2a
- 1 ,
z € U,
0 e R.
We note that for a = 2 in (5.1.8), the distortion result is the same as for the class S (cf. (1.1.6)). The case a = 1 is of special interest because, as we shall see, this is the minimum possible order and any L.I.F. with order 1 must be a subfamily of K. We also mention that if T is a L.I.F. with ord.77 = a < oo, then the estimate (5.1.7) implies that T is a normal family. Proof of Theorem 5.1.8. Taking into account the relation (5.1.1), we deduce that
A Iog[(l-r 2 )/'(re i6? )] dr
r
f'(reie)
2a 1-r 2
5.1. General ideas
169
Integrating both sides of this inequality, we obtain (5.1.6). Furthermore, since for z = retd we have rr Jo
we obtain in view of the upper bound in (5.1.8) that
!/(*)!< [* \f'(pei9)\dp Jo <
rr (i -i- /i^ a ~i
Hence we deduce the estimate (5.1.7). Remark 5.1.10. Let T be a L.I.F. with ordF = a < oo, such that F is a subset of 5. If / € T and \z\ = r € [0, 1), then we also have the estimates (see [Poml], [Cam], [Gon5])
(5.1.10)
| /W |>
and
l/(z)l (51- 'ID -L fifcVl < < i. [fl±iY-il*\ ' 2a i (l + r ) \ * (1 - r»)|/>W| ~ 2a [(l - r ) (5 1 11)
1
Proof. In order to derive the lower estimate in (5.1.10), let
Obviously, the closed disc Upir) is contained in the image of the closed disc Ur. Next, let z\ e U, \z\\ = r, so that |/(zi)| = p(r). It is clear that the closed segment between 0 and f ( z \ ] lies in the closed disc #p( r )- Denote by T this segment and let 7 be the inverse image of F. Then 7 is contained in Ur and moreover, using the lower bound in (5.1.8), we have = f\f(z)\\dz\ J
> 7
170
Linear invariance in the unit disc
Next, we prove (5.1.11). For this purpose, let £ € U and /(^;C) be given by (5.1.2). Then /(z;C) € T and from (5.1.7) and (5.1.10) we obtain
Letting z = — £ in the above, we deduce (5.1.11), as desired. This completes the proof. We remark that Campbell [Cam] extended the results of Theorems 1.1.8 and 1.1.9 to the theory of the universal L.I.F. U(a). The foregoing proof contains the following covering result for L.I.F.'s, due to Pommerenke [Pomlj: Corollary 5.1.11. If F is a L.I.F. with ord T = a < oo and f € F, then f ( U r } contains the schlicht disc Up, where l i ^=— Ci 1p = p(a,r) 2a L V1 + r
for all r € (0,1). In particular, f ( U ) contains the schlicht disc Ui/(2a)We now prove that for each L.I.F. T the order is always at least 1 (cf. [Poml, Folgerung 1.1]).
Theorem 5.1.12. Let F be a L.I.F. and let a - ordT. Then a>l. Proof. Suppose a < 1. Then from (5.1.8) we conclude that if / € T then |/'(z)| -» oo as \z\ —>• 1. This gives a contradiction to the maximum principle for the holomorphic function I//'. Therefore we must have a > 1, as stated. The following result provides another way to generate L.I.F.'s in the unit disc (see [Poml, Satz 1.2]). To this end, let W denote the set of univalent functions (z)\ < 1, z e U, i.e.
Theorem 5.1.13. Let F be a L.I.F. with ord^" = a < oo and let denote the set of functions of the form
where f e F and (p € W. Then M is a L.I.F. and ord M = max{a, 2}.
5.1. General ideas
171
Proof. First we show that if g € M and t/> € W, then A^,(p) e M.. Now
because y? o -0 € W. In particular, this is true if V> € Aut(£7). It is clear that M C £S, and hence M is a L.I.F. Now let 0 = ord M. and with g as above, let
and let A = T o <^» and h = AT-i (/). Then A(0) = 0 and again using the group property of the Koebe transform, we obtain
g(z) = Hence
and
(5.1-12)
'(0)
2
Since --is a function in S and < ....,.., z € 17, it follows from A'(0) A (0) Problem 1.1.1 that 1 A'(0) Using (5.1.12), we conclude that |/(0)| < 2(1 - |A'(0)|) + a|V(0)| < max{a,2}, and therefore (3 < max{o!, 2}. Now let F(z) = z + a,2Z2 + . . . belong to M and choose to be
(1-rz)*'
172
Linear invariance in the unit disc
Then
<£(z) = (1 - r)2z + 2r(l - r) V + ...,
z G U.
(Compare with the Koebe function.) After simple computations, we obtain h
zeU,
and letting r /- 1, we observe that 1
n
Hence (3 > 2. On the other hand, because M D F, we have (3 > a, so /3 > max{a:, 2}. This completes the proof.
5.2 5.2.1
Extremal problems and radius of univalence Bounds for coefficients of functions in linear-invariant families
In Section 5.1 we have seen that the notion of linear invariance plays an important role in several problems of function theory of one complex variable. In this section we are going to study coefficient bounds as well as the radius of univalence, starlikeness, and convexity for L.I.F.'s of finite order. We begin with the following estimate involving the second and third coefficients for functions in compact L.I.F.'s on the unit disc. This result was obtained by Pommerenke in 1964 [Poml]. Theorem 5.2.1. Let T be a compact L.I.F. with a = ord.7-". If f ( z ) — z + azz2 + a^z^ + ... is a function in T such that a% = a, then /(o.z.i) c o i\
« = —(Za. /o ~ 2 4-1). i i\ 03 3 Proof. Let z, £ e U and consider the Koebe transform f ( z ; £) given by (5.1.2). Expanding f ( z ; £) in powers of £ and £ and taking into account (5.1.3) and (5.1.4), we deduce that
/(z; C) = /(z) + (/'(z) - 1 - 2oa/(z))C - *2/'(*)C + 0(|C|2)
173
5.2. Extremal problems and radius of univalence for |£| small. Now expanding in z, we obtain /(*; C) = z + [09 + (3a3 - 2ai)C - ? + O(IC|2)]*2 + • Since / is such that a-i = a, the preceding expansion gives
and hence a + Re [(3a3 - 2a2 - 1)C] + O(|C|2) < a. Considering small values of £, we see that this is only possible if (5.2.1) holds. This completes the proof. Another bound for the coefficients of functions that belong to L.I.F.'s of finite order is given in the following [Poml, Satz 2.4]. (Recall that |o3 — a2,) < 1 for functions in S and |a3 — a|| < 1/3 for functions in K.) Theorem 5.2.2. Let T be a L.I.F. with ord T = a < oo and let A e R. Then (5.2.2) < sup 1103 — Aar,| : / E T, f ( z ) = z + a^z2 + a3z3 + ... | - - A a2 + VZa + 1. o Proof. Since ord T = a < oo we may assume that f is a compact family (see Problem 5.2.4). Choose h € T such that h(z) = z + b-^z2 + b3z3 + ... and 62 = «• Then from Theorem 5.2.1 we deduce that
and hence we obtain the lower bound in (5.2.2). We now let f ( z ) = z + a^z2 + a323 + ... € T, and consider 4r^r = 2a2 + (3o3 - 2a%)z + ...,
z e U.
Applying Lemma 5.1.3, we conclude that 2ir
a +r dd < 2 ~ 1-r 2 '
174
Linear invariance in the unit disc
for 0 < r < 1. The optimal bound on \3a^ — 2a|| is obtained when r = l/\/3, namely
|3a3
€
lal <3>/3 a + 4 = • V V/3/
ity now gives 1^3-
H-A 3
2
2 + a3 - ?a 3
<
2 3~
which completes the proof.
5.2.2
Radius problems for linear-invariant families
We now give some results involving linear invariance and radius problems. In 1964 Pommerenke [Poml] obtained the following remarkable result concerning the radius of convexity rc(.F) of a linear-invariant family F. Recall that rc(F) is the largest number such that every function in the set T is convex on Theorem 5.2.3. Let F be a linear-invariant family with a = ord.F < oo. Then (5.2.3) rc = aRemark 5.2.4. For a = 2 we obtain the familiar radius of convexity for the class S, i.e. rc(S) = 2 - >/3 (see Theorem 2.2.22). We remark that in [Cam-Zi] the authors obtained the radius of close-to-convexity of various linear-invariant families. Proof of Theorem 5.2.3. Let / e jF. Using the relation (5.1.1), we obtain
For z < a — \/a2 — 1 one deduces that (5.2.4) and thus, from Theorem 2.2.3 we conclude that / is convex on Up, where p = p(a) — a — ^o? — 1. Therefore TC(T} > p(ot).
5.2. Extremal problems and radius of uni valence
175
We now show that rc(F) < p(a). To this end, let / e T and let £ with |£| < r = rc(F). Consider the map /(z;C) given by (5.1.2) and set z = — £. Using (5.1.3), (5.1.4) and (5.2.4), we conclude that
Therefore Re [C/"(0)] < 1 + |C| 2 > and since £ is arbitrary chosen on the disc Ur, we deduce by letting |£| -> r that
Consequently, a = sup
1+r" 2r
that is r < a — Vex2 — 1. Thus we obtain rc(^r) = /o(a), as claimed. This completes the proof. Corollary 5.2.5. Let T be a L.I.F. with a = ordF. Then T is a subfamily of K if and only if a = 1 . This result was obtained in [Poml]. Another important consequence of Theorem 5.2.3 is the following (see [Poml, Folgerung 2.5]): Corollary 5.2.6. Let F be a L.I.F. with a = oidf < oo. Then each f G f maps the disc Ui/a conformally onto a starlike domain with respect to zero. Proof. Let ZQ be given with \ZQ\ < I/a and choose £ such that ZQ = 2C/(1 + |C| 2 )- Then it is clear that |£| < a - Va2 - 1. If /(*;C) denotes the function given by (5.1.2), then from Theorem 5.2.3 we know that /(^;C) is convex for \z\ < a — \/o;2 — 1. Therefore /(^;C) ~~ /(~C>C) is starlike with respect to — £ for \z\ < a — Va2 — 1, and hence from Theorem 2.2.2 we deduce that Re
>Q
_ for |C| < \z\ < a — yet2 — 1. Moreover, since ZQ = --j—r^, we obtain Re
176
Linear invariance in the unit disc
Because ZQ is arbitrary, we deduce that / is starlike on Ui/a. We finish this section by obtaining a relation between the radius of univalence of a L.I.F. J7 and its radius of nonvanishing. Let ro denote the largest positive number such that no function in the L.I.F. T has zeros on Uro \ {0}. Also let ri denote the largest number such that every function in the L.I.F. T is univalent on Uri. Then we have the following result, due to Pommerenke [Poml]: Theorem 5.2.7. IjT is a L.I.F. of finite order, then r\ = Proof. Let / e F and r <
1 + v'l - rg
. Also let zi,Z2 be such that
\z\\ < r, \Z2\ < r, z\ 7^ 22- Then 0<
2r
Since f ( z ; z\) 6 T, where f ( z ; C) is defined by (5.1.2), we have
This implies f ( z \ ) ^ f(z2), and thus / is univalent on Ur. Hence
In order to prove the converse, let ZQ, £ e ?7 be such that 0 < \ZQ\ < and ZQ = 1
i |/ , |2 .
r
Then 0 < |C| < ri and r
This implies that rn >
27*1
:C;C)-/(-C;7*n
o > i-e- ri < . ft. This completes the l + ^i 1 + Vl + rg
proof. Remark 5.2.8. Linear invariance is closely related to the concepts of uniform local univalence and uniform local convexity. These latter notions are defined with respect to hyperbolic geometry on the unit disc U. If dh(a,b) is the hyperbolic distance between a, b & U (see (1.1.12)), then Uh(a,p) = {z €
5.2. Extremal problems and radius of univalence
177
U : dh(a,z) < p} denotes the hyperbolic disc in U with center a and radius p, 0 < p < oo. For / e H(U), let p ( z , f ) denote the hyperbolic radius of the largest hyperbolic disc in U centered at z in which / is univalent. Define p(f) = mf{p(zj):
ZEU}.
The function / is called uniformly locally univalent (in the hyperbolic sense) provided p(f) > 0. For a locally univalent function / in U, let pc(f) = sup < p : /is univalent on Uh(a, p) and f(Uh(a, /?)) is a convex domain, for all a € U > . Then pc(f) < p(/)- The locally univalent function / is called uniformly locally convex (in the hyperbolic sense) provided pc(f) > 0. We remark that some of the radii problems we have just studied can be formulated in a more invariant way. Ma and Minda [Ma-Min2] proved the following beautiful result: Theorem 5.2.9. Given p > 0, the L.I.F. /C(p) consisting of functions f in CS such that pc(f) > p coincides with the universal L.I.F. U(coth(2p)). In other words, linear invariance of finite order can be characterized geometrically. Similarly the family S(p) of functions / in CS such that p(f] > p is linearinvariant. Ma and Minda [Ma-Min2] (see also [Poml]) gave upper and lower bounds for its order. Notes. In addition to the basic papers of Pommerenke [Poml], [Pom2], the reader may consult [Cam], [Cam-Cim-Pfa], [Cam-Pf], [Cam-Zi], [Gon5], [Koepl], [Ma-Mini], and [Ma-Min2] for material on L.I.F. 's.
Problems 5.2.1. Let / be a normalized locally univalent function on U. Also let z€U
178
Linear invariance in the unit disc
where {/; z} denotes the Schwarzian derivative of / at z. Show that / G where
/2
(Pommerenke, 1964 [Poml].) (For other relations of this type see [Har3].) 5.2.2. Let F be a family of normalized locally univalent functions on U. For 0 < r < 1, let G(r, F} = G(r) — sup max arg /'(;?), where the argument varies continuously from the initial value of arg /'(O) = 0. Show that if F is a linear-invariant family, then G(r\ — — inf minarg/'f-z). (Campbell and Ziegler, 1974 [Cam-Zi].) 5.2.3. a) Let F be a linear-invariant family of finite order. Let cl(F] denote the closure of F in the topology of uniform convergence on compact sets. Let F(*) = { f ( t z ) / t : f G F, 0 < t < I}. Show that
where G(r, F} is defined in the previous problem. Moreover, show that G(r, F} is an increasing continuous function of r, satisfying 2arcsinr < G(r) for 0 < r < 1. b) If F is any linear- invariant family of normalized convex functions, show that G(r] = 2arcsinr. (Campbell and Ziegler, 1974 [Cam-Zi].) 5.2.4. Let F be a L.I.F. such that ord.F - a. Show that d(F] is a L.I.F. and ord [c/(^")] = a. 5.2.5. Show that the class C of normalized close-to-convex functions on U is a L.I.F. 5.2.6. Let F be a L.I.F. such that ord^" = a < oo. Also let p be the radius of univalence of F. Show that if / £ F, then
5.2. Extremal problems and radius of univalence
179
(Campbell, 1974 [Cam].) 5.2.7. Show that equality in (5.1.8) holds if and only if / is given by (5.1.9). (Campbell, 1974 [Cam].) 5.2.8. Let T be a L.I.F. of finite order and / e F. Show that f ' ( z ) is a normal function. (Pommerenke, 1964 [Poml].)
Part II
Univalent mappings in several complex variables and complex Banach spaces
Chapter 6
Univalence in several complex variables The purpose of the second part of this book is to study the theory of biholomorphic mappings of the unit ball and certain bounded domains in C1 and in complex Banach spaces. The domains will be such that precise results can be given for the solutions of extremal problems for holomorphic mappings - growth, distortion, and covering theorems, coefficient estimates, etc. This means that we shall be primarily interested in the Euclidean unit ball and the unit polydisc in C"1, more generally in the unit ball with respect to an arbitrary norm, and in some cases bounded balanced pseudoconvex domains (see Section 6.1.2), where the growth of holomorphic functions may be measured relative to the Minkowski function. For reasons which are explained in Section 6.1.7, we are obliged to focus on proper subclasses of the normalized biholomorphic mappings of such domains into the ambient space (which we denote by S(B) in the case of the unit ball). As in one variable, such subclasses will frequently be defined by geometrical conditions which admit reformulations as analytic conditions. The most basic conditions of this type are of course starlikeness and convexity, and we shall study criteria for starlikeness and convexity in this chapter and extremal problems for these classes in the next. We shall also discuss a number of other classes which are generalizations of familiar subclasses of S to several 183
184
Univalence in several complex variables
variables - close-to-starlikeness (a generalization of close-to-convexity), spirallikeness, etc. It is to be expected, however, that new subclasses of S(B) which are not direct generalizations of previously-studied classes in one variable will also be of interest. For example, we consider the quasi-convex maps introduced by Roper and Suffridge [Rop-Su2] in Section 6.3 and Chapter 7. A powerful tool for studying extremal problems for the class S in one variable is the Loewner method. We shall give a detailed account of the extension of this method to several variables, including recent improvements of the known existence theorems and new applications. In several variables the set of normalized biholomorphic maps which can be embedded in a nice way as the initial element of a Loewner chain (i.e. such that the initial map has a parametric representation) is a proper subclass of S(B). It is obviously a subclass of special interest and it contains many of the other subclasses considered here. We shall study Bloch mappings in several variables and also linear-invariant families. Finally we shall study extension operators which map particular subclasses of S to particular subclasses of S(B), for the case that B is the Euclidean unit ball in Cn. The most important such operator is the Roper-Suffridge operator, which extends a convex univalent function of one variable to a convex mapping of B and has many other nice properties.
6.1 6.1.1
Preliminaries concerning holomorphic mappings in Cn and complex Banach spaces Holomorphic functions in Cn
We shall begin with a brief discussion of holomorphic functions of several variables. We shall present some of the most important results about such functions, in some cases without complete proofs. Let C71 denote the space of n-complex variables z — (zi,...,zn) where Zj € C, 1 < j < n. Thus Cn is the Cartesian product of n copies of C, and is a complex vector space: addition of two elements as well as the multiplication of an element of Cn by a complex scalar is defined componentwise. As a
6.1. Preliminaries
185
complex vector space C"1 is n-dimensional, but as a real vector space it is 2ndimensional. Naturally, C1 may be identified with E2n via the map (zi + %,...,£„ + iyn] <-»• (a?i, yi, • • - , xn, yn). This identification gives the topology of C1 : all the usual concepts from topology and analysis of the Euclidean space R2n may be transferred to C™ . The Euclidean inner product n
Wj ,
z, w G Cn ,
and the associated Euclidean norm \\z\\ — (ZjZ)12 make C* into an ndimensional complex Hilbert space. If a 6 C™ and r > 0, we let B(a, r) = {z € C™ : \\z — a\\ < r} be the open ball of center o and radius r. The closure of B(a, r) will be denoted by B(a, r) and its boundary by dB(a, r). Instead of J5(0, r) we shall write Br, and the open unit ball B\ will be denoted simply by B and will be called the open Euclidean unit ball of C™ . When it is necessary to mention the dimension explicitly, the ball Br (resp. B) will be denoted by B? (resp. Bn). In the case n = 1 we shall write U instead of B1. We shall also make use of other norms, retaining the notation B for the corresponding unit ball (except in certain special cases). An open polydisc of center a = (ai, . . . , a n ) and polyradius R — (ri, . . . , r n ) is a subset of C™ of the form P(o, R) = P(ai , . . . , OB, n, . . . , r n ) = < z 6 C1 : \Zj-aj\
1 < j < n >.
Thus P(a, R) is the Cartesian product of n discs. When n = . . . = rn = r we denote the polydisc P(a,R) by P(a,r). Instead of P(0, r) we shall write Pr. The unit polydisc P(0, 1) will be denoted by P and is the unit ball with respect to the maximum norm || • ||oo5 INloo = max \Zj\. l<j
186
Univalence in several complex variables
The set of bounded complex-linear operators from C71 into C™ will be denoted by Z^C^C"); the identity in £(€",0*) is denoted by /. If a = (QI, ... , a n ) is an ordered n-tuple of nonnegative integers c^, we shall use the standard multi-index notations ai H
Q
nn _,a z~l 1 • • • z~oc n = z ,
l-a n = |a|
/ /pn z = (zi,... , z n )\ €.,- (L .
Writing the coordinates ^ of a point z € Cn in the form zy = Xj + < j < n, we introduce the differential operators
a
i/a
dzj
2 \dxj
.d\
- i—dyjj
,
d
i ( d ^. d
and -— = h I-TT&Zj 2 \dxj
which act on complex- valued functions of class C1 . There are several equivalent ways of defining a holomorphic function on a domain in C71. Definition 6.1.1. Let ft be a domain in Cn and / : fi -> C. We say that / is holomorphic if / is continuous on £1 and holomorphic in each variable separately, i.e. for each ZQ — (z®, . . . , z%) e 17 and j € {1, . . . , n}, the function
is holomorphic on the open set {(, € C : (z°, . . . , Zj_v C, Zj+l, . . . , z%) e ft}. Such a function obviously satisfies the Cauchy-Riemann equations
~(z) = 0, OZj
2 e f t , 1< j
(The assumption of continuity in Definition 6.1.1 can be omitted, but it simplifies matters to make this assumption. See [Kran].) Definition 6.1.2. The function / : ft -> C is holomorphic if for each point a € ft there is a mapping in L(C n ,C), denoted by £>/(a), such that f ( z ) = /(a) + Df(a)(z - a) + o(z - a) o(z — a) near a, where hm -^-TT- = 0. z-^a
2- a
6.1. Preliminaries
187
Definition 6.1.3. A function / : fi —>• C is holomorphic if for each point a e fi there is an open neighbourhood V of a such that / can be represented as a multiple power series expansion oo
(6.1.1)
/(*)=
cai...an(Zl-air...(zn-an}a-
£ ai,...,an=0
that converges absolutely and uniformly on each compact subset of V. (One can choose V to be any poly disc centered at a that lies in D.) Using multi-index notation, (6.1.1) can be written as
We denote the set of holomorphic functions on a domain Q in C1 by If / is a holomorphic function in a neighbourhood of a closed polydisc P(a,R), then Cauchy's integral formula can be generalized as follows: l
f
J
= (2^
-
|Cn-an|=rn
|Cl-ai|=
doP(a,R)
This formula is known as Cauchy's integral formula on the polydisc. Note that if n > 1, the integration in (6.1.2) does not extend over the full boundary of P(a, R), but only over a subset d<)P(a,R) of real dimension n called the distinguished boundary. (There are other versions of Cauchy's integral formula for domains in C" with smooth boundary in which the integration is taken over the full boundary, but we shall not need them in this book. See [H6r2], [Kran], [Ran].) By expanding the kernel in (6.1.2) as a multiple geometric series, one sees that the coefficients in (6.1.1) are given by (0.1.6) cai...an -(613) c C a --C
1
(2wi)n
JI doP(a,R)
/(Cl,....CnXl...<*Cn
188
Univalence in several complex variables
On the other hand, by differentiating under the integral sign in (6.1.2) we obtain, using multi-index notation, (6.1.4)
(z} —
:—
/
'''''
———r,
d0P(a,R)
for any ordered n-tuple a = (ai,... ,a n ) of nonnegative integers ai and z £ P(a,R}. Comparing this formula with (6.1.3), one deduces that
i
-
d
^f(
Therefore, the expansion (6.1.1) is unique. Moreover, holomorphic functions have derivatives of all orders. Some additional consequences of the Cauchy integral formula on the polydisc are given in the following theorems: Theorem 6.1.4. (Cauchy estimates on the polydisc) Let fJ be an open neighbourhood of a closed polydisc P(a, R) in Cn and let f : 0, —> C be a holomorphic function. If\f(z)\ < M for all z 6 P(a,R) and a = (ai,... ,an) € NQ (where N0 = N U {0}), then
or in multi-index notation, (6.1.5)
d\a\f
< M(a\)R~a,
where R~a = r^ai • • • r~ Q ". Theorem 6.1.5. (Maximum modulus theorem) Let J7 be a domain in C71 and f : fi! —> C be a holomorphic function. If \f\ achieves its maximum at an interior point of Jl, then f is constant. 6.1.2
Classes of domains in Cn. Pseudoconvexity
We begin with domains which have certain types of rotational symmetry.
6.1. Preliminaries
189
Definition 6.1.6. Let ft be a domain in C™. (i) We say that ft is circular if el0z & £1 whenever z 6 ft and 0 € R. (ii) We say that ft is complete circular or balanced if Az € ft whenever z € ft and A e J7. (iii) We say that ft is a Reinhardt domain if, whenever (zi,...,z n ) € ft and 0j e R, j = 1,..., n, we have (ei6lz\,..., ei6>nzn) € ft. (iv) Finally, we say that ft is a complete Reinhardt domain if, whenever (zi,..., zn) € ft and Xj € t/", j = 1,..., n, we have ( A i z i , . . . , A n z n ) e ft. Obviously the complete Reinhardt domains in C" are those domains which can be written as the union of poly discs centered at 0. Such domains arise in the study of power series expansions: the largest domain in which a power series ^D aaza converges absolutely is a complete Reinhardt domain. For balanced domains in C", there is an important auxiliary function known as the Minkowski function: Definition 6.1.7. Suppose that ft is a balanced domain in C71. The Minkowski function of ft is defined for z € C" by h(z) = inf h > 0 : - e f t } . Clearly ft = {z € Cn : h(z) < 1}. It can be shown that h : Cn -» [0, oo) is upper semicontinuous, and it is clear that h(Xz) = \X\h(z) for A 6 C, z € C", and h = 0 iff ft = C". When ft is a bounded balanced convex domain in Cn, it is well known that h is a norm. We shall also consider certain pseudoconvex domains in this book. Pseudoconvexity is a basic concept in several complex variables and is denned in terms of plurisubharmonic functions. Definition 6.1.8. Let D be a domain in C and let (p : D —>• [—00, oo) be upper semicontinuous. We say that (p is subharmonic if for every z 6 D and p > 0 such that U(z,p) C D and for every function g, continuous on U(z,p) and harmonic on U(z,p) such that (p < g on dt/(z,/o), we have (p < g on U(z,p). Definition 6.1.9. Let ft be a domain in C" and let (p : ft —>• [—00, oo) be an upper semicontinuous function on ft. We say that (p is plurisubharmonic
190
Univalence in several complex variables
(psh) on O if for each z 6 $1 and w e C" the function of one complex variable C >->•
is subharmonic where it is defined. We note the following properties of psh functions: Theorem 6.1.10. Let Cl C C* and ft' C C" be domains. (i) If f € H(£l,C) then \f\a, where a > 0, and log |/| are psh on 0. (ii) Let
is positive semi- definite at each point zof£l. (Hi) If ip is psh on fJ' and f : J7 -» fi' is holomorphic then ip o / is psh on ft. Definition 6.1.11. Let ft be a domain in C™ and for z 6 fi let Sfi(z) denote the Euclidean distance from z to dft. We say that J7 is pseudoconvex if the function — log<5f}(-) is psh on £l. The importance of pseudoconvexity stems from the study of the Levi problem (one of the main lines of development of the early theory of several complex variables) and its solution: a domain Q C Cn is pseudoconvex iff it is a domain of holomorphy, i.e. the natural domain of definition of some holomorphic function. For further details, see [Gun-Ros], [H6r2], [Kran], or [Ran]. For balanced domains, pseudoconvexity can be characterized in terms of the Minkowski function: Theorem 6.1.12. Let fJ be a balanced domain in C" with Minkowski function h. Then H is pseudoconvex iff h is psh. For a proof, see [Dinl, Lemma 2]. As shown by Hamada and Kohr and as we shall see in this book, quite a number of results in geometric function theory on the unit ball in C71 can be generalized to balanced pseudoconvex domains in Cn whose Minkowski function is C1 on Cn\{0}. (Caveat: A balanced domain with C1 boundary need not have a Minkowski function which is C1 on Cn\{0}, or even continuous. See [Ham4].)
6.1. Preliminaries
6.1.3
191
Holomorphic mappings
Let ft be a domain in C71 and let / : ft —» C™ be a mapping. By writing / = (/i,..., fm) and fk = Uk + ivk, 1 < k < m, where Uk, vj, : ft —>• M, we may identify / with the mapping from ft C R2n into R2m given by ^3^1, yi} • • • 5 «En.5 Un) ' ' \^1» ^1? • • • i ^TTM ^m)'
The mapping / is called holomorphic on ft if each of its components / ! > • • • » /m is holomorphic on ft. In this case the differential D f ( z ) at z € ft is a complex linear mapping from C" into C7", which can be identified with the (complex) Jacobian matrix
(6.1.6)
Df(z) =
and the relation
holds for h in a neighbourhood of the origin of C" . We denote the set of holomorphic mappings from 0 into a domain fi' C C" by H(ft, ft'). We shall write H(£l) in place of #(ft,C") (the equidimensional case). If 0 € ft, a mapping / € JJ(ft) will be said to be normalized if /(O) = 0 and Df(0) = I. Since our objective is to study univalent mappings, we begin by recalling the inverse mapping theorem. If / : ft C C" —>• C" is holomorphic, we say that / is nonsingular at ZQ 6 ft if D/(ZO) is invertible. The mapping / is nonsingular on ft if it is nonsingular at each z € ft. Theorem 6.1.13. (Inverse mapping theorem) //ft is a domain in C71 and / : ft —)• C71 is a holomorphic mapping such that D/(ZQ) is invertible for some point ZQ of ft, then there are neighbourhoods V of ZQ and W of f(zo) such that f is a one-to-one map of V onto W whose inverse is holomorphic on W.
192
Univalence in several complex variables
If / : ft C Cn —> Cn is a holomorphic mapping on the domain ft, let Jf(z] = det D f ( z ] , z e ft, denote the (complex) Jacobian determinant of / at z. Also let D r f ( z ) be the real Jacobian matrix of / at z € ft, that is
du\
du\ dyn ov\
dx\ (6.1.7)
dyi
dvn dyn . and Jrf(z] = det D r f ( z ) . Then elementary computations using matrix theory in (6.1.6) and (6.1.7) give the relation (6.1.8)
dvn
dvn
L dxi
dyi
Jrf(z) = | J f ( z ) \ 2 > 0,
'"
zeft.
We say that / is locally biholomorphic on ft if Jf(z) =£ 0, z € £2. In this case, / preserves orientation. We say that / : ft C Cn —> Cn is biholomorphic on D if / is a holomorphic map from ft onto a domain JV C Cn and has a holomorphic inverse defined on ft'. In this case the domains ft and ft' are called biholomorphically equivalent. Let 5(J5) be the set of normalized biholomorphic mappings from the unit ball B of C71 intoC n . If ft is a domain in Cn and / : ft —> Cn is an injective holomorphic mapping, we say that / is univalent on ft. Such a mapping is necessarily nonsingular on ft, and is thus biholomorphic on ft. For this reason, there is no confusion in using the term univalent mapping instead of biholomorphic mapping. (In infinite dimensions we must be more careful.) The chain rule of multivariable calculus can be expressed in complex form as follows: if ft C Cn and ft' C Cm are domains, if g: ft ->• ft' and / : ft' -> Cp are C1 maps, and if C — 9(z}i
and
w
— /(C)i then
6.1. Preliminaries
193
In particular, if / and g are holomorphic then so is fog, and D(fog)(z) Df(g(z))Dg(z). If (p is a C1 function on f2 we shall sometimes write
=
d(p _ =
Moreover, if (p is a C2 function on 17, let
0V x _ r &v < xi
A
&v , x _~ r
The theory of normal families generalizes to several variables, and we have the following basic theorem: Theorem 6.1.14. I f f is a locally uniformly bounded family of holomorphic mappings from a domain 17 C Cn into C"1 then F is normal. A related result which is very useful in the study of Loewner chains is Vitali's theorem. To state it we need the following definition: Definition 6.1.15. Let 17 be a domain in C1. A subset A of 17 is called a set of uniqueness for the holomorphic functions on 17 if whenever / € H(17, C) and f\A = 0 then / = 0. We note that there exist countable sets of uniqueness, for example any countable dense subset of 17 is a set of uniqueness. Of course, if A is a nonempty open subset of 17, then -A is a set of uniqueness for the holomorphic functions on J7. The reader may show that if J = {1/k : k = 2,3, . . .} and A = (JxJx . . . x J) n B, then A is a set of uniqueness for the holomorphic n— times
functions on the unit ball B of C" .
Theorem 6.1.16. (Vitali's theorem in several complex variables) Suppose that ft is a domain in C" and A C fi is a set of uniqueness for the holomorphic functions on fi. Let {fk}keN be a sequence of functions in H(tl,C) which is locally uniformly bounded and which is such that {/fc(^)}fceN converges for all z € A. Then there exists f € H(O,C) such that fk —* f locally uniformly on 17 as k —> oo. We note that this theorem also applies to holomorphic mappings whose target space is C" . There is also an n-dimensional version of Hurwitz's theorem:
194
Univalence in several complex variables
Theorem 6.1.17. Suppose that ft is a domain in C™ and that {/fc}fceN C H(fl) is a sequence of biholomorphic maps such that /& —> / locally uniformly on £1. Then either f is biholomorphic on ft or Jf = 0. This is complemented by the following convergence result: Theorem 6.1.18. Suppose that ft is a domain in C1 and that f e H(fl) is a biholomorphic mapping. Suppose {fk}k&w C H(ft) is a sequence of mappings such that f k ~ * f locally uniformly on ft. Let K be a compact subset of f t . Then there exists ko > 1 such that fk\K is injective for k > kQ. Proof. Suppose the contrary. Then there exists a subsequence of {/fc}fceN) which we denote again by {/fc}fceN> and there exist sequences {afc}^eN and {fyfelfceN of points in K such that Ofc ^ 6^ and fk(o>k) — fk(bk)- By passing to subsequences we may assume that a^ —> a and bk —>• b as k —)• oo, where a and 6 are points in K. If a =£ b then we obtain /(a) = /(b), which is a contradiction. Hence a = b. We have flfc - bk
K-6,|| (Dfk(bk)-Df(bk^
\\Vk-bkl
Passing to subsequences again, we may assume that there is a unit vector f € C71 such that -^ —-^ -> v as fc -> oo. But then Df(a)(v] - 0, which is ||a* -Ofcll a contradiction. There are two basic theorems on holomorphic mappings in several variables due to H. Cartan [Cartl]. The first is his theorem on fixed points: Theorem 6.1.19. Suppose that f2 is a bounded domain in C1, and that f € H(ft, H) /ias a /ized pom< 2:9 a^ which Df(zo) = I. Then f is the identity map. This theorem may be viewed as a generalization of the Schwarz lemma. Prom it we obtain Theorem 6.1.20. Suppose that fZi and X72 are bounded circular domains in C" containing 0, and that f is a biholomorphic map from fii onto f^2 such that /(O) =0. Then f is linear.
6.1. Preliminaries
195
This theorem shows that biholomorphic mappings in several variables are very rigid. In particular we obtain the following result, which was first proved by Poincare [Poi]: Theorem 6.1.21. When n > 2 the Euclidean unit ball B and the unit polydisc P are not biholomorphically equivalent. Proof. Theorem 6.1.20 shows that there is no biholomorphism of these domains which takes 0 to 0. We may reduce to this case using the automorphisms of P or of B. (See Section 6.1.4.) Consequently, the Riemann mapping theorem fails in several complex variables. There is another generalization of the Schwarz lemma which will be useful [Haml] (see also [Dinl]): Lemma 6.1.22. Let f2 be a balanced pseudoconvex domain in C™ with the Minkowski function h. Suppose that f : fi —>• $7 is a holomorphic mapping and that /(O) = 0. Then h(f(z)) < h(z) for all z e f t . Proof. Let v e C1 be such that h(v) = 1 and consider the mapping g € H(U,£l) given by g(£) = f ( £ v ) . Then
6.1.4
Automorphisms of the Euclidean unit ball and the unit polydisc
Theorem 6.1.20 may be used to determine the biholomorphic automorphisms of the unit ball and the unit polydisc in C71. For more details, the reader may consult [Rudl-2]. To this end, let U denote the set of unitary transformations of C™ . If fi is a domain in C™ , let Aut(£2) =
196
Univalence in several complex variables
Let B be the unit ball of Cn with respect to the Euclidean norm. We have Theorem 6.1.23. Up to multiplication by a unitary transformation, Aut(.B) consists of mappings (6.1.9)
<]> a a(z
a
V I - (z,a)J
,
,
,
where TQ = I and for a ^ 0, Ta is the linear operator given by (6.1.10)
Ta(z) = ^^-a + saz,
zeC1,
and sa = ^/l - INI 2 -
(6.1.11) That is,
Aut(B) = lv<j)a : a € £, V e WJ = {<&>W : b € B, W e Proof. The fact that 0a is an automorphism of B follows from parts (ii) and (iii) of the following lemma. The fact that all automorphisms of B have the indicated form follows from Theorem 6.1.20. As we shall remark in the next section, Theorem 6.1.23 also gives the automorphisms of the unit ball in a complex Hilbert space. Some of the basic properties of the mappings in Aut(JB) are as follows (see [Rud2, p.25-30]): Lemma 6.1.24. Let a e B and let (f)a be given by (6.1.9). Then (i) 4>a(a} = 0.
(iii) O^rH*) = 4>-a(z), ztB. (iv) $a extends to a homeomorphism of B onto B . (v) J/-0 € Aut(£?) and a = ip~l(Q], then (6.1.12) Moreover, (6.1.13)
W 2 )
|
f =
s ln+1 __ ,
zeB.
6.1. Preliminaries
197
We note that Aut(5) acts transitively on J5, and thus B is a homogeneous domain in C™. In fact the action is transitive on directions at a given point. If the unit ball B is replaced by the unit polydisc P of Cn , then we have Theorem 6.1.25. Up to multiplication by a diagonal unitary transformation and a permutation of the coordinates, the set Aut(P) consists of mappings (6.1.14) ll>(Z) = $a(z) =
frM*l),
• • - ,^o n (Sn)), * = (*1, . . - , * „ ) G P,
where a = (ai, . . . , a n ) € P and ^aj(zj) = ,Zj~-aJ .
(6.1.15)
l<j
^j^j
That is, Aut(P) = 01, . . . , On 6 R, \a.j\ < 1, 1 < j < n, a is an arbitrary permutation of (1, . . . , n) >. Again we note that P is a homogeneous domain; however this time the action of Aut(P) is not transitive on directions at a given point.
6.1.5
Holomorphic mappings in complex Banach spaces
Let X be a complex Banach space with respect to the norm || • |j . Again we shall denote the open ball of radius r centered at ZQ e X by B(ZQ, r). Instead of .8(0, r) we shall write Br, and for the unit ball B\ we shall write simply B. As always, U denotes the open unit disc in C. The closure of the ball B(zQ^r) will be indicated by B(zQ,r) and its boundary by dB(zo,r). Now let X and Y be two complex Banach spaces with respect to the norms || • ||i and || • || 2 respectively. For simplicity, we denote both norms by || • ||, when there is no possibility of confusion. Let L(X, Y) denote the Banach space of all continuous complex-linear operators from X into Y with the standard operator norm ||:||z|| = l},
A€L(X,Y).
198
Uni valence in several complex variables
The identity mapping in L(X, X) will be denoted by /. Let ft be a domain in X and let / : ft —> Y be a mapping. Then / is called holomorphic on ft if for any z 6 ft there is a mapping Df(z] € I/(X, F), called the Prechet derivative of / at 2, such that hm -rr-Tj- = 0, .e.
Let H (ft, ft') be the set of holomorphic mappings from a domain ft C X into a domain ft' C F, and let #(ft) = ff(fi, X). If / e H(ft,y) and z e ft, then for each k = 1,2,..., there is a bounded symmetric ^-linear mapping k k D f(z] : TT X —> y , called the /et/l-order Frechet derivative of / at z such j=i that 00 (6.1.16)
/H
for all ty in some neighbourhood of z. If for some u & X the disc {z + Cw : |£| < r} is contained in ft then the expansion is valid for all w in this disc. If ft — z is balanced, the series converges uniformly to / in some neighbourhood of each compact subset of ft. If / is bounded on the ball B(z, r) C ft, then we have uniform convergence on B(z,s] for 0 < s < r (see [Muj, Theorems 7.11 and 7.13]). It is understood that for h€X, D°f(z}(h°) = f ( z ) and for k > 1, Dkf(z}(hk] =
Dkf(z)(h,h,...,h). k— times
Moreover, if / 6 H (ft, Y) and the closed segment [a, a + h] is contained in ft, then the Taylor formula with remainder (6.1.17)
/(a + h) = /(a) + Df(a)(h) + ... +
±Dkf(a}(hk)
6.1. Preliminaries
199
holds for all k e N. The terms in the expansion (6.1.16) can be expressed in terms of Cauchy's integral formula. We begin with the case of a holomorphic map from a domain in the complex plane into X. If D is a domain in C and 7 : [a, b] —> D is a piecewise Cl curve, and if q : D —> X is a continuous function, then the complex line integral of q along 7 is defined by
We then have the following Cauchy integral theorem and Cauchy integral formulas for mappings in H(D,X): Theorem 6.1.26. If D is a domain in C and g € H(D,X), then
I for each simple closed piecewise C1 contour 7 in D such that the interior of 7 is contained in D. If £Q e D and r > 0 are such that U((o,r) is contained in D, then
/
These formulas lead to the Cauchy estimates: ll<7 (fc) (Co)|| < Mr~kk\,
A; = 0,1,2,...,
whenever g is a holomorphic mapping from a domain D D U(^Q,r) into X such that ||0(0|| < M for C € 17(Co,r). Going back to the general case, if SI is a domain in X, z £ fi and w e X, the set
is an open subset of C, and the function fZjW : Q,z,w —>• F, given by /2,u;(C) = f ( z + £iy), is holomorphic on Slz,w. We may therefore apply the foregoing discussion to fZjW on each connected component of £lz>w.
200
Univalence in several complex variables Let r be sufficiently small that the Taylor series expansion
is valid in a neighbourhood of B(z,r). Then the term which is homogeneous of degree k may be expressed as (6.1.18)
k\
k ^Dkf(z}(h ' v JV '
2m J
C fc+1
for all h 6 X with \\h\\ < 1 and k = 0,1,2,.... Moreover, if \\f(w)\\ < M for w 6 B(z,r], then the above formulas lead to the analog of the classical Cauchy estimates: (6.1.19) \ \ D k f ( z ) ( h k ) \ \ < M(k\}r~k, whenever h 6 X with \\h\\ < 1 and k = 0,1,2,.... In fact when h is fixed, \\h\\ — 1, we may take r in (6.1.18) and (6.1.19) to be any positive number such that fz>h is holomorphic in a neighbourhood of Ur (adjusting M as needed). Cauchy's integral formula allows us to extend other results in complex function theory to infinite dimensions, for example the maximum modulus theorem: Theorem 6.1.27. Let D C X be a domain and f : D —> Y be a holomorphic mapping. Then \\f(z)\\ can have no maximum in D unless \\f(z)\\ is of constant value throughout D. We also have the following version of the Schwarz lemma (see [Dinl], [Harrl], [Hil-Phi], [Pra-Ve]): Lemma 6.1.28. Let M > 0 and f : B —> Y be a holomorphic mapping such that /(O) = 0 and \\f(z)\\ < M for z 6 B. Then \\f(z)\\ < M\\z\\, z € B. Further, if there is a point ZQ € B \ {0} such that ||/(^o)|| = -^ll z o||> then ||/(Czd)|| = M||C20||, for all C e C, |C| < l/||*d||. Moreover, if Df(Q) = 0 , . . . ,Dk~lf(0) = 0, k € N, k > 2, then \\f(z)\\ < M\\z\f for \\z\\ < 1 and ^Dkf(Q}(wk}\\
< M for \\w\\ = I.
Proof. Fix z € B \ {0} and define g(Q = /(C*)/C, |CI < Vlkll- Then g is a holomorphic mapping from |£| < I / \ \ z \ \ into Y (it is easy to see that
6.1. Preliminaries Dg(Q)(h) = l/2D2f(0)(hz,z)
201 for h € C). For 0 < r < l/\\z\\ and |C| = r,
we have ||0(C)|| < M/r, i.e. ||/(C«)/C|| < M/r. In view of Theorem 6.1.27, we deduce that ||/(C*)|| < Af|C|/r for |C| < r. We now let r /• l/p||, and this yields that \\f(C,z)\\ < M\C\\\z\\ for |C| < l/||z||. For C = 1, the conclusion follows. If ||/(zo)|| = -^Ikoll) then from Theorem 6.1.27 and the above reasoning, one deduces that ||/(Czo)/CII = MIN|, for all C, |CI < r, 0 < r < I/\\ZQ\\. Letting r /* l/||zo||, one obtains the claimed result. For the last statement, it suffices to use similar reasoning as above for the map MO = /(C*)/C fc , ICI < 1/NI, where z € B \ {0}. As in one complex variable, a mapping v £ H(B) is a Schwarz mapping if v(0) = 0and \\v(z)\\ < 1, z € B. If 0 e fi and / 6 H(ft), we say that / is normalized if /(O) = 0 and D f ( 0 ) = I. A mapping / € H($l,Y) is called locally biholomorphic on the domain fi if for each z € 0 there are neighbourhoods V of z and W of f ( z ) such that / is a one-to-one map of V onto W whose inverse is holomorphic on W. It is known that / is locally biholomorphic on f) if and only if the Frechet derivative Df(z] has a bounded inverse at each z 6 fi. If -X" = Y = Cn, the condition that D f ( z ) has a bounded inverse is just the condition that J f ( z ) ^ 0 for z 6 0. A mapping / 6 H(f), Y) is said to be biholomorphic on the domain 17 if /(fi) is a domain in Y and the inverse f~l exists and is holomorphic on /(fi). As in the finite dimensional case, let S(B) denote the set of normalized biholomorphic mappings from B into X. If X = C" and / is a univalent mapping from B into C71, then / is a biholomorphic mapping from B onto the domain f ( B ) in C". However, in the case of complex Banach spaces, this result is not necessarily true. Heath and Suffridge [Hea-Su] gave an example of a univalent (holomorphic and injective) mapping of the unit ball B of a complex Banach space such that /-1 is not holomorphic on f ( B ) , f ( B ) contains an open set, but f ( B ) is not open. Hence we must be careful to specify that we are studying biholomorphic maps rather than univalent maps. As observed by Franzoni and Vesentini [Fra-Ve], Cartan's theorem on fixed
202
Univalence in several complex variables
points (Theorem 6.1.19) may be generalized to complex Banach spaces. There is a generalization of Theorem 6.1.20 as well [Pra-Ve]: If fi is a bounded circular domain in a complex Banach space containing 0, and if / 6 Aut(Q) is such that /(O) = 0, then / is (the restriction of) a linear isomorphism of X. It follows that the automorphism group of the unit ball in a complex Hilbert space is given by Theorem 6.1.23 (with the appropriate notion of unitary transformation). Notes. The main sources that have been used in this section for holomorphic mappings in Banach spaces are [Pra-Ve], [Hil-Phi], [Muj], [Su3-4]. For further results, the reader may consult [Boc-Sic], [Din2], or [Herv].
6.1.6
Generalizations of functions with positive real part
We wish to present some basic results concerning a class of mappings in higher dimensions related to the Caratheodory class. The definition makes use of linear functionals. For further discussion see [Gur], [Su3-4], [Rop-Su2]. For the complex Banach space X, let X* denote the dual of X (i.e. X* = L(X, C)). For each z € X \ {0}, we define (6.1.20)
T(z) = [lz e X* : lz(z] = ||*||, ||iz|| = l},
where
By the Hahn-Banach theorem, T(z) is nonempty. This set plays a central role in the study of biholomorphic mappings of the unit ball in a complex Banach space. A case of particular interest is that of the p-norm in C1, i.e. 1/P
(6.1.21)
E
• .IP "
max \Zj\,
--
-oo
p = oo.
Let B(p) denote the unit ball with respect to || • ||p. When p — oo, B(p) reduces to the unit polydisc P of C71. It is known that if X = C71 with respect to a
6.1. Preliminaries
203
p-norm, 1 < p < oo, and lz e T(z), then
E WX/*,(6.1.22) (6.1.23)
lz(w) =
{J: j
* *°}
- for K p < o o ,
lz(w) =
for
=
where 7,- € C, |7j| < 1, for all j e {1,..., n}, (6.1.24)
' ' *
^
w
***
where each tj > 0 and
ij = 1. Of course, when p = 2, (6.1.22) reduces to lz(w) = (w, 7\—r,}* \\z\\ ' The following families play a key role in our discussion: A/-0 = {/ : B -> X : fe H(B), /(O) = 0, Re [«,(/(*))] > 0,
= / e A/*0 : Re
0, * € JB \ {0},
When X = C" with respect to a p-norm, the set M. is sometimes denoted by A4 (p), and consists of those normalized holomorphic mappings w : B(p) —>• C" such that for z = (*i, . . . , zn) e B(p] \ {0}, Re
(6.1.25)
oo
^- > 0,
Re
Re
Wj(z) Zi
> 0, pHoo = M > 0 (1 < j < n),
Again we note the special case p = 2, for which we have Re(w(z),z)>0,
z£
p = oo.
204
Univalence in several complex variables
In the case X = C, if / 6 A/b then Re |Xf(2)] > 0 on U, or equivalently Re [f(z)/z] > 0 on U. Thus, by the minimum principle for harmonic functions, either Re [f(z)/z] = 0 or Re [f(z)/z] > 0. Hence a function / has the property that / 6 A/b \ A/" if and only if f ( z ) = iaz for some real a. However, in dimensions greater than 1, the set A/b\A/" can be larger. The following example is due to SufTridge [Su4j. Example 6.1.29. Let X = C2 be the Euclidean space of two complex variables. Let w : B C C2 -> C2 be given by w(z) = (-22, z\). Then w 6 H(B), u;(0) = 0 and Re lzwz
= Re
Thus w 6 A/b \ A/". For X = C, the above characterization of A/b implies that if w G A/b and a € C, |a| < 1, then iu(az)/a e A/b- (It is understood that w(az)/a denotes the limiting value w'(Q)z when a = 0.) Moreover, w G A/" unless Re w/(0) = 0 and in this case w ( z ) / z = constant. In the case of normed spaces the analogous result is the following (see [Su3]): Lemma 6.1.30. If w G A/b and \a\ < 1 then w(az)/a € A/b (w(az)/a is understood to have the limiting value Dw(Q)(z] when a = 0). Further, if lz € T(z), 0 < ||;s|| < 1, tfienRe [lz(w(z)}} = 0 if and only if Re [lz(Dw(G)(z))] = 0, and m 0. Thus w(az)/a E A/b for |o;| < 1. Further, in view of the fact that Re [lz(w(az)/a)] is a nonnegative harmonic function of a for fixed z with Jo;) < 1/||*||, we conclude by the minimum principle that either Re [lz(w(az)/a)] > 0 or Re [lz(w(az}/a)} = 0. This completes the proof.
6.1. Preliminaries
205
The following lemma, due to Gurganus [Gur], will be applied in several situations in forthcoming chapters. Lemma 6.1.31. Let X be a complex Banach space and h € Ai. Then for allzeB\ {0} and lz e T(z], (6 L26)
*11 * * ['*(ft(2))1 *
'
«*»•
Proof. Let z 6 B \ {0} and lz € T(z) be fixed. Also let
Since /i(0) = 0, p is a holomorphic function in the unit disc 17, and since h € M. it is clear that Re p(£) > 0 on U. Indeed, since there is a one-to-one correspondence between T(az) and T(z) given by l a z(') = — ^z(')? for each a a € C \ {0}, one deduces for 0 < |£| < 1 that
Re
= Re i
Therefore Re p(^) > 0 on U, and p(0) = 1 from the normalization of h. From Theorem 2.1.3 we therefore deduce that
Substituting £ = |jz|| in the above, we obtain the relation (6.1.26), as desired. This completes the proof. We shall see in Theorem 6.1.39 that a stronger result holds, namely the set M. is compact when X = U1 with respect to an arbitrary norm. We note the following refinement of Lemma 6.1.31 (originally proved by Pfaltzgraff [Pfal]) in the case X = C71 with the Euclidean structure. Lemma 6.1.32. If h £ M then (6.1.27)
N !
2
4 < ^ (h(z),z) < N !
2
4
z 6 B.
206
Univalence in several complex variables
We next consider generalizations by Suffridge [Su3-4] of two theorems of Robertson [Robe3] to complex Banach spaces (see Theorems 2.1.7 and 2.1.8). In fact Lemmas 6.1.33 and 6.1.35 are slight modifications of Suffridge's results. Lemma 6.1.33. Let v : B x [0, 1] —>• B be a mapping such that v ( - , t ) is holomorphic on B for each fixed t € [0, 1], v(0, t) = 0 and v(z, 0) = z. Suppose that (6.1.28) exists and is holomorphic on B for some p > 0. Then w G A/bProof. Fix z e B \ {0} and lz e T(z). In view of (6.1.28), we deduce that lim v(z,t) = z. Prom Lemma 6.1.28 we obtain that ||v(z,£)|| < ||z||, and t-»o+ hence fz-v(z,t)\\_\\z\\--R*[lz(v(z,t))]
Using the continuity of lz and this relation, one deduces by letting t —> 0+ that Re [lz(w(z))} > 0, as desired. The following example of SurTridge [Su3] shows that there are situations in which the mapping w obtained in Lemma 6.1.33 belongs to A/b \ A/". Example 6.1.34. Let X — C2 be the Euclidean space of two complex variables z = (zi, 22)- Let v : B x [0, 1] —> B be given by V (z, t) = ( A/1 — t2Zi + tZ%, —tZi + V i — t2Z2 j .
Obviously v satisfies the requirements of the above lemma and lim
z-v(z,t) t
.
.
Hence w(z) = (—zz, z\] and from Example 6.1.29 we have w e A/o \ A/". Lemma 6.1.35. Let f : B C X —»• Y be a biholomorphic mapping of B onto a domain f ( B ] of Y such that /(O) = 0. Assume F : B x [0, 1] -> Y is holomorphic on B for each t G [0,1], F(z,0) = f ( z ) , F(0,t) = 0, and F(B, t) C f ( B ] for each t € [0, 1]. Moreover, let p > 0 6e such that
6.1. Preliminaries
207
exists and is holomorphic on B. Then g(z) = D f ( z ) w ( z ) on B, where w G NQ. Proof. Since F(B,t) C f ( B ) for t € [0,1], there is a mapping v : B x [0,1] ->• B such that F(z,t) = /(v(z,t)) for all z € B. Then v(z,t) = f~1(F(z,t)) is holomorphic on B for fixed t, v(Q,t) = 0 for t e [0,1] and v(z,0) = 2, 2 € -B. Prom the Schwarz lemma (Lemma 6.1.28) we obtain that IK*,*) II < 11*11 for z € B and t € [0,1]. On the other hand, from (6.1.29) we deduce that lim F(z, t) = f ( z ) and hence, lim v(z,t) = z. z € B. v t^0+ ^ ' ^ ' ' t^o+ Next, fix z G B \ {0}. Then expanding / in a Taylor series, we have f ( v ( z , t}) = f ( z ) + D f ( z ) ( v ( z , t) - z) + R(v(z, t), z), where ,, ™'z'^ -> 0 as lly - zll -> 0. Therefore for t G (0, 1], we obtain .L30) (6 V
^
tP
tP
p
We next prove that ||z — v(z,t)||/t is bounded as t —>• 0+. Assuming the contrary, we may suppose for a certain sequence {£m}meN? ^m —>• 0+, that —» co as m —» co.
Obviously, from (6.1.30) we have k
= lim
\
••-*/
/
\
\ n ii
\z —
\\z-v(z,tm}\\
On the other hand, since v(z, tm) —>• z and
„
,— .,, —t 0 as ra
||s-t;(3,*m)||
co, the preceding equality yields that lim
m—>-oo
(_£ !fe \ll*-f(*. ;
However, this relation is impossible since / is biholomorphic (thus D f ( z ) is invertible). Consequently, we deduce that \\z — v(z,t}\\/tp must be bounded as t —> 0+. Hence, taking into account the fact that the limit (6.1.29) exists and is holomorphic, using (6.1.30) and the invertibility of D f ( z ) , we conclude that the limit z-v(z,t) lim —- = w(z)
208
Univalence in several complex variables
exists and is holomorphic on B. It now follows from Lemma 6.1.33 that w € A/oThe equality g(z) — D f ( z ) w ( z ) is clear from (6.1.30), and this completes the proof. We conclude this section with an important result, which establishes the compactness of M. on the unit ball of C™ with an arbitrary norm || • ||. This result has recently been obtained in [Gra-Ham-Koh] (see also [Ham-KohlO] for the infinite dimensional case). In the next chapter we shall give another proof of this result, using coefficient estimates for mappings in .M. We need to introduce some notions about the numerical radius and numerical range of holomorphic functions in complex Banach spaces [Harr2] . Definition 6.1.36. (i) Let h : B —> X be a holomorphic mapping such that h has a continuous extension to B, again denoted by h. Let {lz(h(z}}:lz£T(z},\\z\\ z =
=
be the numerical range of h (taken with respect to T) . (ii) Let h : B —>• X be a holomorphic mapping. Define hs(z) = h(sz) for 0 < s < 1 and z e B. The numerical radius of h is the number
\V(h)\ = limsup ||A| : A € V(h,)\. - -
"•
J
Clearly if h has a uniformly continuous extension to B, then sup{|A| : A e V(h}}. Harris [Harr2] proved the following result, which provides an estimate of the norm of a homogeneous polynomial in terms of the numerical radius. (Abstract homogeneous polynomials are discussed in Section 7.2.3.) Lemma 6.1.37. Let Pm : X -> X be a continuous homogeneous polynomial of degree m > 1 . Then (6-1.31)
||Pm|| < km\V(Pm)\,
where ||Pm||=sup{||Pm(*)||: ||*|| 2, and ki = e. Further, let
L(h) = limsup JRe [lz(h,(z))] : lz € T(z), \\z\\ - ll, s-»l-
^
J
6.1. Preliminaries
209
where h : B —>• X is a holomorphic mapping. Using Lemma 6.1.37, Harris, Reich, and Shoikhet [Harr-Re-Sh] recently obtained an interesting distortion result for holomorphic mappings with restricted numerical range. Lemma 6.1.38. Let h : B -» X be a holomorphic mapping such that /i(0) = 0 and L(h) is finite. Then Q|U||2
(6.1.32)
\\h(z)-z\\< v-
1
*""z
(L(h)-l),
z£B.
\\ \\)
We are now able to prove the following theorem on the unit ball of C™ with an arbitrary norm (cf. [Ham-KohlO], [Gra-Ham-Koh]). In Chapter 7 we shall give another proof which uses Lemma 6.1.37 directly. Theorem 6.1.39. The set M. is compact. Proof. First we prove that M. is locally uniformly bounded on B. For this purpose, let h € Ai. In view of (6.1.26) we have Re [lz(h(tz}}\ < i±^t, J. — t
lz € T(z), \\z\\ = 1, 0 < t < 1.
Let Jit(z) = h(tz)/t for 0 < t < 1. Then it is obvious that Jit € M. Let L(ht) = limsup/Re [lz(ht(sz))} : lz G T(z), \\z\\ = ll. a-»l-
L
J
It is clear that
L(ht) < \^- < oo, -L
J/
0 < t < 1,
by the above arguments. Further, using the relation (6.1.32), one deduces that
Next, for any r with 0 < r < 1, let t be such that r < t < 1. Also let 5 = r/t. Then 0 < s < I and QS2
Of 2
- t z \ \ < - . - ,
\\z\\ <s.
Consequently, we have proved that for each r € (0, 1), there exists a constant M = M(r) > 0 such that \\h(z)\\ < M(r) for all h € M and z € Br, as claimed.
210
Univalence in several complex variables
We next prove that M. is closed. For this purpose, let {pfc}fceN De a sequence in M. such that p^ —>• p locally uniformly on B. Then p e H(B], p(0) — 0 and Dp(Q) = /, by the normalization of pk, k G N. Next, fix z 6 B \ {0} and lz € T(z). Since Re [^z(pfc(z))] > 0 for A; € N, we deduce by taking limits and using the continuity of lz that Re [/ z (p(z))j > 0. Since p is normalized, we must have Re [lz(p(z))] > 0, by Lemma 6.1.30. Thus p e M and this completes the proof. 6.1.7
Examples and counterexamples
There are many results in univalent function theory in one complex variable which cannot be extended (at least not without restrictions) to higher dimensions. We have already noted the failure of the Riemann mapping theorem. The analog of the class S on the unit ball in Cn is S(B) = {/ € H(B] : f is biholomorphic on B, /(O) = 0, Df(Q) = /}. It is easily seen that S(B) is not a normal family when the dimension is greater than one, and that there can be no growth or covering theorems or coefficient bounds for the full class. Indeed, if g is an arbitrary holomorphic function on the unit disc such that #(0) =
/(si, z2) = (zi, z2 + g ( z i ) ) ,
(zi, z2) e B,
belongs to S(B}. Examples of this type were given by H. Cartan [Cart2] in his appendix to Montel's book on univalent functions, published in 1933. Partly on the basis of such examples, Cartan conjectured that the Jacobian determinant of a mapping in S(B) should satisfy bounds depending only on ||z||. (We recall that |J/(^)|2 gives the infinitesimal magnification factor for the volume.) However, if h is an arbitrary holomorphic function on U such that h(Q) =0, the mapping (6.1.34)
f ( z ) = (zi, z
2
e ) ,
z = (Zl, z2) e 5,
6.1. Preliminaries
211
belongs to S(B) and since Jf(z) = eh(Z1\ Cartan's conjecture is false. (This example was given by Duren and Rudin [Dur-Rud]; the special cases
were considered by Barnard, FitzGerald, and Gong [Bar-Fit-Gon2].) Cartan also suggested that particular subclasses of S(B), such as the starlike and convex mappings, should be singled out for further study. Indeed, many of the results of univalent function theory do have extensions to higher dimensions for these classes. The failure of Bloch's theorem for normalized holomorphic maps / : B -> C™, n > 2, is a consequence of elementary examples of Harris [HarrS], Graham and Wu [Gra-Wu], and Duren and Rudin [Dur-Rud]. Graham and Wu showed that a normalized mapping / € H(B] such that «// = 1 could have an image of arbitrarily small volume. In particular, there can be no lower bound, independent of /, for the supremum of the radii of balls contained in the image of /•
Example 6.1.40. Let n = 2 and let Q be the square of side-length equal to 2 centered at the origin in C, i.e. Q = {z € C : (Re z\ < 1, |Im z\ < I } . Fix a positive integer k and let /: Q x Q —>• C2 be given by exp(27rfczi), 22 exp(-27rfczi) Then J f ( z i , z z ) = 1 and / is a covering map onto its image. Using the periodicity of the exponential function, it is easy to see that / is a 2fe-to-l map (except for a set of measure 0), and hence 1 8 volume(/(<5 x Q)) = — volume(<5 x Q) = -. ZK K
Restricting / to the unit ball of C2 , we see that by taking k large we can make volume(/(B)) arbitrarily small. The example of Harris [Harr3] and Duren and Rudin [Dur-Rud] shows that even the univalent Bloch theorem fails in C", n > 2. The example is a simple polynomial map of degree 2.
212
Univalence in several complex variables Example 6.1.41. If 8 > 0, then the map / : B C C2 ->• C2 given by f ( z ) = (zi,z2 + (zi/<5) 2 ),
z = (zi,z2) € B,
belongs to S(B), but f ( B ) contains no closed ball of radius 8. Proof. We show that there is no (1^1,102) S C2 such that f ( B ) contains the circle {(wi + 8ei0,w2) : -TT < 0 < TT}. To see this, fix (101,102) € C2. If (wi + X,w2) €. f ( B ] , then we must have 1
2
So if {(wi + 8ei9,w2) : -TT < 0 < TT} C f ( B ) , then the inequality \82w2-w2l-2w18ei0-82e2ie\<82 holds for all 0. However, the L2-norm of the function on the left-hand side of this inequality is oJ|2 + |2iwi<5|2 + <5 4 )] 1
> (27r)1/2<52,
and this is a contradiction. Other interesting examples and counterexamples are given in the recent monographs of Gong [Gon4-5]. Although it is too large a subject to include in this book, we mention that the theory of entire biholomorphic mappings is much different in higher dimensions then in one variable. In one variable an entire univalent function must be linear, and in particular any automorphism of C must be linear. Both statements are completely false in higher dimensions. If the functions g and h in (6.1.33) and (6.1.34) are entire, the resulting maps / give automorphisms of Cn. Moreover, in higher dimensions Cn can be biholomorphic to a proper subset of itself. This has been known since the 1920's when examples were given by Fatou and Bieberbach (see [Ros-Rud]), and there has been much study of Fatou-Bieberbach maps in recent years. Recent years have also seen many developments in the theory of univalent mappings of bounded domains, and it is to these that we now turn.
6.2. Criteria for starlikeness
6.2 6.2.1
213
Criteria for starlikeness Criteria for starlikeness on the unit ball in C™ or in a complex Banach space
In the following we are going to present some generalizations to higher dimensions of the well-known criterion for starlikeness on the unit disc (Theorem 2.2.2). We begin with the Euclidean unit ball in Cn, and then give generalizations to the unit ball with respect to an arbitrary norm in C" and the unit ball of a Banach space. We also consider bounded balanced pseudoconvex domains in Cn, and we briefly mention a class of domains introduced by Kikuchi. Finally we obtain starlikeness criteria for mappings of class C1 (which need not be holomorphic). Before going further let us define the notion of a starlike mapping, which will recur. Definition 6.2.1. Let X and Y be two complex Banach spaces and let fi be a domain in X. Also let / 6 H(£t,Y} and ZQ e fi. We say that / is starlike on 17 with respect to ZQ if / is biholomorphic on fi and /(^) is a starlike domain with respect to /(ZQ) (thus (1 — t)f(zo) + tf(z) € /($7) for all z € fi and t e [0,1]). We shall use the term starlike to mean starlike with respect to 0. That is, a starlike mapping / on a domain fi C X with 0 € f7 is a biholomorphic mapping from fi into Y such that /(O) = 0 and /(fJ) is a starlike domain in Y. If B is the unit ball of X, we let S*(B) denote the subset of S(B) consisting of normalized starlike mappings from B into X. We begin with the Euclidean unit ball in C1. In 1955 Matsuno [Mat] obtained (essentially) the following basic result: Theorem 6.2.2. Let /:#—>• C™ be a locally biholomorphic mapping such that /(O) =0. Then f is starlike if and only if (6.2.1)
Re ([Df(z)]-lf(z)t
z) > 0, z € B \ {0}.
Proof. We shall prove the necessity here; the proof of sufficiency will be given in a more general context (Theorem 6.2.5). Assuming that / is starlike, we may define a map v: B x [0,1] —>• B by v(z, t) = /-1((1 — t)f(z)). Then for fixed t, v is holomorphic in z and v(0, t) =
214
Univalence in several complex variables
0, so the Schwarz lemma gives ||i;(2,<)|| < \\z\\, 0 < t < 1, z € B. For fixed z we obtain 2 2
d
t
dt "
v
,
' '"
t=0
= -2Re
and this shows that Re ([Df(z)]~lf(z),z) > 0. To see that the inequality is strict, let g(z] — [D f (z)}~1 f (z) for z € B. Since g is normalized holomorphic on B and Re (g(z), z} > 0 for z e B\{0}, it follows from Lemma 6.1.30 that 5 e .M, and hence Re (g(z], z} > 0 on B\{0}. Therefore (6.2.1) is necessary for starlikeness. In 1970 Suffridge [Su2] considered the case of the unit polydisc P in C* and obtained the following necessary and sufficient condition for starlikeness in this context: Theorem 6.2.3. Let f : P —>• Cn be a locally biholomorphic mapping such that /(O) = 0. Then f is starlike if and only if (6.2.2)
f(z) = Df(z)w(z],
z € P,
where w 6 Ai. We recall that for the case of the maximum norm || • ||oo in C", the class M consists of those normalized holomorphic mappings w : P —» Cn that satisfy Re
l^)] > o,
11*1^ = \Zj\ > 0,
l<j
More generally, in the same paper [Su2] Suffridge obtained the following necessary and sufficient condition for starlikeness on the unit ball B(p) with respect to a p-norm |j • ||p, with 1 < p < oo. Theorem 6.2.4. Let f : B(p) —» C11, 1 < p < oo, be a locally biholomorphic mapping such that /(O) = 0. Then f is starlike if and only if (6.2.3)
/(z) = Df(z)w(z)
on B(p),
where w € M.(p). The extension of these results to the case of an arbitrary norm in C" was also carried out by Suffridge [Su3]. Gurganus [Gur] gave a different proof using
6.2. Criteria for starlikeness
215
a differential equations argument. We shall present a version of Suffridge's proof. Theorem 6.2.5. Let f : B —>• C™ be a locally biholomorphic mapping with /(O) =0. Then f is starlike if and only if there is w € M. such that (6.2.4)
f ( z ) = Df(z)w(z) on B.
Proof. Necessity. First assume that / is starlike. Let F(z, t) = z€ B, te [0, 1]. Since
t
(l-t)f(z),
.
we may apply Lemma 6.1.35 to conclude that there is a mapping w € A/b such that f ( z ) = Df(z)w(z) on B. This equality implies that Dw(0) = /, and in view of Lemma 6.1.30 we conclude that w € M, as desired. Sufficiency. Conversely, assume that / is locally biholomorphic on the unit ball B and there is a map w 6 M. such that equality (6.2.4) holds on B. We shall show that / is starlike. We first observe that f ( z ) ^ 0 if z ^ 0. Otherwise, f ( z ) = 0 would imply that Df(z)w(z} = 0, and hence w(z) = 0, using the fact that / is locally biholomorphic on B. On the other hand, since z ^ 0 and w € .M, we have Re [lz(w(z))] > 0 for lz € T(z). However, this is impossible if w(z) = 0. First step. For z e B \ {0}, let V be a neighbourhood of z on which / is biholomorphic, and let /-1 denote the corresponding inverse of / on f ( V ) . Also let v(z, t) = f~1((l — t)f(z}) for \t\ < e, where e is a positive number such that (1 - t)f(z) e f ( V ) for \t\ < e. We shall show that ||u(^,t)|| is a strictly decreasing function of t for \t\ sufficiently small. First we observe that v(z, t) = v(z, 0) 4- [ D f ( z ) ] ' l ( - t f ( z ) )
+ o(t) = z- tw(z] + o(t],
and hence for t < 0 with \t\ sufficiently small and lz € T(z), we obtain IK*,*)!! > Re [lz(v(z,t))] = \\z\\ - tRe [lz(w(z))} + o(t] > \\z\\.
216
Univalence in several complex variables
Moreover, applying this argument to y — v(z,t) (in this case we consider v(y,s) = f~l((l — s}(l — t}f(z))}, we deduce that ||u(z,£)|| is strictly decreasing for \t\ sufficiently small, as claimed. Second step. Next, for each r € [0, 1] consider the line segment
Let T =
0
We wish to argue that 7(2) converges to a point ZQ £ B as t —> TQ . But this follows from a Cauchy sequence argument using the fact that ||7(t)|| is a decreasing function of t and 7 has finite arc length. We note that the length of 7 is given by the (improper) integral
This is finite because |J7(£)|| < \\z\\ for t € [0, TO) and for each r e (0,1) there exists a positive number M = M(r) such that \\[Df(z)]-l\\<M(r),
\\z\\
Is is now clear that f~l can be analytically continued along LTQ and /-1((1 — TQ)/(Z)) = ZQ. Hence T is closed, as asserted. Third step. Finally we show that / is univalent. If we assume that this is not true, then there are distinct points z,z' € B such that f ( z ) — f ( z ' ) . Now /-1 can be analytically continued along the segment L = {(1 — t)f(z) : 0 < t < 1} with either f ~ l ( f ( z ) ) = z or f ~ l ( f ( z ) ) = z' as the initial value. But these analytic continuations must coincide in a neighbourhood of 0 € Cn,
6.2. Criteria for starlikeness
217
because /(£) = 0 if and only if £ = 0. Hence they must coincide everywhere along L and z = z' . Since this is a contradiction, the proof is complete. Suffridge [Su3] observed that his proof remained valid for the unit ball of a complex Banach space if one explicitly added the assumption that for each r, 0 < r < 1, there exists M = M(r] > 0 such that ||[D/(2)]~1|| < M(r) when || z ||
/(z) = Df(z)w(z),
z € B,
where w e M. .
6.2.2
Starlikeness criteria on more general domains in C"
It is natural to study starlike mappings of domains which are somewhat more general than the unit ball with respect to some norm. The first person to do so was Kikuchi [Kik] in 1973. His assumptions on the domain are formulated in terms of the Bergman kernel function, which is defined for example in [Ber], [Kran], [Ran]. Theorem 6.2.7. Let ft be a bounded pseudoconvex domain in C™ for which the Bergman kernel function Kfi(z, z) assumes its minimum value only at z = 0, tends to oo at all boundary points, and satisfies Kn(g(z),g(z)) < Kn(z,z) for any holomorphic mapping g of ft into ft such that g(0) = 0. A locally biholomorphic mapping f : ft —>• C™ such that /(O) = 0 is starlike if and only if
Re <W*)]-V(z),
(*, *)
> 0, z 6 ft \ {0}.
However, in practice it may be difficult to check whether a given domain satisfies Kikuchi's assumptions. Starlike mappings on Reinhardt domains were studied by Gong, Wang, and Yu [Gon-Wa-Yu2]. However, perhaps the most natural class of domains
218
Univalence in several complex variables
to consider in this context is the class of bounded balanced pseudoconvex domains whose Minkowski function is of class Cl in C71 \ {0}. The following result is due independently to Liu and Ren [Liu-Ren2] (necessity) and Gong, Wang, and Yu [Gon-Wa-Yu5] (sufficiency), and to Hamada [Ham4]. Other proofs appear in [Gon4], [Ham-Koh3], and [Ham-Koh-Licl]. Theorem 6.2.8. Let ft be a bounded balanced pseudoconvex domain in C71 whose Minkowski function h is of class Cl in C1 \ {0}. Let f : ft —> C™ be a locally biholomorphic mapping such that /(O) = 0. Then f is starlike if and only if (6.2.6)
Re ( [ D f ( z ) ] - l f ( z ) , \
~(z}\ > 0, oz /
z € ft \ {0}.
Proof. Assume that / is starlike. Since (1 — t)f(z) 6 /(ft) for all z e ft and t € [0,1], we may define a mapping v : ft x [0,1] —*• ft by v(z,t] = f-l((l-t)f(z)). Then v(-,i) is holomorphic on ft, v(0, t) = 0 for each t <E [0,1], and from Lemma 6.1.22 one concludes that h(v(z,t}} < h(z) for z € ft and t G [0,1]. Fixing z € ft \ {0}, we therefore obtain
*(„(,. *))-*(,) =d
t)
~
= -2Re
making use of the chain rule. At this point we have shown that Re l[Df(z}]-lf(z\ \
^(z}\ > 0, oz i
z G ft \ {0}.
Now since h?(£w) = \^\'2h2(w) for C € C and w 6 C1, it follows that C/*2H,
C € C \ {0},
and setting £ = 1 in the above, we have
W
e C" \ {0},
6.2. Criteria for staxlikeness
219
For fixed w <E dft, let
where g(z) = [Df(z}\ l f ( z ) , z e ft. Since p is holomorphic on ft,
(Cti;),
(<»
> 0,
0 < |C| < 1.
Since qw(Q) = 1 and Re qw(C) > 0 on C7, we must have Re qw(£) > 0 on 17, by the minimum principle for harmonic functions. Consequently, we have proved that Re
g(Cw),
(C«0\ > 0,
0 < |C| < 1.
The desired conclusion (6.2.6) follows, completing the proof of necessity. The proof of sufficiency is similar to that in Theorem 6.2.5 and we leave it for the reader. (The argument is sketched again in Theorem 6.2.10.) Further criteria for starlikeness in several complex variables may be found in the recent monograph of Gong [Gon4].
6.2.3
Sufficient conditions for starlikeness for mappings of class Cl
We next indicate how the idea of starlikeness may be extended to the case of mappings of class C1 (which are not necessarily holomorphic). In this context we define a starlike map as follows. Definition 6.2.9. Let ft be a domain in C1 such that 0 6 ft and let Cl(ft) denote the set of mappings of class C1 from fl into Cn . If / € C1 (ft) we say that / is starlike if / is injective on ft, /(O) = 0 and /(ft) is a starlike domain in C71 (with respect to zero). For a C1 mapping / from a domain ft in C" into C" , let ./,/(«) -d
,yi,...,xn,yn)
Univalence in several complex variables
220
denote the determinant of the real Jacobian matrix of / at 2, where Zj — Xj + iyj and f j = U j + i v j , j = l,...,n. When / is holomorphic on fi we have
jrf(z) = \Jf(z)\2, z e n.
Let / 6 C71(f2) and suppose Jrf(z) 7^ 0, z e fJ. Then for each z € Ct there exists a neighbourhood Vz d Q, of z such that / is a diffeomorphism of class C1 from Vz onto f ( V z ) . Therefore we can define the matrices
Dwf~l(w) =
dwk
•H
dwk
•H
and
where w = f(z). Recently Kohr [Koh4] obtained a sufficient condition for starlikeness for mappings of class C1 on domains for which the Bergman kernel function satisfies Kikuchi's conditions. Also Hamada and Kohr [Ham-Koh3] obtained the following result on bounded balanced pseudoconvex domains for which the Minkowski function is C1 on Cn \ {0}: Theorem 6.2.10. Let 17 be a bounded balanced pseudoconvex domain in Cn whose Minkowski function h is of class C1 in Cn \ {0}. Let f 6 Cl(ty be such that /(O) = 0. // Jrf(z) ^ 0, z € $1, and if
(6.2.7) Re
[Dwf-l(f(z))}f(z}
+
>0
on fJ \ {0}, then f is starlike. Proof. We shall give only a sketch of the proof, since it is similar to the proof of sufficiency in Theorem 6.2.5. Since Jrf(z) ^ 0 on 0, / is a local diffeomorphism . As in the first step in Theorem 6.2.5, we may define v(z,t) = f ~ 1 ( ( l — t)f(z)) for z e ft \ {0} and |t| sufficiently small. Using (6.2.7) and the chain rule, we deduce that h2(v(z,t}) is strictly decreasing for \t\ sufficiently small. The second and third steps may also be carried out as before. We are no longer dealing with analytic continuation, but we may still speak of continuing
6.2. Criteria for starlikeness
221
f~l along a line segment since / is a local diffeomorphism. This completes the proof. Of course if / e H(fi), then Theorem 6.2.10 reduces to the sufficiency condition in Theorem 6.2.8. Further, if ft is the unit ball B(p) with respect to a p-norm || • ||p, where 1 < p < oo, then h(z) = \\z\\p, z € C", and we obtain the sufficient condition for starlikeness in Theorem 6.2.4. On the other hand, if n = 1 and 1) is the unit disc, then Theorem 6.2.10 gives the following result of Mocanu [Moc2]: Corollary 6.2.11. Let f 6 Cl(U) satisfy the conditions (i) /(Q) = 0 and f(z) ^ 0, z e U \ {0}, (ii) Jrf(z) >Q,ze U,
(in) Re
/(*)
Then f is starlike. Proof. For all £ e U there exists a neighbourhood V$ C U of £ such that : / ^C ~^ /(^c) ig a diffeomorphism of class C1. Let g(w) — f ~ l ( w ] on After simple computations, one deduces the relations
_ dw
_ Jrf
_ dz
_ dw
Jrf
&z'
The result now follows from Theorem 6.2.10, making use of the fact that h(z) = \z\ for all z € C.
6.2.4
Starlikeness of order 7 in Cn
In Chapter 2 we considered functions in S which are starlike of a given order, and proved the Marx-Strohhacker theorem which states that a normalized convex function is starlike of order 1/2. This result can be generalized to several complex variables [Cu2], [Koh2] (also see [Ham-Koh-Lic2]), as we shall see in Section 6.3.3. In this section we shall introduce the necessary definitions. Let B be the unit ball of C" with respect to an arbitrary norm.
222
Univalence in several complex variables
Definition 6.2.12. Let /: B —>• Cn be a locally biholomorphic mapping such that /(O) = 0, and let 7 e (0,1). We say that / is starlike of order 7 if 1
Equivalently we may express this as
or as
for all a e U \ {0}, u <E C n , ||u|| = 1, and lu € T(u). It is easy to see that in one variable the condition (6.2.8) reduces to
Re <
> > 7, that is the usual condition for starlikeness of order 7 in
the unit disc U. Notes. There are now many references for starlike mappings in higher dimensions. For more information see the recent monographs [Gon4], [KohLic2]. Some of the original papers are [Bar-Fit-Gonl], [Gon-Wa-Yul,2,4,5], [Gra-Kohl], [Gra-Koh-Koh], [Gur], [Ham4], [Ham-Kohl,3], [Kik], [Kohl,4,7], [Liu-Ren2], [Pfa-Su3], [Por5], [Rop-Su2], [Su2,3,4], [Chu2], [Kub-Por]. Problems 6.2.1. Let 1 < p < oo and let / : B(p) C C2 -»• C2 be given by f ( z ) = (zi + azz, z%), z = (z\,z<2) 6 B(p). Show that / is starlike if and only if
(Suffridge, 1975 [Su4], Roper and Suffridge, 1999 [Rop-Su2].) 6.2.2. Let 1 < p < oo and / : B(p) C C2 -> C2 be given by f ( z ) = (z\ + aziZ2,Z2), z = (21,^2) € B(p). Show that / is starlike if and only if M < 1. (Roper and Suffridge, 1999 [Rop-Su2].)
6.3. Criteria for convexity
223
6.2.3. Let A(*) = ( 71—rr\5» 7!—T^2 )
and
/2(2) =
for 2: = (zi, 22) e J5. Show that / = (/i + /2)/2 is not starlike on the Euclidean unit ball of C2. (In fact / does not belong to S(B).) 6.2.4. Let X be a complex Hilbert space with the inner product (•, •} and the induced norm || • ||. Let / be a normalized starlike function on the unit disc U. Also let ZQ € X with ||;zo|| = 1 and let F : B —>• X be given by
Show that F is starlike. (Suffridge, 1975 [Su4].) Hint. Apply Theorem 6.2.6. 6.2.5. Let /i, . . . , fn be normalized starlike functions on the unit disc U. Let 1 < p < oo and / : B(p) -XC* be given by f ( z ) = (fiM, . . . , fn(zn)), z = (21, . . . , zn) € B(p). Show that / is starlike. 6.2.6. Let PJ > 1, j = 1, . . . , n, and let B(pi, . . . ,p n ) C C" be the domain
Show that if / : B(pi, . . . ,pn) ->• C" is given by Zl
Z
then / is starlike. (Hamada, 2000 [Ham4].)
6.3
Criteria for convexity
6.3.1
Criteria for convexity on the unit polydisc and the Euclidean unit ball
The present section deals with the study of certain necessary and sufficient conditions for convexity on the Euclidean unit ball and unit polydisc of C" . In
224
Uni valence in several complex variables
Section 6.3.2 we shall study the notion of convexity on the unit ball of complex Hilbert and Banach spaces. We begin with the definition of a convex mapping from a domain in a complex Banach space X into another complex Banach space Y. Definition 6.3.1. Let X and Y be two complex Banach spaces and £2 C X be a domain. Also let / : 17 —> Y be a holomorphic mapping. We say that / is convex on fl! if / is biholomorphic on fi and /(fi) is a convex domain in Y (thus (1 - t)f(z) + tf(w) 6 /(ft) for z,w € ft and t € [0, 1]). If B is the unit ball of X, we let K(B] denote the subset of S(B) consisting of normalized convex mappings from B into X. In 1970 Suffridge [Su2] obtained a remarkable structure theorem for convex maps of the unit polydisc P of Cn . (Thus we consider Cn with the maximum norm || • ||oo-) This result gives a complete characterization of convexity in the polydisc. Theorem 6.3.2. Let f : P —>• Cn be a locally biholomorphic mapping such that /(O) = 0. Then f is convex if and only if there exist convex functions (pj, 1 < j' 5: n> on the unit disc U such that (6.3.1)
f(z) = M((p!(zi), . . . , (pn(Znl),
Z = (zi, . . . , Zn) 6 P,
where M £ L(C n ,C n ) is a nonsingular linear transformation. Proof. It is obvious that if / satisfies equality (6.3.1) with y i , . . . , y ? n convex functions on U, then / is convex. Therefore, we have only to show the converse. Assume that / = (/i, . . . , fn) is a convex mapping of P. We shall show d2f that ——m— = 0 on P. for rn, fc, / — 1, . . . , n, k ^ 1. For Aj > 0, 1 < j < n, let ozkdzi At(z) = (ZleiAlt, Z2eiA*t, . . . , zneiAnt),
F(z,t) = \ [ f ( A t ( z } } + f ( A - t ( z ) ) ] t
-!<*
z£P,
0 < t < 1,
then F(-,t) G F(P), F(0,t) = 0, t > 0, and F(z,0) = /(z), z e P. Moreover, since / is convex, F(P,t) C /(P). Straightforward computations yield that
6.3. Criteria for convexity
225
the limit
't} = g(z) = Wz),...,<M(Z)),
t*
z
e P,
exists, where (6.3.2)
2, n k-1
Since this limit is holomorphic on P, it follows from Lemma 6.1.35 (for p = 2) that there is a mapping w G A/b such that g(z) = Df(z)w(z),
zeP.
Let Jf(z) - detDf(z), z € P. Then w,- = --- where D& is obtained Jf from Df by replacing the jth column by g written as a column vector. Now for fixed k, I < k < n, let A^ = 1 and A/ = 0, / ^ fe, 1 < / < n. For z € P such that ||^||cx) = \%j\ > 0, j ^ k and 2^ = 0, we have g(z) = 0 and hence Wj(z) = 0. Since Re [wj/Zj] > 0 for ||z||oo = \Zj\ > 0 and Wj/Zj assumes the value 0, we deduce by the minimum principle for harmonic functions that Wj = 0. Consequently, we conclude for 1 < j < n and 1 < k < n that (6.3.3)
dzj.
dzk
ozk
(replacing Wk by hk), where Re [hk(z)/Zk] > 0 for H^Hoo = \zk\ > 0. With k as before, fix /, 1 < / < n, I ^ k, and let Ak = 1, A/ = e > 0 and Am = 0, 1 < m < n, m 7^ fc, /. In view of (6.3.3) we deduce that there is a (different) mapping w £ A/b such that
where Gj is obtained from J/ by replacing the jth column by the column , 1 < m < H. Letting e —^ 0 we see that Re
L 2;j«'
> 0 for H*!^ = \Zj\ > 0,
226
Univalence in several complex variables
and since Re -—- = 0 for ZkZi = 0, we deduce that Gj = 0 for j ^ k. In fact this relation is true for j = k also, for we may interchange the roles of I and k without affecting the value of Gj . On the other hand, because the system of equations
V^ ®fm i d f / TT— ^7 = ~ nm i "
1< m < n ~ ~
has the solution -^ = 0, 1 < j < n, J f
we deduce that
d2fJm
= 0,
1 < m < n.
Solving this equation, we obtain (6.3.4)
fm(z)
where (pjm is a holomorphic function on the unit disc U. Substituting (6.3.4) into (6.3.3), we obtain + ckmZkV'km = ckmh>kv'krm fc, m = 1, . . . , n.
We may assume that / is normalized since this may be achieved by multiplying / by a nonsingular matrix. Then we may take Cmm = 1 and ^m(0) = 1, and we obtain z
m(f>mm + zvn(Pmm ~ hmfmm-
Using the fact that ip'mm ^ 0 and Re (hm(z)/zm) > 0 when ||z||oo = \zm\ > 0, it is not hard to deduce from this equation that hm is a function only of zm and that (f>mm is convex. It remains to show that
6.3. Criteria for convexity
227
and since we may take y>fcm(0) = 0> ¥>fcm differs from ip^k at most by a multiplicative constant, which we may absorb into Cfcm. We remark that Liu and Ren [Liu-Ren3] obtained an extension of Suffridge's theorem to the case of products of bounded convex circular domains in C". We only state their result and we leave the proof for the reader. Theorem 6.3.3. Let J7i c C711 , . . . , ft* C C"fc be bounded convex circular domains, whose Minkowski functions /ii,...,/ifc are real-analytic (except for a lower dimensional set). Let Z^ = (z^,...,^) denote the coordinates in C1', j = !,...,&. Suppose that f : QI x • • • x fifc ->• Cni+'"+nfe is a locally biholomorphic mapping such that /(O, . . . , 0) = 0. Then f is a convex mapping if and only if there exist convex mappings
1 - Re ([Df(z}rlD*f(z}(v,
v\ z) > 0,
for all z € B and v G C1 with \\v\\ = 1 and Re (z, v} = 0. Proof. First assume that / is a convex mapping. We need to make use of the fact that in this case f(Br) is a convex domain for any r 6 (0, 1). This we
228
Univalence in several complex variables
shall prove more generally for convex maps of the unit ball of a Banach space (Lemma 6.3.7). We now use an argument of Hamada and Kohr [Ham-KohlO]. Let z e 5\{0} and v € C^O} be such that Re ( z , v ) = 0. Also let r = \\z\\ and define
= (r\f(*) + tDf(z)v), r\fw + for t in a neighbourhood of zero. Then #(0) = \\z\\2 = r2 and (/(O) = 2Re ( z , v ) = 0. If g"(Q) < 0, then we would have g(t) < 0(0) = r2 for t ^ 0 near zero, and f ( z ) + t D f ( z ) v € f(Br) for nonzero t near 0. However, since f\Br is convex, this would imply that f ( z ) e f(Br) which is false. Therefore, we must have g"(0) > 0. This gives
0"(0) = 2|H|2 + 2Re ( D 2 f - \ f ( z } } (D f (z)v , D f (z)v) , z) = 2|H2 - 2Re ([Df(z)]-lD2f(z)(v,
v ) , z } > 0,
which is equivalent to (6.3.5) except that we have not shown that the inequality is strict. To obtain strictness of the inequality, we argue as follows: Since Re ( z , v } = 0, one can replace z ^ 0 by aZ and v by aV7|a:| with 0 < |a| < 1, \\Z\\ = \\V\\ = 1 and Re (Z,V} = 0, to obtain l-Re((Df(z)]-lD2f(z)(v,v),z) =
l-Re[a([Df(aZ)]-lD2f(aZ)(V,V),Z)}.
Since Re $a([Df(aZ)]-lD2f(aZ)(V, V), Z)\ is a harmonic function of a, one concludes using the minimum principle for harmonic functions that the inequality (6.3.5) is strictly satisfied for all z € £\{0} and v e C1, ||v|| = 1 such that Re (2,1;) = 0. Before proving the converse, we make the following additional remark on the proof of necessity: Letting Sr = dBr for 0 < r < 1, we note that f ( S r ) is a hypersurface with defining function ^{w} = ||/-1(^)||2-r2, i.e. f ( S r ) = {w 6 f ( B ) : i>(w) = 0}. Moreover, if z G Sr and v varies over unit vectors which are tangent to Sr at
6.3. Criteria for convexity
229
z, i.e. such that Re (z,v) = 0, then Df(z)v varies over all directions in the (real) tangent space to the hypersurface /(5r). Hence the condition #"(0) > 0, for all such v, says precisely that the (real) Hessian of ip is positive definite on the real tangent space to f ( S T ) at /(z). Equivalently, the second fundamental form of f ( S r ) is positive definite. In other words, f(Sr) is a strongly convex hypersurface. We now prove that the condition (6.3.5) is also sufficient for convexity. To this end, we shall use an argument similar to that in [Gon-Wa-Yu3]. First, we show that if / is biholomorphic on Br with 0 < r < 1, then f(Br] is a convex domain. Without loss of generality, we may assume that /(O) = 0. Since / is biholomorphic on Br, /(S^) is a real hypersurface for any p, with 0 < /z < r. Reversing the argument in the first part of the proof, we conclude from (6.3.5) that for any z € 5M and v € C1, \\v\\ = 1, satisfying Re (z, v} = 0, we have 0. Hence /(5M) is a strongly convex hypersurface for any /z, 0 < /j, < r. Therefore f(Br) is a convex domain (see [Kran, Propositions 3.1.6 and 3.1.7]). Next we show that if / is biholomorphic on Br then / is injective on Br. It is elementary to show that f(Br) is a convex set, in particular a starlike set. If / is not injective on Br, then we can find distinct points z,z' G Br and w E f(Br) such that /(z) = f ( z ' ) = w. Since / is univalent on Br and is locally biholomorphic on B, it follows that w ^ 0. The argument in the third step of the proof of Theorem 6.2.5 now shows that z = z', contrary to the assumption. Finally we consider the set K = < r e (0,1] : /is biholomorphic on Br >. We shall show that K = (0,1]. It is clear that K is nonempty, and it is elementary to see that K is closed in (0,1]. The proof will be complete if we can show that 72. is open in (0,1]. If 72. is not open, then there exists r € 72., a sequence {ep}peN5 £P > 0> lim £p = 0, and two sequences {zp}peN and {z'p}p^ such that zp, z'p 6 Br+£p, zp ^ z'p, and f ( z p ) = f(z'p}, for all p = 1,2,.... Since {zp}p& and {z'p}p& are bounded sequences, there exist subsequences {^Pfc}fceN> {zpfe}fceN of {^
230
Univalence in several complex variables
and {Zp}PeN5 which converge to z and z' respectively. It is clear that z, z1 € Br and that f ( z ) = f ( z ' ) . If z ^ z', we obtain a contradiction to the fact that / is injective on Br. However if z = z' we have a contradiction to the assumption that / is locally biholomorphic on B. Hence we must have K = (0,1] as claimed, and / is biholomorphic on B. We remark that in the case n = I the condition (6.3.5) reduces to the usual condition for convexity on the unit disc U, given in Theorem 2.2.3. Indeed, let z € U and v G C with |i>| = 1 and Re [zv] = 0. Then ~zv + zv = 0 and substituting from this equation in (6.3.5), we obtain 0 < 1 - Re
i.e.
6.3.2
Necessary and sufficient conditions for convexity in complex Banach spaces
We next obtain some necessary and sufficient conditions for convexity on the unit ball of a complex Banach space. First we note a condition considered by Suffridge [Su3] and Roper and Suffridge [Rop-Su2], which is necessary but not sufficient for convexity. (On the Euclidean unit ball in Cn it says that the second fundamental form of f(dBr) at f ( z ) is positive in the direction of Df(z)(iz}.) Theorem 6.3.5. Let X and Y be two complex Banach spaces and let B denote the unit ball of X. If f : B —>• Y is convex then (6.3.6)
D 2 f ( z ) ( z , z) + D f ( z ) z = D f ( z ) w ( z ) ,
z£B,
where w € .M. Proof. Without loss of generality, we may assume that /(O) = 0. Let F(z,t) = ±[f(e*z)+f(e-itz)],
z & B,
te[0,l].
6.3. Criteria for convexity
231
Then F(-,t) is holomorphic on B, F(z,0) = /(z), F(0,t) = 0, and F(z,t) 6 f ( B ) for z € B, t € [0, 1]. Moreover, two applications of L'Hopital's rule yield
/M-
[/(<*"*) + /(«-"*)]
Hence F(z,t) satisfies the hypotheses of Lemma 6.1.35 with p = 2, and (6.3.6) follows with w € M>- It is clear from (6.3.6) that Dw(0) = I, and Lemma 6.1.30 yields that w € M.. This completes the proof. An elementary example of Suffridge [Su3] shows that the condition (6.3.6) is not sufficient for convexity in higher dimensions: Example 6.3.6. Let C2 be the space of two complex variables with the maximum norm || • \00, and let / : P C C2 -» C2 be given by /(*) = (zi + 22/2,22),
z = (21,22) € P.
Then / satisfies the relation (6.3.6). However / is not convex on P, because / is not of the form (6.3.1). A basic result about convex maps is the following lemma due to Suffridge [Su3]. The notations are the same as in Theorem 6.3.5. Lemma 6.3.7. /// : B —> Y is a convex mapping then f(Br) is a convex domain, for all r e (0, 1). Proof. The space X x X is a Banach space with respect to the norm ||(2,z')ll = max{||*||,||;s/||}. The unit ball of this space is B x B. For fixed t € [0, 1] we consider the map B given by
Then
232
Univalence in several complex variables
we do not know whether the assumption (6.3.7) (which is automatically satisfied in the finite dimensional case) and the assumption that f be biholomorphic are essential. Theorem 6.3.8. (i) Let f : B —>• Y be a convex mapping. Then \\v\f - Re ( ( D f ( z ) } - l D 2 f ( z ) ( v , v ) , z )
>0
for z e B, v € X \ {0} with Re (z, v) = 0. (ii) Let f : B -4 Y be a biholomorphic mapping on B. Assume that for each r e (0,1), there exists a constant M = M(r] such that (6.3.7)
\\(Df(z)}-1(f(z)
- /H)ll < M(r),
z,w e Br.
If
\\v\\2 - Re ([Df(z)\-lD*f(z)(wM
>0
for z E B and v € X \ {0} with Re (z, v} = 0, then f is convex. Proof. The argument in the first part of the proof of Theorem 6.3.4 proves the first statement, and indeed was originally given in the context of complex Hilbert spaces in [Ham-KohlO]. To prove the second statement, let r G (0,1) be given and let Or = f(Br). It suffices to prove that Qr is convex. For this purpose, let S = {(P,Q) € Sir x O- : [P,Q] C O-}, where [P, Q] denotes the straight line segment joining P and Q. It is obvious that S is open and nonempty. Since fJ r is connected, the proof will be complete if we can show that S is a closed subset of fir x fi r . Now if we suppose that S is not closed, then we can find points P, Q € Or and P^, Qv € fir such that PV —>• P, Qv —> Q, [PjyjQ,,] C fir and [P, Q] is not contained in ftr. In this case there exists a point w = f(z] € d£lr fl [P, Q]. Indeed, let t0 = sup {r € [0, !]:(!- t)P + tQ e fiP) 0 < t < r}. Since P, Q e f2 r , to > 0 and (1 - t)P + iQ e fir for 0 < t < t0. Let
iQ),
0 < t < tQ.
6.3. Criteria for convexity
233
Then \\v(t)\\ < r for 0 < t < t0. Since = [Df(vm-\Q - P) = -t(Df(v(t))}-\'f(v(t)}
- P),
it follows that ||dv/dt|| is integrable on [to/2, to) by (6.3.7). Let to/2 < si < «2 < to. Then 82
dv(t) dt
\dt<M(r)log(s2/si).
Let {Tfc}fceN De an increasing sequence such that Tk > 0 and Tk —>• toA Cauchy sequence argument shows that there is a point z 6 X such that vfrk) —>• z as fc —» oo. Moreover, since ||v(7fc)|| < r we have ||z|| < r, and by the continuity of /, we must have f ( z ) = (1 — to)P + toQ- We now see that 1 1 z 1 1 = r, for otherwise we obtain a contradiction with the definition of toLet for t in a neighbourhood of 0. Then
= 2|H|2 - 2Re ([Df(z)}-lD2f(z)(v,v),
z} > 0,
which implies that g(t] > r 2 for t ^ 0 near 0. However, this is a contradiction with g(t) < r 2 . Thus S must be closed, as claimed. This completes the proof. (If we assume only that / is locally biholomorphic, the conclusion is that /(B) is a convex domain.) In the finite-dimensional situation, we have the following result of Hamada and Kohr [Ham-Koh7]: Theorem 6.3.9. Let B be the unit ball o/C" with respect to a norm which is of class C2 in C" \ {0}. Denote this norm by h. Suppose that f : B —>• C71 is a locally biholomorphic mapping. Then f is convex if and only if
234
Univalence in several complex variables
Re
for allz£B\ {0} and v e C1 \ {0} wtifc Re -(z) = 0. We leave the proof for the reader. Note that the quantity on the left-hand side of this inequality is just the second fundamental form of the hypersurface dBr (up to a scalar factor), where r = \\z\\. We now consider a two-point condition for convexity, analogous to (2.2.3), obtained by Suffridge [Su3,4] for the unit ball of a complex Banach space. Theorem 6.3.10. Let f : B —>• Y be a holomorphic mapping. (i) If f is convex then f(x) - f(y) = Df(x)w(x, y},
x,y£B,
where Re {lx(w(x,y})} > 0 for all x,y € B, \\y\\ < \\x\\ and lx € T(x). (ii) Assume f is locally biholomorphic and f ( x ) - f ( y ) = Df(x)w(x,y),
x,y&B,
where Re {lx(w(x,y))} > 0 for all x,y e B, \\y\\ < \\x\\ and lx € T(x). Also assume that for each r 6 (0, 1), there exists M = M(r) > 0 such that \\[Df(x)]~l\\ < M(r] when \\x\\ < r. Then f is convex. Proof. Necessity. Because (1 - t)f(x) + tf(y) € f ( B ) for all x,y e B and t € [0,1], we can find a mapping v : B x B x [0, 1] —> B such that f ( v ( x , y , t ) ) — (1 — t)f(x) + tf(y). As in the proof of Lemma 6.3.7, we have Hx,y,t)|| < max{||x||, ||y||}, x,y e B, t 6 [0, 1]. On the other hand, f ( v ( x , y, t)} = f(x) + Df(x)(v(x, y, t) - x} + R(v(x, y, t), x), where R(v(x,y,t),x)/t —> 0 as t —> 0+. Hence
t->-o+
Df(x)
6.3. Criteria for convexity Since the limit w(x,y) = hm
235
x-v(x,y,t]
t
exists and is holomorphic, we conclude in view of Lemma 6.1.33 that Re [lx(w(x,y))] > 0, for x,y 6 B, ||y|| < ||x||, x ^ 0, and lx € T(x). Moreover, since Dw(Q, 0)(x, y) = x — y, we deduce as in the proof of Lemma 6.1.30 that Re [lx(w(x,y))] > 0 for ||y|| < ||x|| and lx 6 T(x). This completes the proof of necessity. Sufficiency. Since w(x, 0) € AT it follows from Theorem 6.2.6 that /(x) — /(O) is starlike and in particular biholomorphic. Fix x and y e B and let
where to is chosen such that (1 — t)/(x) -ft/(y) 6 /(B) when 0 < t < to- Using similar arguments as in the proof of Theorem 6.2.5, we obtain
v(x, y, t) = x - tw(x, y) + o(t), and from this the fact that ||v(x,y, t)|| is a decreasing function of t when ||v(x,y, t)|| > ||y||. Next, for each r G [0,1] we consider the line segment LT = {(1 - t)/(x) + t/(y) : 0 < t < r} and the set T = < r € [0,1]: f~l can be analytically continued along LT with the initial value f~1(f(x)) = x > . As in the proof of Theorem 6.2.5 we can show that T is nonempty, open, and closed, and this completes the proof. We remark that Suffridge [Su3,4] gave infinite-dimensional versions of the characterizations of convex maps of the polydisc and of the unit ball with respect to the 1-norm in C": Theorem 6.3.11. a) Let X — f1 be the space of summable complex sequences and let Y be a complex Banach space. If f : B C £l —» Y is convex then f ( z ) — /(O) is linear. b) Let X = i°° be the space of bounded complex sequences and let Y be a complex Banach space. If f : B C i°° —> Y is convex then f ( z ) - /(O) = £>/(0)(0i(zi), - . .,&,(*»), • •.),
236
Univalence in several complex variables
where gk(zk) is o, normalized convex function on the unit disc \Zk\ < 1, k € N. Theorem 6.3.10 is of course useful in n dimensions as well as in the infinitedimensional case. An interesting application is the following theorem of Roper and Suffridge [Rop-Su2], which gives a sufficient condition for convexity on the Euclidean unit ball in C". Such a result cannot hold for normed linear spaces in general because of Theorem 6.3.11. Since the proof is long and rather complicated, we omit it. Theorem 6.3.12. Let f : B —>• C™ be a normalized holomorphic mapping 00 fc2 on the Euclidean unit ball of C n . Assume 2^. -77\\Dkf(0)\\ < 1. Then f is k=2kconvex. The examples below of convex mappings, due to Suffridge [Su4] (see also [Rop-Su2]), make use of Theorem 6.3.10 or Theorem 6.3.4. Example 6.3.13. Let B be the Euclidean unit ball in C2 and let / : B C C2 —> C2 be given by f ( z ) = (z\ + az 2 ,22)5 z — (21,22) € B. Then / is convex if and only if |a| < 1/2. First proof. We first make use of Theorem 6.3.10 to prove the above assertion. For this purpose, we have to show that Re([Df(z)]-1(f(z)-f(u^z)>Q for z = (zi,z-2) and u — (1*1,^2) € C2 with \\u\\ < \\z\\ < 1. Short computations yield that / is locally biholomorphic on B and Re ([Df(z)]-l(f(z)
- /(u)), z} = \\z\\2 - Re {*,«) - Re {azfa - u2)2} > \\z\\2 — Re (z,u) — \a\ • \z\\ • \z% — U2\2
= \\z\\\l - \a\ • |z!|) - Re (z,u)(l - 2\a\ • \Zl\) - \a\ • |^|(||u||2 - \z, - m| 2 ) > |H|2(1 - |a| • |zi|) - Re (z,u)(l - 2|a| • |zi|) - |a| • ki|(||^||2 - \z, 2
2
Ul\
)
2
= (\\z\\ - Re <*,tO)(l - 2|a| • |*i|) + |a| • \zi\ • |zi - ui| > 0 for |a| < 1/2 and z,u e J5, ||u|| < ||z||. If |a| > 1/2 we may find z\ G U such that ~z\a > 1/2. Choosing z G B with first coordinate z\,u\ = z\ and tt2 = —22 S R> we deduce that Re((Df(z)}-l(f(z)-f(u)),z)
6.3. Criteria for convexity
237
= ||z||2 - Re (z, u} - Re {azi(z2 - n2)2} < ||z||2 - Re {zizi - Z2z2} -
2
2)
= 0.
The result now follows from Theorem 6.3.10. Second proof. In the second proof we use Theorem 6.3.4. Straightforward computations yield the relation [Df(z)]-lD2f(z)(v,v)
= (2ai&0),
z € B,
v = (wi.t*) € C2.
Therefore for |o| < 1/2 we obtain 1 - Re ({Df(z)]-1D2f(z)(v,v),z}
= 1 - Re poZivJ]
> l-2|a||2i)H 2 > l-2|a| > 0 for all z e 5, v 6 C2, ||v|| = 1 and Re (z, v) = 0. Hence in view of Theorem 6.3.4 we conclude that / is convex. As in the first proof, if |a| > 1/2, there is z\ € U such that ~z\a > 1/2. Moreover, if z = (z\, 0) and v = (0, 1), then Re (z, v} = 0 and 1 - Re ([Df(z)]-lD2f(z)(v,v),z)
= 1 - 2azi < 0.
This completes the proof. With similar arguments we can verify the following ([Su4], [Rop-Su2]): Example 6.3.14. Let B be the Euclidean unit ball of C2 and let / : B ->• C2 be given by f(z) = (z\ + Q,Z\ZI, z2), z = (z\,z2) 6 B. Then / is convex if and only if |a| < l/A/2. Remark 6.3.15. These examples can easily be used to show that Alexander's theorem (/ € K if and only if zf'(z) 6 5*, see Theorem 2.2.6) is not true in higher dimensions. If / is a normalized convex mapping on the unit ball B of Cn, n > 2, then Df(z)z need not be starlike on B, except for certain special norms. (For example, it is true for the 1-norm since convex maps are linear in this case, and it is true for the polydisc as a simple consequence of Theorem 6.3.2 and Problem 6.2.5 (see [Lic3] and [Koh-Koh]).) To see this, consider the mapping in Example 6.3.14. Then Df(z)z = (zi + 2aziz2,z2), z = (zi,z2) E B, and
238
Univalence in several complex variables
for a = l/\/2 this is not a starlike map in view of Problem 6.2.2. In fact, Df(z)z is not even univalent, because (z\ (l + V^z-z) ,22) = (0, — 1/\/2) for all 2 = (zi,-l/V2) eB. Similarly, if Df(z)z is starlike then / need not be convex on the unit ball of C™, n > 2. To see this, it suffices to let n = 2 and C2 be the Euclidean space of two complex variables. Also let f(z] = (z\ + (a/2)z|,^2) for z = (21,22) e &• Then Df(z)z = (z\ + az^z?} and in view of Problem 6.2.1, we deduce that Df(z)z is starlike for \a\ < 3\/3/2. However, if \a\ > 1, / is not convex because of Example 6.3.13. This example was considered in [Rop-Su2].
6.3.3
Quasi-convex mappings on the unit ball of Cn
For various reasons, it is appropriate to consider new classes of univalent mappings in several variables which are not simply analogs of familiar subclasses of S. We have seen that the convex mappings of the unit ball in C" have a very rigid structure for certain norms. On the other hand, even for the Euclidean norm it may not be easy to verify whether a given mapping is or is not convex. It is also surprisingly difficult to construct convex mappings of the Euclidean unit ball from convex functions on the unit disc. See problems 6.3.2 and 6.3.3 (i). (We shall return to this question in Chapter 11.) In particular, in contrast to the situation of the polydisc, if /i, . . . , fn e K, n > 2, then the mapping F: B —> C71 defined by F(z) = (h (*i), • • • , /«(*»)),
z = (*i, . . . , zn) € 5,
is in general not convex. Let / : U —> C be a normalized holomorphic function. As shown in Theorem 2.2.4, / € K if and only if
Trying to generalize this idea in several variables, Roper and Suffridge [Rop-Su2] introduced the following class of mappings on the unit ball of C" with respect to an arbitrary norm \\ • \\. Let £S(B) be the class of normalized locally biholomorphic mappings on B. Recall that C denotes the extended complex plane.
6.3. Criteria for convexity
239
Definition 6.3.16. Let u € C" with \\u\\ = 1, and let lu € T(u). For / € £S(B) define Gf : U x U -* t by (R i &\ (6.3.8)
-
lu([Df(au}]-i(f(au]_f(pu}}}
a_p>
We define the class of quasi- convex mappings of type A by Q = {/ 6 £S(B) : Re Gf(a,/3) > 0 for all a,/3 e C/, |/3| < |a|, lu e T(u), and u € C1, ||w|| = 1 1.
Actually this is a slight modification of [Rop-Su2, Definition 3]. Lemma 6.3.17. Let f e £S(B). Also letueC1, \\u\\ = l,lue T(u), and e U, a^Q. Then (6.3.9)
lim Gf(a,0) = I +
alu([Df(au)]-lD2f(au)(u,u))
In particular, Gf(a,/3) is continuous in /3 at fi = a. Proof. Expanding f(0u) about au, we obtain f(0u) = f(au) + Df(au)((P - a)u)
- o)ti, 09 - «)«) + 0((fi - a)3). Hence (Df(au)}-l(f(au)
-
Taking into account the above equality, we deduce that _ 2o_ lu((Df(au)]-i(f(au) -
a + (3 ~
(a - /
(a -
- a)) J
240
Univalence in several complex variables The first equality in (6.3.9) now follows if we let (3 —»• a. To obtain the
second equality we use the fact that lau(-} = — l u ( - } € T(au) for u e Cn, \\u\\ = 1, a n d a e U\{0}. Roper and Suffridge [Rop-Su2] proved that K(B) C Q C S* (B}. In fact we shall show that if / € Q, then / is starlike of order 1/2. These results are given in the next two theorems. Theorem 6.3.18. Let / : £ ? — > • C™ be a normalized convex mapping. Then
fee. Proof. Since / is convex, it follows that Re
l z ( ( D f ( z } ] - l (1 f (^) z ) -- /(«))) /(«)))}} > 0,
|M| < ||z|| < 1, J, €
by Theorem 6.3.10. Let u € C", ||w|| = 1, and let lu 6 T(u). By the preceding inequality, we have (6.3.10) Re [lau((Df(au}]-l(f(au}
- f((3u}}}\ > 0,
\(3\ < \a\ < 1.
Using the one-to-one correspondence between T(au) and T(u) given by \a\ lau('} = —^u(')> we obtain from (6.3.10) that a
Re
lu([Df(au}}-(f(au)
- f(/3u))}
> 0,
or equivalently, 2a
Letting \/3\ — \a\ < 1 with (3 ^ a, we obtain AXC S
, r ^^
-x
X1
i / */
\
j « / ^ » \ \ \ i
^•
L 'u(
Moreover, if|a| = |/3|,a7^/S, then it is easily checked that Re < -- > = 0, and therefore we obtain
/ \ lu( [ D f ( a u ) } - i ( f ( a u }
- /(/3u)))
a-
6.3. Criteria for convexity
241
Consequently we have shown that Re Gf(a,(3) > 0 for |a| = \/3\, a ^ (3. Keeping a € U \ {0} fixed, we note that Gf(a, •) is analytic on the disc U\a\ and, using Lemma 6.3.17, continuous on Z7|a|. Moreover, ReG/(a,a) > 0 by an argument similar to the proof that strict inequality occurs in (6.3.5). Therefore, by the minimum principle for harmonic functions, we conclude that Re G/(a,/3) > 0 when \/3\ < \a\ < 1, i.e. / € <7, as desired. This completes the proof. The next result is a generalization of the Marx-Strohhacker theorem (see Theorem 2.3.2) to several complex variables. Theorem 6.3.19. /// € Q then f is starlike of order 1/2. Proof. Since / € Q, Re Gf(a,/3) > 0 for u e C1, \\u\\ = 1, lu € T(u), and \(3\ < \a\ < 1. For ft = 0 one deduces that Re (?/(ct,0) = Re , n^£l , ,,—rr - 1 > 0, /v lu([Df(au)}-if(au)) and therefore
But this is exactly the condition (6.2.9) for 7 = 1/2, so we are done. The class Q is defined by a two-point condition. There is a related one-point condition which we have already seen in Theorem 6.3.5. Roper and Suffridge [Rop-Su2] made use of it to introduce another subclass of locally univalent mappings on the unit ball in C71 which extends at least some of the properties of K to several variables. Definition 6.3.20. For / e £S(B), z € B\{0}, and lz € T(z), let Ff(zt lt) = lz([Df(z)}-l(D2f(z)(z,
z)
The class of quasi-convex mappings of type B is defined by = {/ e CS(B) : Re Fff(z, l z ) > 0, for all z € B \ {0} and lz € T(z)\. Roper and Suffridge [Rop-Su2] proved the following inclusion: Theorem 6.3.21. // / € Q then f 6 7. Proof. This follows from Lemma 6.3.17.
242
Univalence in several complex variables
It is easier to work with the two-point condition defining the class Q than with the definition of F, but we do not know whether the inclusion in Theorem 6.3.21 is proper. Roper and Suffridge proved that in the Euclidean unit ball of Cn the upper and lower bounds on the growth of mappings in Q are the same as for normalized convex maps. We shall return to this result in Chapter 8, and also we shall investigate other properties of these holomorphic mappings in higher dimensions. We remark that in contrast to the situation of convex mappings, if /i(zi),..., fn(zn) are functions in K, then F(z) = (/i(*i), • • • , f n ( z n } } is a mapping in Q and T on the unit ball of C"1 with respect to a p-norai, 1 < P < oo (see Problem 6.3.8). Hence K(B) § Q and K(B) g T in C n , n > 2. Notes. In both Sections 6.2 and 6.3 we have studied some well known and also some new criteria for starlikeness and convexity for holomorphic (or nonholomorphic) mappings in the unit ball and certain pseudoconvex domains in C™. Hamada and Kohr [Ham-Koh7] obtained necessary and sufficient conditions for convexity of locally biholomorphic mappings on bounded balanced convex domains whose Minkowski function is of class Cl in C™ \ {0}. For some similar results, the reader may consult [Gon4j. For other results about starlikeness and convexity, the reader may use the following references: [Su3], [Su4], [Rop-Su2], [Gon2], [Gon4], [Ham-Koh7], [Ham-Koh3], [Ham-Koh2], [Gon-WaYu2-3], [Koh2], [Koh7-8], [Gra3-5].
Problems 6.3.1. Let f ( z ) = (z\ + a,Z2,Z2), z = (21,22) € C2. Prove the following assertions using Theorem 6.3.10: a) if ||2||i = \zi\ + \zz\, then / is convex on the unit ball J5(l) C C2 with respect to this norm if and only if a = 0. b) if || z || oo = max{|2i|, |z2|} then / is convex on the unit polydisc P of C2 if and only if a = 0. (Suffridge, 1970 [Su2] and 1975 [Su4].) 6.3.2. Let B(p) be the unit ball of C2 with respect to a p-norm, where
6.3. Criteria for convexity
243
1 < p < oo. Let / : B(p) ->• C2 be given by f ( z ) = (zi/(l - zi),z 2 ), z = (21,22) € B(p)- Show that / cannot be convex. (Suffridge, 1975 [Su4].) 6.3.3. (i) Let B denote the Euclidean unit ball of C", n > 2, and let Z
Z
Z
£( \ l % n \ r> /(*)= ((\;— — > ,— I-''"' i— T ' z =i (zi,...,z\n^)eB. 1 - Zl 1- Z2 1 - Z J n
Show that / is not convex. (However, / is convex on the unit polydisc of C1.) (Gong, Wang and Yu, 1993 [Gon-Wa-Yu3], Roper and Suffridge, 1995 [RopSul].) (ii) Show that if / : B -> C2 is given by
where B is the Euclidean unit ball of C2 , then / is convex. (Roper and Suffridge, 1995 [Rop-Sul].) 6.3.4. Show that /(z) = z/(l — zi), z = (zi,...,z n ) e B, is a convex mapping on the Euclidean unit ball B of C" . 6.3.5. Let X be a complex Hilbert space with inner product {•, •) and let jj u be a unit vector in X. Show that f ( z ) = ----, z € B, is a convex 1 - (z,w) mapping on B. (Hamada and Kohr, 2002 [Ham-Koh9].) 6.3.6. Let B(p) be the unit ball of C" with respect to a p-norm, where 1 < p < oo. Let F : B(p) -» C" be a mapping with one of its coordinate maps, /fc, a function of one complex variable only. Show that it is a necessary condition for F e T that fk € K. (Roper and Suffridge, 1999 [Rop-Su2].) 6.3.7. Let B(p) denote the unit ball of C2 with ap-norm, where 1 < p < oo. Also let / : B(p) ->• C2 be given by f ( z ) = (zi + az$, z2), z = (zi, z2) € B(p). Show that / e Q (or / e T} if and only if
-
4
(Roper and Suffridge, 1999 [Rop-Su2].)
p-i
244
Univalence in several complex variables
6.3.8. Let / : B C C* -> C" be given by f ( z ) = (/i(*i), ...,/„(*„)), z = (zi, . . . , zn] e B, where f j E . K , j — l,...,n. Show that / G Q in any absolute norm. (That is, any norm for which \Zj\ < \Wj\ for each j implies
INI < IMIO (Roper and Suffridge, 1999 [Rop-Su2].) 6.3.9. Let B(p) denote the unit ball of C2 with respect to a p-norm, 1 < p < oo, and f ( z ) = (z\ + aziz^z-z), z = (z\,Z2) G B(p). Show that / € Q (or / € ^ if and only if |a| < ^- (p + l) (Roper and Suffridge, 1999 [Rop-Su2 .) 6.3.10. Let B be the unit ball of a complex Hilbert space X and suppose that /: B —>• X is a locally biholomorphic mapping. Show that the condition Re ( [ D f ( x ) ] - ( f ( x )
- /(y)),x> > 0,
||y|| < ||x|| < 1,
implies the condition ||u|| 2 -Re ( { D f ( x ) } - 1 D 2 f ( x ) ( v , v ) , x )
>0
for x e B, v € X \ {0}, and Re {z, v) = 0. 6.3.11. Consider the class T introduced in Definition 6.3.20. Must any / G F be biholomorphic on the unit ball of C" , n > 2?
6.4
Spirallikeness and $- likeness in several complex variables
In this section we are going to study the sets of spirallike and 3>-like mappings in the unit ball of Cn . In 1975 Gurganus [Gur] gave a beautiful extension of the theory of $-like holomorphic functions in the unit disc to locally biholomorphic mappings on the unit ball of C1 and complex Banach spaces. The case in which $ is linear leads to the notion of spirallikeness with respect to a linear operator, considered by Gurganus and (in somewhat greater generality) by Suffridge [Su4]. More recently, Hamada and Kohr [Ham-Kohl, 4] considered spirallike mappings of type a, which corresponds to the case in which the linear operator is
6.4. Spirallikeness and ^-likeness
245
a scalar multiple of the identity. This case is probably the most natural generalization of spirallikeness to several variables; however, interesting phenomena arise when one considers a more general linear operator. Kohr [Koh3] and Hamada and Kohr [Ham-Koh3] also gave some sufficient conditions for diffeomorphism and spirallikeness for mappings which are locally diffeomorphisms of class C1 on the unit ball, as well as on bounded balanced pseudoconvex domains in C". We consider the Euclidean structure on C" and we let B denote the unit ball of C". As usual, by {•, •} we denote the Euclidean inner product in C1. We recall that for the Euclidean case the class A/" consists of those holomorphic mappings h : B -» C1 such that /i(0) = 0 and Re (h(z), z) > 0, z e B \ {0}. Gurganus [Gur] gave the definitions below of a $-like holomorphic mapping and a $-like region in C1. These definitions generalize Brickman's notion [Bri] of ^-likeness in the unit disc (see Section 2.4.3). Definition 6.4.1. Let / : B —> C71 be a normalized locally biholomorphic mapping. Let <1> 6 H(f(B)) be such that 3>(0) = 0. We say that / is $-like if (6.4.1)
[Df(z)]-^(f(z))eM,
In the case n = 1 we do not need the assumption /'(C) ^ 0, C G U, but in higher dimensions the open mapping theorem fails (see Problem 6.4.4), and this is the reason why in Definition 6.4.1 we assume that / is locally biholomorphic on B. Definition 6.4.2. Let £} be a region in C" containing 0 and let <£ € H(£l) be such that $(0) = 0 and Re (D$(G)z, z) > 0 for all z e C1 \ {0} (i.e. D$(Q) + D$(Q)* is a positive definite operator). We say that fi is $-like if for each WQ £ fi, the initial value problem (6.4.2)
-^ = -$H, at
tw(0) = WQ
has a solution w(t) defined for allt > 0 such that w(t) e fi for alH > 0 and w(t) —>• 0 as t —*• oo. Remark 6.4.3. The local existence and uniqueness of solutions of the initial value problem (6.4.2) follows from standard theorems in ordinary differential equations. The existence of solutions defined for all t > 0 for a given
246
Univalence in several complex variables
differential equation requires special features of the equation, for example the assumption h € A/" in Lemma 6.4.4. We shall return to this question in Chapter 8. We now study some properties of 3>-like mappings on the unit ball of Cn. For generalizations of some of these results to the case of complex Banach spaces, the reader may consult [Gur]. We begin with the following lemma of Gurganus [Gur]: Lemma 6.4.4. Let h € A/". Then for each z 6 B, the initial value problem dv _ = _/!(,,),
(6.4.3)
„(()) = *,
has a unique solution v(t) = v(z,t) defined for all t > 0. For fixed t, Vt(z) = v(z,t) is a univalent Schwarz mapping which satisfies the inequality (6.4.4)
\\v(z,t)\\ <
where kh = min Re (D/i(0)z, z) > 0. Proof. We only make some remarks on the proof since we do not need the full force of the lemma. The local existence and uniqueness of solutions is clear; the existence of solutions defined for all t > 0 is established in more generality in Theorem 8.1.3. (The normalization Dh(Q) = I assumed there plays no role in the existence and uniqueness theorem.) r\
Since «-||t;(;M)l|2 = -2Re (h(v),v) < 0, u(z,0) = z, and v(0,t) = 0, we C/Z' see that vt(z) = v(z,t) is a Schwarz mapping for any fixed t > 0. We do not need the univalence of vt(z] at this stage, but we do need to know that v(z,t) —)• 0 as t —> oo for any fixed z € B. This follows from the estimate (6.4.4); to prove it we make use of a generalization of Lemma 6.1.32 to the class A/". We leave this as an exercise for the reader. Gurganus [Gur] showed that, as in one variable, ^-likeness is equivalent to univalence. The reader may compare the following results with Theorems 2.4.16, 2.4.18 and Corollary 2.4.17. Theorem 6.4.5. Let f be $-like. Then f is univalent on B and f ( B } is a &-like domain.
6.4. Spirallikeness and ^-likeness
247
Proof. First step. Since / is $-like, there exists a mapping h € J\f such that
*(/(*)) = Df(z)h(z),
z e B.
Moreover, by the normalization of / we obtain D$(0) = D/i(0), and hence Re (D$(Q)(z),z) > 0 for all z € Cn \ {0}, by Lemma 6.1.30. Next, fix z e B. In view of Lemma 6.4.4, the initial value problem dl)
^ = -M«), *><>> w(o) = *, has a unique solution vz(t) = v(z,t). Let wz(t) = w(t) = f ( v z ( t ) ) , t > 0. Then Wz(Q) = f ( z ) and a simple computation yields
^ = Df(vz(t))^ Hence (6.4.5)
= -Df(vz(t))h(vz(t))
^j£ = _*(„,,),
= -*(/(V,(t))).
u,,(0) = /(z).
It is clear that wz(i) & /(-B), and in view of (6.4.4) it follows that vz(t) —> 0 as t —)• oo. Therefore lim w z (t) = lim /(«*(*)) = /(O) = 0.
t—MX>
Consequently, f ( B ) is a 3>-like domain. Second step. We now prove that / is univalent on B. To this end, let a, 6 € B and assume /(a) = f ( b ) . Then the solutions wa(£) and Wb(t) of the initial value problem (6.4.5) satisfy wa(0) = Wb(0). Using the uniqueness of solutions of this equation, we deduce that wa(t) = Wb(t) for all t > 0, and consequently f(va(t)) = f(vb(t)) for all t > 0. Now, since / is locally biholomorphic on B, / has a local holomorphic inverse at zero and since va(t) —t 0, Vb(t) —} 0 as t —>• oo, there exists to > 0 such that va(t) = Vb(t) for all t > IQ. However, since va(t) and Vb(t) are solutions of the initial value problem fii)
— = -h(v), at
v(t0) = va(t0) = vb(tQ),
we conclude by the uniqueness of solutions that va(t) = Vb(t) for t > 0. Hence a = va(0) = Vfe(O) = b and this completes the proof of Theorem 6.4.5.
248
Univalence in several complex variables
Corollary 6.4.6. Let f : B —> C™ be a normalized locally biholomorphic mapping. Then f is univalent on B if and only if f is $-like for some $. Proof. In the previous theorem we have seen that if / is $-like, then / is univalent. Thus it suffices to prove the converse. For this purpose, let h € A/" be arbitrary and consider the equation *(/(*)) = D f ( z ) h ( z ) ,
z e B.
Since / is biholomorphic on B, this equation determines a mapping $ e H ( f ( B } } such that Re ( ( D f ( z } } - l $ ( f ( z ) ) ,
z) > 0, z E B \ {0}.
Thus / is $-like. The next result of Gurganus [Gur] is the converse of Theorem 6.4.5. Theorem 6.4.7. Let f : B —>• Cn be a normalized univalent mapping on B such that /(-B) is a $-like domain in C". Then f is $-like. Proof. Let h(z) = [ D f ( z ) ] - l $ ( f ( z ) ) , z G B. Then h 6 H(B), h(Q) = [D/(0)]-1$(/(0)) = 0, Dh(Q) = JD*(0) and we claim that h € JV. To see this, fix z € B and let wz(t) = w(z,t) be the unique solution of the initial value problem (6.4.6)
^* =_$(„,,),
t>0>
wz(0) = f ( z ) .
By hypothesis, we have wz(t) € f ( B ] for t > 0 and wz(t] —t 0 as t —>• oo. Also let u z («) = u(z,«) = /-^^(i)). Clearly, if z = 0 then wz(t) = 0 for t > 0 by the uniqueness of solutions of (6.4.6). If z ^ 0 then wz(t) ^ 0 for all t > 0, again in view of the uniqueness of solutions. Hence in this case vz(t) ^ 0 for z ^ 0 and t > 0. Further, since $(z) and f ( z ) = wz(0} are holomorphic mappings, it follows as in the proof of Theorem 2.4.18 that w(-, t) is holomorphic in z for fixed t. Since / is univalent on B, vz(t) is also holomorphic in z for fixed t, and moreover ||v2(t)|| < I , z €. B,t >Q. Consequently, in view of the Schwarz lemma, ||vz(£)|| < ||z|| for all z e B and t > 0. We therefore obtain dt t=0 Moreover, since
= lim
t^0+
^iu, (+\\\
t
t->0+
t
6.4. Spirallikeness and ^-likeness
249
we must have Re (h(z),z) >0,
z£B.
Finally, in view of Lemma 6.1.30 and the relation Re (Dh(Q)z, z) = Re (D$(Q)z, z) > 0, we deduce that Re (h(z), z} > 0 on B \ {0} . This completes the proof. Theorems 6.4.5 and 6.4.7 may be used to obtain an alternate proof of the necessary and sufficient condition for starlikeness given in Theorem 6.2.2, and also to characterize spirallikeness in higher dimensions. The proof below is du& to Gurganus [Gur]. Corollary 6.4.8. Let f : B —>• C™ be a locally biholomorphic mapping such that /(O) = 0. Then f is starlike if and only [Df(z)]~1f(z) € M. Proof. We may assume that Df(Q) = I. Obviously if we take $ = / then the solution of (6.4.2) is w(t) = WQe~l. Hence a domain fi containing the origin is J-like if and only if fl is starlike. Now assume that / : B —»• C" is a locally biholomorphic mapping such that /(O) = 0 and [Df(z)]~1f(z) e M. Then Theorem 6.4.5 implies that / is starlike. Conversely, if / is starlike then Theorem 6.4.7 implies that [Df(z)]~1f(z) € A/". Using the fact that [Df(z)]~1f(z) is a normalized mapping, we conclude that [Df(z)]~1f(z) £ M. and we are done. We next present some ideas about spirallikeness on the unit ball of C™. Gurganus [Gur] introduced the notion of spirallikeness with respect to a normal linear operator (i.e. an operator A € Z^C^C") such that AA* = A*A) whose eigenvalues have positive real part. Suffridge [Su4] enlarged the class of operators considered and also extended the definition to complex Banach spaces. We shall give Sufrridge's definition, but we consider only the Euclidean unit ball in C". For a linear operator A e L(C n ,C n ) we introduce the notation (6.4.7)
m(A) = min JRe (A(z),z} : \\z\\ = 1\.
Definition 6.4.9. Let / : B -» C1 be a normalized univalent mapping on B. Let A E Z^C^C") be such that m(A) > 0. We say that / is spirallike
250
Univalence in several complex variables
relative to A if e~tAf(B) C f(B] for all t > 0, where e-tA fer=0
The following generalization of Theorem 2.4.10 is due to Suffridge [Su4]; the important special case when A is normal was considered by Gurganus [Gur]. Theorem 6.4.10. Let A e £(€",€") be such that m(A) > 0 and let f : B —>• C™ be a normalized locally biholomorphic mapping. Then f is spirallike relative to A if and only if (6.4.8)
[Df(z)}-lAf(z)eM.
Proof. First step. Assume that / is locally biholomorphic and satisfies (6.4.8). Then there exists a mapping h € Af such that Dh(Q) = A and Df(z}h(z)
= Af(z),
zeB.
In view of Lemma 6.4.4, for each z G B there exists a unique solution vz(t) = v(z,t) to the initial value problem (6.4.9)
^ = _/»(„,), at
w,(0)
= «.
Prom (6.4.4) we conclude that ||vz(£)|| ->• 0 as t -» oo. Hence / is a $-like mapping with $ = A. Therefore / is univalent on B, and using the result of Theorem 6.4.5 we deduce that f ( B ) is a <&-like domain in C". If we let u(z,t) = f ~ l ( e ~ t A f ( z ) ) forz£B and t > 0 sufficiently small such that iA e~ f(z) e /(J5), then u — u(z,t) is a solution of the initial value problem (6.4.9). In view of the uniqueness of solutions of this equation, we conclude that u(z,t) = v(z,t) for z e B and t > 0. Since f ( B ) is a
zeB, t > 0,
which means that /(B) is a spirallike domain. Consequently / is spirallike, as desired.
6.4. Spirallikeness and ^-likeness
251
Second step. Conversely, assume that / is spirallike relative to A. Let F(z,t) = e~tAf(z), z € B and t > 0. Then F(-,t) € H(B), F(B,t) C f ( B ) , F(z, 0) = /(z), z e B, and F(0, t) - 0 for all t > 0. Since
.. F(z, 0) - F(z, t) .... hm v ' ' —^—L-L = A f ( z ) , t-»o+ t
zeB,
we conclude from Lemma 6.1.35 that there exists a mapping h € A/b such that A/(z) = D f ( z ) h ( z ) ,
zeB.
Since D/i(0) = ^ and Re (A(z),z) > 0, z € Cn \ {0}, we deduce in view of Lemma 6.1.30 that h e JV. Thus [D/(z)]~1A/(2;) 6 A/" and this completes the proof. Remark 6.4.11. As already noted, Gurganus [Gur] originally denned spirallikeness in several variables as spirallikeness with respect to a normal matrix A whose eigenvalues AI, . . . , An have positive real part, or equivalently such that m(A] > 0. If A is a normal matrix, there exists a unitary matrix V such that WIV"1 is diagonal with diagonal entries A i , . . . , A n . In this case a domain fi in C11 containing 0 is spirallike with respect to A if and only if whenever w = (wi,... ,wn) 6 V"(fi), the spiral (w\e~Xlt,... ,wne~Xnt), t > 0, is contained in V(17). In particular if A = e~ial for some o: e (—7r/2,7r/2), we obtain a class of mappings studied by Hamada and Kohr [Ham-Koh4] and called spirallike of type a. In this case the condition (6.4.8) reduces to (6.4.10)
e-ia[Df(z)]-1f(z)
€ M.
This is the most straightforward generalization of the notion of spirallikeness to several variables, and it is this class whose behaviour is closest to that of spirallike functions of one variable. In particular we shall see in Chapter 8 that spirallike mappings of type a can be characterized in terms of Loewner chains. The following example of Hamada and Kohr [Ham-Koh4] shows that the class of spirallike maps with respect to an arbitrary linear operator need not be a normal family, even when the operator is diagonal. Example 6.4.12. In the Euclidean space C2, let f ( z ) = (zi+az^, z%) and let A(z) = (2*1,22), z = (zi,z2) € C2. Then for all o 6 C, [Df (z)}~1 Af (z} = A(z). Since A € A/", it follows that f is spirallike with respect to A for all a € C.
252
Univalence in several complex variables
Open Problem 6.4.13. Find conditions on the linear operator A such that the class of spirallike maps relative to A be a normal family on the unit ball of C1, n > 2 . Notes. Recently Hamada, Kohr, and Liczberski [Ham-Koh-Licl] extended the above notions and results concerning ^-likeness and spirallikeness in the unit ball of C71 to bounded balanced pseudoconvex domains whose Minkowski function is of class Cl in C™ \ {0}. Also in [Ham-Koh3] the authors obtained some conditions for diffeomorphism and spirallikeness for a local diffeomorphism of class C1 on the above domains. Kohr [Koh3] has recently studied the notion of spirallikeness on bounded domains fi in C1 whose Bergman kernel function becomes infinite everywhere on the boundary. For further material concerning spirallike and $-like holomorphic mappings in higher dimensions, the reader may consult the instructive papers [Gur], [Ham3] and [Su4], Also see the monograph [Koh-Lic2].
Problems 6.4.1. Let fj(zj),
j = 1, . . . , n, be a spirallike function of type a on the n
unit disc U, where a e (-7r/2,7r/2). Let \j > 0, y^Aj = 1. Show that
3=1
is spirallike of type a on the Euclidean unit ball of C" . (Cf. Pfaltzgraff and Suffridge, 1999 [Pfa-Su3]; Hamada and Kohr, 2001 [HamKoh6].) 6.4.2. Let / : B C C2 ->• C2 be given by z
=
where a € (— 7r,7r). Show that / is spirallike relative to e~ ta / 2 / on the Euclidean unit ball of C2 (i.e. / is spirallike of type a/2). (Suffridge, 1975 [Su4].)
6.4. Spirallikeness and ^-likeness
253
6.4.3. Let fj(zj), j = l,...,n, be a spirallike function of type a e (-7r/2,7r/2) on the unit disc U. Show that f ( z ) = (/i(zi),... ,/n(2 n )), z = (21,..., 2n) £ B, is spirallike of type a on the Euclidean unit ball B ofC71. (Suffridge, 1975 [Su4].) 6.4.4. Let B be the Euclidean unit ball in C2 and let /(z) = (2 z = (zi, 22) e B. Show that the image of B is not open.
Chapter 7
Growth, covering and distortion results for starlike and convex mappings in Cn and complex Banach spaces The aim of this chapter is to give growth, covering and distortion results for subclasses of normalized biholomorphic mappings on the unit ball in Cn, and to some extent on the unit ball of a Banach or Hilbert space or on certain bounded pseudoconvex domains in Cn. Also we shall consider bounds for the coefficients of starlike and convex mappings on the unit ball of C1. Such bounds are expressed in terms of the kth Frechet derivative of the mapping, using either the norm or the linear functionals introduced in Section 6.1.6. We have seen in Section 6.1.7 that the full class S(B) of normalized biholomorphic mappings on the unit ball in C" is not a normal family. Moreover, in higher dimensions the growth and covering theorems fail for the full class S(B). Similarly, there is no distortion theorem involving the differential D f ( z ) or the Jacobian determinant Jf(z) of a mapping / G S(B}. Finally, there are no bounds for the Taylor series coefficients of order two or higher for mappings in S(B). Because of such examples, we are obliged to restrict to proper subclasses 255
256
Growth and distortion results for starlike and convex maps
of S(B) in studying growth, covering, and distortion theorems or coefficient estimates. In this chapter we shall focus primarily on the normalized starlike mappings and the normalized convex mappings.
7.1
Growth, covering and distortion results for starlike mappings in several complex variables and complex Banach spaces
7.1.1
Growth and covering results for starlike mappings on the unit ball and some pseudoconvex domains in Cn. Extensions to complex Banach spaces
We begin this section with the growth theorem for normalized starlike mappings on the unit ball of C™ with the usual Euclidean structure. This result was obtained by Barnard, FitzGerald and Gong [Bar-Fit-Gonl] using the analytical characterization of starlikeness, and by Kubicka and Poreda [Kub-Por] (and later Chuaqui [Chu2]) using the method of Loewner chains. We shall give a proof using the analytical characterization of starlikeness; the idea is due to Liu and Ren [Liu-Ren2] (cf. [Gon4]). In the next chapter we shall give some generalizations to the case of normalized biholomorphic mappings which have parametric representation on the unit ball of Cn . Theorem 7.1.1. Let f : B —> Cn be a normalized starlike mapping on the Euclidean unit ball B. Then
"•'
z
•"
B
'
These estimates are sharp. Consequently, f(B) D -Bi/4Proof. Fixing ZQ £ B \ {0}, we consider the curve 7 in B parametrized by
Thus 7 is the inverse image under / of the straight line segment joining 0 to /(ZQ). We have
= Drl(tf(zQ))f(z0)
=
-t[Df(z(t))]-lf(z(t)),
7.1. Starlike mappings and hence (7.1.2)
257
jt\ = ?Re ( [ D f ( z ( t ) ) } - l f ( z ( t ) ) ,
*(*)),
t € (0,1].
Since / is starlike, ( D f ) ~ l o / e .A/1, and hence
(7.1.3) NI2^|| < ^ (p/^)]-1/^)^) < INI2^jp| « e B, by (6.1.27). Combining (7.1.2) with (7.1.3) and using the fact that
A we obtain
- \\z(t)\\ ' or
1
< d* <
- IkMII
Integrating both sides of this inequality from e to 1, where 0 < e < 1, we deduce that , Poll . \\z(e)\\ 1
-log
log
log
-log or equivalently,
and \\*(e)
<
Since lim —^—- = /(ZQ) by the normalization of /, we obtain (7.1.1) by e-»0+ £ letting e —> 0+ in the last two inequalities. In order to show that these estimates are sharp, let
258
Growth and distortion results for starlike and convex maps
From Problem 6.2.5, we know that / is starlike on B. Moreover, for r € T
[0,1) and z = (r,0,... ,0), we have \\f(z)\\ = \
= /
,},,., \
M
M
/
for r € [0,1) and z = (-r,0,... ,0), we have ||/(*)|| = —^ = L \
i ')
J1*1'
\L < \\z\\)
This completes the proof. The same example shows that the covering result given in Theorem 7.1.1 is sharp. Thus the Koebe constant for normalized starlike mappings of the Euclidean unit ball in C1 is 1/4, as in one variable. When / isfc-foldsymmetric (i.e. exp(—27ri/k)f(e 2 m / k z) = f ( z ) for z € B, where A; is a positive integer), we obtain the following refinement of the above result, due to Barnard, FitzGerald and Gong [Bar-Fit-Gonl]. We leave the proof of Corollary 7.1.2 for the reader. Corollary 7.1.2. // / : B —>• C™ is a normalized starlike mapping which is k-fold symmetric, then
(7.1.4)
„ , !!*!'„» < ii/coii < —JW
These estimates are sharp. Consequently, f ( B ) contains a ball centered at zero and of radius 2~ 2 / fc . This result leads to another proof [Bar-Fit-Gonl] of the well known inequivalence between the ball and the poly disc in Cn, n > 2 (Theorem 6.1.21). Corollary 7.1.3. The image of the unit ball B under a normalized starlike mapping f is a balanced domain in C™ if and only if f ( B ) = B. Consequently, the Euclidean unit ball and the unit polydisc o/C n , n > 2, are not biholomorphically equivalent. Proof. First assume that f ( B ) is a balanced domain. Then this domain is fc-fold symmetric for all k e N. Letting k —> oo in (7.1.4), we deduce that ||/(z)|| = ||z|| on B. Together with the normalization of /, this implies that f ( z ) = z. (Without using Cartan's theorem on fixed points, this can be seen as follows: Let u e C n , \\u\\ = 1, and let TTU denote the orthogonal projection of Cn onto the subspace CM. Let ipu: C -» Cn be given by V'u(C) = Cw- Then the Schwarz lemma in one variable implies that •0~1 o TTU o /(£M) = £, C € U. Hence TTU o /(CM) = Cu, which implies that f(£u) = CM, for otherwise we could not have ||/(CM)||-||CM||.)
7.1. Starlike mappings
259
The converse is obvious. Next suppose that F : B —> P is a biholomorphic mapping such that F(B) = P. By composing with an automorphism of P we may assume that F(0) = 0. There exist unitary matrices W and V such that W[DF(Q)]V is diagonal. After a linear change of coordinates in the target space we may assume that W[Z)F(0)] V = I. Now the mapping G = WoFo V is a normalized biholomorphic mapping of B onto a balanced domain, and hence G(z) = z. On the other hand, the image of B under G is W(P). Hence W(P) = B, which is impossible. A similar growth theorem holds for starlike mappings on the unit polydisc P in C" . We leave the proof for the reader. The growth theorem for starlike mappings has been studied in other situations, for example on the unit ball for a p-norm, 1 < p < oo ([Gon-Wa-Yul], [Pfa3]), and classical bounded symmetric domains ([Gon-Wa-Yu4], [LiuT2]; see also [Gon4]). Further results for starlike maps including generalizations of the covering theorem can be found in [Gon4]. A case we shall consider in more detail is the case of bounded balanced pseudoconvex domains whose Minkowski function h is of class Cl in C1 \ {0}. (It is easily seen that h is continuous at 0 and nonzero except at 0 for any bounded balanced domain.) This case was studied by Liu and Ren [Liu-Ren2] using the analytical characterization of starlikeness (see also [Gon4, Theorem 3.4.1]), and by Hamada [Ham4] using the method of Loewner chains. Theorem 7.1.4. Let ft be a bounded balanced pseudoconvex domain whose Minkowski function h is of class C1 on C™ \{0}. /// : ft —>• Cn is a normalized starlike mapping, then
<•» Consequently, /(ft) 2 l/4ft. Proof. Let z G ft \ {0} and u = z/h(z] e 5ft. Also let
, o
U, 2
/ dh \ Using the fact that (w, -^z-(w)) = /i2(u) = 1 (see the proof of Theorem 6.2.8)
260
Growth and distortion results for starlike and convex maps
we see that g is holomorphic on U. We also note that the Minkowski function satisfies
This allows us to rewrite g in the form
9(0 = j£p (p/CCt*)]-1/^), 7
) , C € u \ {0},
and using Theorem 6.2.8 we see that Re p(£) > 0, C € £7. Hence g £ P, and by Theorem 2.1.3 we have
Letting £ = h(z) in the above, we obtain
Now let z(£) = /~ 1 (t/(z)), 0 < t < 1, be the inverse image of the straight line segment between 0 and f ( z ) . Then z(t) is a curve of class C1 contained in the domain ft such that z(0) = 0 and z ( l ] = z. A short computation yields that
\ dt ' dz
dt
)J,2
t
v v
' \
,te(o,i],
and thus we obtain the relation
Integrating both sides of the above inequality from e to 1, with 0 < e < 1, and using the fact that dh2(z(t)) = 1h(z(t}}dh(z(i)}, we deduce that
L M*(*) (i f1
l-
After short computations, we obtain the relations h(z)
.
h(z(e}}
7.1. Starlike mappings
261
h(z)
h(z(e))
I "I __ hi y\\^ \ V / y
I 1 — h( ylc"\\\~ V \ \ y / /
or equivalently,
M*)
M^(g))
MZ)
h(z(e))
and
Since lim
= h(f(z)) by the normalization of /, we obtain (7.1.5) J
V
+
'
by letting e —> 0 in the two previous relations. This completes the proof. Finally we remark that the 1/4-growth result given in Theorem 7.1.1 can be extended to the case of normalized starlike mappings on the unit ball of complex Banach spaces (see [Don-Zha], cf. [Gon4]). Theorem 7.1.5. Let X be a complex Banach space and let B be the unit ball of X. Let f : B —t X be a normalized starlike mapping. Then
z
zeB
'
Remark 7.1.6. When X is a complex Hilbert space the growth result (7.1.6) is sharp. To see this, let / € S* and let u e X be a unit vector. Also let (•, •) be the inner product of X and consider the holomorphic map Fu : B -4 X given by
(7.1.7)
Fu(z) = f((z,u))U+y/f'((z,U))(z
- <*,«)«),
zeB.
Graham and Kohr [Gra-Kohl] proved that Fu is a starlike mapping of B. (This extension operator was introduced by Roper and Suffridge [Rop-Sul] and will be the subject of Chapter 11.) If we begin with the function /(£) = jcomputation in (7.1.7) yields that
-^, £ e [7, a straightforward
262
Growth and distortion results for starlike and convex maps
If z = ru, where r € [0,1), then ||z|| = r and \\Fu(z}\\ = „ T ., = ,, , , ' 9 . (1-»T (l-||z||)^ On the other hand, if z = —ru, r € [0,1), then \\z\\ = r and ||Fu(z)|| = r (1 + r)2
7.1.2
\\z\\
— . . 9 . This completes the (1 + ||,
Bounds for coefficients of normalized starlike mappings in Cn
We next consider some bounds for the power series coefficients of normalized starlike mappings on the unit ball of Cn. In this section we work with an arbitrary norm || • || in C™. As usual, B will denote the unit ball with respect to this norm. The coefficient bounds are expressed in terms of bounds for the fcth Frechet derivative of the mapping at 0. We begin by obtaining some estimates for the fcth Frechet derivative of mappings in the class M (cf. [Koh6], [Gra-Ham-Koh]). These results are of independent interest; in particular the norm estimates lead to another proof of the compactness of the class M [Gra-Ham-Koh] (cf. Theorem 6.1.39). Theorem 7.1.7. If p e M then (7.1.8)
lw(±Dkp(Q)(wk))\<2,
k>2,
\\w\\ = I ,
lw£T(w).
These estimates are sharp when B is the unit ball of Cn with respect to a p-norm, I < p < oo. Moreover, (7.1.9)
< 2km,
m!
m > 2,
\\w\\ = 1,
where km = mm^m~1^ . Consequently, M is a compact set. Proof. Fix w G Cn with ||w|| = 1 and let lw <E T(w). Also let
1,
C = o.
Since p is normalized and holomorphic on B, it is clear that q is a holomorphic function on U. Moreover, since p € M it follows that Re q(C,} > 0 on U. \a\
Indeed, since law(-) = —- l w ( - ) € T(aw) for each a e C \ {0}, it follows that a
7.1. Starlike mappings
263
for £ 6 U \ {0} we have
and hence Re g(0 > 0 on U. Therefore q e P, and using Theorem 2.1.5 we conclude that (7.1.10) J. g W(o)|< 2 > fc>l. On the other hand, expanding p(C^)/C in a power series in £ we deduce that
Hence by the uniqueness of Taylor series expansions, we conclude that (7.1.11)
q(k-V(Q) = lw±Dkp(Q)(wk)",
k>2.
Using (7.1.10) and (7.1.11), we obtain (7.1.8), as desired. We next prove that these estimates are sharp on the unit ball of C" with respect to a p-norm, 1 < p < oo. To this end, let A G C, |A| = 1, and let p : B —> C™ be given by / \
p(z) =
/
1 i AZl
1 ~T AZn \
V*1 I^M ''''' Z n T^\z~) '
for z = (zi,..., z n ) € B. Then p 6 H(B), p(0) = 0, Dp(0) = I, and it is easy to deduce that Re lz(p(z)} > 0 for z € B \ {0} and lz 6 T(z) (see Section 6.1.6). Hence p € M.. On the other hand, a short computation yields that I k\
t n
for w = (wi,..., wn) e C" and k > 2. Therefore, for w = (1,0,..., 0) and lw € T(w), we obtain '«) ( TTJ
We next prove (7.1.9). For this purpose, let Pm(z) = —-D m p(Q)(z m ). Then 771! Pm(z] is a homogeneous polynomial of degree ra and recalling Lemma 6.1.37, we have (7.1.12) ||Pm|| < km\V(Pm}\,
264
Growth and distortion results for starlike and convex maps
where km = m™/^-1) for m > 2. We recall that \\Pm\\ = sup {\\Pm(z)\\:z€B}, and since Pm is continuous on B, the numerical radius of Pm is just \V(Pm)\ = sup {\lz(Pm(z})\ : lz € T(z\ \\z\\ = l}. Taking into account (7.1.8) and (7.1.12), we easily deduce that |V(P m )|<2,
m>2,
||Pm|| < 2fcm,
m > 2.
and consequently,
We finally prove the compactness of M.. For this purpose it suffices to show the local uniform boundedness of jM, since the fact that M is closed has been proved in Theorem 6.1.39. Indeed, from (7.1.9) we deduce that
\\Pm(z}\\ < r(l + 2 m=2
< r(l + 4 T mr"1-1) = M(r) < T T Oz , ^—' v(1 — r) m=2 '
||«ll < r < 1,
making use of the fact that fcm < 2m for m > 2. This estimate yields the fact that M. is locally uniformly bounded on B, and hence it is a normal family, as claimed. As an application of Theorem 7.1.7 we shall obtain some estimates for the second order Prechet derivative of a normalized starlike mapping (cf . [Koh6] , [Gra-Ham-Koh]). It seems to be more difficult to estimate the higher order Frechet derivatives by this method. Theorem 7.1.8. /// e 5*(B) then (7.1.13) This estimate is sharp when B is the unit ball of C71 with respect to a p-norm, 1 < p < oo.
7.1. Starlike mappings
265
Proof. Let p(z) = [Df(z)]-lf(z), and from (7.1.8) we have
z € B. Then p 6 M since / is starlike,
lw(D2p(Q)(w2))
(7.1.14)
Fix w € C" \ {0}. Since f ( z ) = Df(z)p(z), z £ B, a. simple computation yields that (7.1.15) D2f(0)(w2) = -D2p(0)(w2). To this end, it suffices to expand the mappings f ( z ) and Df(z)p(z) in power series and then to compare the second order terms. Using (7.1.14) and (7.1.15), one deduces (7.1.13), as desired. In order to see that (7.1.13) is sharp on the unit ball of C™ with respect to a p-norm, 1 < p < oo, it suffices to consider /(*) =
Then / is a normalized starlike mapping on B (see Problem 6.2.5), and for w = e\ — (1,0,..., 0) and lw € T(w) we easily deduce that
This completes the proof. By a similar argument, one can also estimate the norm of the second order Frechet derivative of a normalized starlike mapping on the unit ball [Gra-HamKoh]. Theorem 7.1.9. If f e S*(B) then
Proof. Again letting p(z) = [Df(z)]~lf(z), ±D2p(0)(w2)\\<2k2 = 8, A
we have \\w\\ = I ,
\\
from (7.1.9), and together with (7.1.15) this gives the desired bound. For higher order Frechet derivatives of starlike mappings on B, we have the following result due to Poreda [Por5].
Growth and distortion results for starlike and convex maps
266
Theorem 7.1.10. Let f : B —>• Cn be a normalized starlike mapping. Then (7.1.16)
il
for all w G Cn, ||w|| = 1, and k G N, k > 2. Proof. Fix it; € C™ with ||iy|| = 1 and k > 2. Using the Cauchy integral formula for vector-valued holomorphic functions
|C|=r
we deduce that \\f(rei0w]
l
-d8<
where the last step uses the growth theorem for starlike maps (Theorem 7.1.5). Now, if we let i
:, 0
mn
:0
e(k and hence we obtain the desired estimate (7.1.16). The following could be regarded as a generalization of the Bieberbach conjecture to several complex variables. More precisely, it would generalize Theorem 2.2.16. Conjecture 7.1.11. Let / : B —> Cn be a normalized starlike mapping on the unit ball of C n , n > 2. Then
(~Dkf(0)(wk)} \K\
I < /c, k > 2, |H| = 1, lw e T(w}.
/I
Remark 7.1.12. The corresponding estimate for the norm of the fcth order Frechet derivative of a starlike mapping on the unit ball of C n , n > 2, i.e. (7.1.17)
\\w\\ = 1,
fc>2,
7.1. Starlike mappings
267
does not hold in general. To see this, let n = 2 and let C2 be the space of two complex variables with the Euclidean norm. Also let
f(z) = (zi + az|, z2),
z = (zi, z2) € B.
If \a\ = 3V3/2 then / € S*(B) by Problem 6.2.1. However for w = e2 = (0, 1), we have
It is still possible that the norm estimates (7.1.17) are true in the case of the polydisc. Gong [Gon5, Theorem 5.3.1] gave a proof that (7.1.17) holds for k = 2, 3, and formulated the following: Conjecture 7.1.13. If n > 2 and /: P —> C1 is a normalized starlike mapping of the unit polydisc then l-Dkf(0}(wk)\\
fc>2,
Hloo = l.
It follows easily from Theorem 6.3.2 that (7.1.17) is true for all fc € N, k > 2, for maps of the form f ( z ) = Dg(z}z, z € P, where g: P —>• C" is a normalized convex map.
Problems 7.1.1. Show that if h : B(p) ->• C1 is given by L I -v h(z)=
where 1 < p < oo and |A| = 1, then h e M.. 7.1.2. Prove the estimate (7.1.17) for k = 2,3 in the case of normalized starlike mappings of the unit polydisc. 7.1.3. Prove Corollary 7.1.2. 7.1.4. Prove Theorem 7.1.5.
268
7.1.3
Growth and distortion results for starlike and convex maps
A distortion result for a subclass of starlike mappings in Cn
We finish this section with an estimate for the Jacobian determinant of a particular subclass of the normalized starlike mappings on the Euclidean unit ball in C1, n > 2. This result is due to Pfaltzgraff and Suffridge [Pfa-Su3], and they conjectured that it is valid for the full class S*(B). (In this subsection, B denotes the Euclidean unit ball in C1.) First we discuss a procedure from [Pfa-Su3] for generating mappings in S*(B) from mappings in S*. We have already seen one method of doing this in Problem 6.2.5. Theorem 7.1.14. Suppose that g 6 H(B,C) and g(Q) = 1. Let F(z) = zg(z), z € B. Then F e 5*(B) if and only if (7.1.18)
R
Proof. For the purposes of this proof, vectors in C1 will be written as column vectors, and the transpose of such a vector is a row vector. Note that if F € S*(B) or if (7.1.18) holds then g(z) + 0, z e B. It is easy to see that
where L(z) = —r~r~i z & B. Hence using the formula 9\z)
we obtain
From this it follows that
and the condition Re (w(z),z) > 0 is equivalent to Re {1 + L(zYz} > 0, which is (7.1.18). (Moreover, if this condition is satisfied then DF(z) is clearly invertible.)
7.1. Starlike mappings
269
Theorem 7.1.15. For each j = l,...,n, let fj be a normalized starlike n
function on the unit disc U. If Xj > 0 and Y^ Aj = 1, then
(7.1.19)
F(z) = z f[ (&M\ ',
is a starlike mapping on B. Proof. Let
TT (
j
e
z=(z1,...,zn
U, j = l,...,n, and let g(z) =
and L(z) = Dg(z)/g(z), z 6 B. By logarithmic differentiation,
we have r»
7 r > V>n(Zn)J
and hence the condition (7.1.18) becomes
Since V" Aj; = 1, this is equivalent to
Re
This condition is clearly satisfied since /i, . . . , fn € S* and Aj > 0, j = 1, . . . , n, and the proof is complete. Let S*(B) denote the set of starlike mappings F defined by (7.1.19). When n=l,S$(U) = S*, but in higher dimensions S*(B) g S*(B). To see this, let n = 2 and let /(z) = (zi+az$, z2) for z ~ (zi, z2) € B, where 0 < |o| < 3-V/3/2. Taking into account Problem 6.2.1, one concludes that / € S*(B), however
fts:(B}.
Pfaltzgraff and Suffridge [Pfa-Su3] obtained the following bounds for the Jacobian determinant of mappings in S*(B): Theorem 7.1.16. If F e S*(B) then
Growth and distortion results for starlike and convex maps
270
These estimates are sharp.
n
Proof. Let F e S*(B). Then there exist Xj > 0, ^ \j = 1, and normalj=i ized starlike functions fj(zj), j — 1, . . . , n, such that F ( z ) = 2
=
and
+ zL(zY},
z e B,
where
L(zf = (X Note that the n x n matrix zL(zY has proportional columns, and hence its rank is 1. Therefore, JF(Z) = j=i
or equivalently (7.1.21) On the other hand, because fj is starlike on U, j =• 1,..., n, Theorem 2.2.7 gives (7.1.22)
< ^ M1 _
v
Ifj
|-y.
- \\z\
^ • 2 ' Also since -~—^- e"P, ?' = !,...,n, Theorem 2.1.3 implies that
(7.1.23)
1- llzl
7.2. Convex mappings
271
Finally, from (7.1.21), (7.1.22) and (7.1.23), we obtain the bounds (7.1.20), v as desired. Equality holds in (7.1.20) if we take F(z) = -. ^ for z = i 1 C € C/", which (zi,..., zn) € B. To see this, let AI = 1 and /i(C) = 7' "• — rTTTT, (i-O\tyield in (7.1.21) that
Mz) = TT^I Therefore, for r € [0,1) and z = (r, 0, ...,0), we obtain equality in the upper bound in (7.1.20), and for r e [0,1) and z = (—r, 0 , . . . , 0), we obtain equality in the lower bound in (7.1.20). This completes the proof. The problem of finding the sharp bounds for | JF(-Z)| f°r *^e ^u^ c^ass S*(B) is still an open problem in several complex variables. However, Pfaltzgraff and Suffridge [Pfa-Su3] proposed the following: Conjecture 7.1.17. The distortion bounds (7.1.20) hold for all normalized starlike mappings of .0, when B is the Euclidean unit ball of Cn, n > 2.
7.2
7.2.1
Growth, covering and distortion results for convex mappings in several complex variables and complex Banach spaces Growth and covering results for convex mappings
In this section we shall prove a number of growth and covering results for normalized convex mappings on the unit ball of C71 and complex Banach spaces. We first discuss the case of the unit polydisc P of C" (i.e. the unit ball with respect to the maximum norm || • ||oo). Using Suffridge's criterion for convexity of biholomorphic maps of the polydisc (Theorem 6.3.2), it is not hard to show the following (see [Licl] and [Gon4]): Theorem 7.2.1. If f : P —t C" is a normalized convex mapping, then
(7.2.1)
-Mj> J- ~T I I ^ H O O
< H/f,)^ < JMk- „„ P. -1 ~ || Z I|00
272
Growth and distortion results for staxlike and convex maps
These estimates are sharp. Consequently, f ( P ) 2 -Pi/2Proof. Since / is normalized convex, it follows from Theorem 6.3.2 that / has the form f(Z)
= (/I(*l), - - - , fn(zn)),
Z = (Zi, . . . , Zn) € P,
where /i, • • • , /n are normalized convex functions on U. Therefore from Theorem 2.2.8 we have
'~J'
< \f.(z.\\ < _£:
Z 1 -I1 -f U |Zj|I — 'J3\ 3)\ — i1
and thus max
. =
< ~ max
< m a x J^ = ^ i
~l^l
-L-Flloo
as desired. Sharpness follows by considering the mapping / : P —> C71 given by
which is a normalized convex mapping of P. For z = (r, 0, . . . , 0), 0 < r < 1, we have HzHoo = r and ||/(«)||oo = r/(l - r), and for z = (-r,0, . . . ,0), 0 < r < 1, we have ||^||oo = f and ||/(^)||oo = r/(l + r). This completes the proof. When B is the Euclidean unit ball of C", we have the following result obtained with different methods by Suffridge [Su5], FitzGerald and Thomas [Fit-Th], and Liu [LiuT2]. (See also [Lic2] for the upper bound in (7.2.2).) In the next chapter we shall give another proof of this result, using the method of Loewner chains. Kohr [Koh8] gave a two-point version of this theorem. Theorem 7.2.2. Let f : B —>• C" be a normalized convex mapping. Then
These estimates are sharp. Consequently, f(B] ~D Bi/2.
7.2. Convex mappings
273
The above results show that the Koebe constant for normalized convex mappings of the unit ball and the unit polydisc of C™, n > 2, is equal to 1/2, as in the case of one complex variable. We shall give the proof of this result in the context of complex Banach spaces (Theorem 7.2.3). In Section 6.3 we have introduced the class of quasi-convex mappings of type A, as another natural extension to higher dimensions of the normalized convex functions on the unit disc U. Roper and Suffridge [Rop-Su2] showed that the 1/2-growth result, given in Theorem 7.2.2, remains true for this larger class, and Graham, Hamada and Kohr [Gra-Ham-Koh] have recently shown that the result remains true on the unit ball of C" with respect to an arbitrary norm. In fact it is true for normalized starlike mappings of order 1/2, as we shall see in the proof of the theorem below. We now prove the extension of Theorem 7.2.2 to the case of complex Banach spaces. This result was stated without proof by Gong [Gon4, Theorem 4.3.3]. Complete proofs of the growth result for normalized convex mappings in complex Banach spaces were given by Hamada and Kohr [Ham-Koh9] and Chen [CheHB]. The covering result was considered by Hamada and Kohr [Ham-KohlO]. Theorem 7.2.3. Let X be a complex Banach space with norm \\ • \\ and let B be the unit ball of X. If f : B —> X is a normalized convex mapping, then
1 + \\z\\
1 \\z\\
Consequently, f ( B ) D £1/2Proof. Step 1. We begin by observing that the Marx-Strohhacker theorem (a convex mapping is starlike of order 1/2) remains valid on the unit ball of a complex Banach space. To see this, we first remark that the definitions of starlikeness of order 7 (Definition 6.2.12), and of quasi-convex mappings of type A (Definition 6.3.16), extend easily to complex Banach spaces. The proofs of Lemma 6.3.17, Theorem 6.3.18, and Theorem 6.3.19, in which we showed that a quasi-convex mapping of type A is starlike of order 1/2, are likewise valid on the unit ball of a complex Banach space.
274
Growth and distortion results for star like and convex maps The condition of starlikeness of order 1/2 can be expressed using (6.2.8)
as
Equivalently, we may write this as a 6
for all a e *7 \ {0}, u e X, ||u|| = 1 and Zu e T(u). Now fix u e X, ||w|| = 1, and let /u e T"(w). Also let
g(a) =
([D/Cau)]'1/^)),
a
Then p is holomorphic on 17 and g(0) = 1. The above inequality is equivalent to \g(a) - 1| < 1,
aeU,
and since g(0) = 1 we deduce from the Schwarz lemma that (7.2.3)
\g(a) - 1| < |a|,
a € U.
The relation (7.2.3) implies that 1 - H < Re g(a) < I + \a\
on 17,
and hence (7.2.4)
(1 - ||*||)||;z|| < Re { l z ( [ D f ( z } ] - 1 f ( z ) } }
< (I + ||z||)||z||,
for all * G B \ {0} and lz € T(*). Now fix ZQ e B \ {0} and let 7 be the curve in B parametrized by
We easily compute that
7.2. Convex mappings
Since lim
275
= /(zo)5 -77 is continuous on [0,1], and hence there exists a at \dz(t) constant C > 0 such that < C for t € [0,1]. Therefore, for 0 < dt ti < 1, we have at
'*2 dz(t) -dt dt
ll*(*2)|| - W
/
t ,
d It follows that \\z(t)\\ is absolutely continuous on [0,1], and hence — \\z(t}\\ exists a.e. and is integrable on [0,1]. When -r:||z(£)|| exists, it is given by at (7.2.5)
= iRe
{lz(t]((Df(z(t)}}-lf(z(t}}}}
by [Kat2, Lemma 1.3]. Indeed, let t be such that -H dt Re {lz(7)(z(t))}<\\z(t)\\
t=t
exists. Then
-
and
and therefore **{l,Q(z(t)-z®)}<\\z(t)\\-\\z®\\. Dividing both sides of this inequality by t — t and letting t —>• t from above and from below respectively, we obtain (7.2.5), as claimed. We now use the relations (7.2.4) and (7.2.5) to obtain (7.2.6)
(1 -
In view of (7.2.6) we easily deduce that d\\z(t}\\
^ dt
t
d\\z(t}\\
276
Growth and distortion results for starlike and convex maps
and integrating both sides of this inequality between e and 1, where 0 < e < 1, we obtain
*i5b-*iSii^ MOII
Now it is obvious that (7.2.7) is equivalent to the relations
INI
\\z(£)\\
1 + zo and
11*0011
Nil
Using the fact that lim —— = /(ZQ), we obtain the growth result by letting e-»-0+
£
e —> 0+ in the last two inequalities. Step 2. We next prove the covering result, using similar arguments as in [Ham-KohlO]. It is clear that there exists p > 0 such that / maps the ball Bp biholomorphically onto a neighbourhood V of 0. Given w € V \ {0}, there exists a unique z € Bp such that f ( z ) = w. Let r, p < r < 1, be fixed and let to = sup{t > 0 : tf(z) e f(Br)}. Using the growth theorem we see that I < to < oo. Also if we let v(t) = f ~ l ( t f ( z } } , then the convexity of f(Br) implies that \\v(t)\\ < r for 0 < t < to, and we have Since / € K(B) it follows that [Df^t))]'1/^)) e M, and in view of Theorem 6.1.39 we deduce that there exists L = L(r) > 0 such that
Therefore ||dv/dt|| is integrable on [to/2, to)- Next, let si, 52 be such that to/2 < si < «2 < to- Then
7.2. Convex mappings
277
2 dt < —L(r)(s2 - s\) < oo. II dt to Moreover, if {£fc}fceN is an increasing sequence of positive numbers such that tfc —>• to as k —>• oo, then we obtain
\\dv(t)
\\v(tk) - v(tm}\\ ->• 0
as
fc,m-»oo.
Therefore {^(tjOKeN is a Cauchy sequence and there exists a point z'Q € X such that v(tk) —> z'Q as k —>• oo. Since ||v(£fc)|| < r and / is biholomorphic on Br we must have ||ZQ|| = r, and the continuity of / at z'Q gives f(z'Q] = tof(z). Using the first step of the proof, we have ||to/(z)|| > **/(! + r), and thus f(Br) contains the ball Br/(i+ry Letting r /* 1, we conclude that /(.B) contains the ball .61/2, as desired. This completes the proof. Remark 7.2.4. When X is a complex Hilbert space with inner product {•, •), the above result is sharp. To see this, we remark that Roper and SufTridge [Rop-Sul] proved that if / : U —> C is a convex function and u 6 X is a unit vector, then the mapping Fu : B -»• X given by (7.1.7), i.e. Fu(z] = f ( ( z , u))u
is a convex mapping of B. We shall discuss this result in Theorem 11.1.5. If we take the function f(Q = £/(! — £), £ e 17, a straightforward computation using the above mapping yields that Fu(z] = ~ 7 r, 1 - (z, U)
2 € S.
If z = ru where r € [0,1), then \\Fu(z)\\ = r/(l-r) = \\z\\f(1 - \\z\\), while ifz = -ru, r e [0,1), then ||FU(«)|| = r/(l +r) = \\z\\/(I + \\z\\). On the other hand, since —w/2 ^ ^u(-B), the constant 1/2 cannot be replaced by a larger constant in the covering result. This completes the proof. Remark 7.2.5. Various results obtained in the finite dimensional case are special cases of Theorem 7.2.3. Gong and Liu [Gon-Liul] obtained the growth theorem for convex mappings of the unit ball B(p) with respect to a p-norm, 1 < p < oo. Also Hamada [Ham2] obtained the upper bound in the growth result of Theorem 7.2.3 when X = C71 with respect to an arbitrary norm.
278
Growth and distortion results for starlike and convex maps
Let 0 be a bounded balanced convex domain in C™ . Then the Minkowski function of 0 is a norm on Cn and 0 is the unit ball of Cn with respect to this norm. Therefore, Theorem 7.2.3 holds for f2. Liu and Ren [Liu-Ren4] obtained this result, and the corresponding covering theorem, in the case when the Minkowski function is Cl except for a lower dimensional manifold in fi. Their theorem is also presented in [Gon4]. In particular, this result applies to the complex ellipsoid -B(pi, . . . ,p n ) where pi, . . . ,pn > 1.
7.2.2
Covering theorem and the translation theorem in the case of nonunivalent convex mappings in several complex variables
We now come back to the case of convex mappings on the Euclidean unit ball B of Cn. We have seen that the Koebe constant for normalized convex mappings of the ball B is 1/2. This result may be deduced as a direct consequence of the lower estimate (7.2.2). There is also a more general covering result [Gra4] which does not require univalence. To state it, let 17 denote the convex hull of a set fJ C C™ . Theorem 7.2.6. Let f : B —> Cn be a normalized holomorphic mapping. Then f ( B ) contains the ball BI/^. Proof. Let u e dB and let TTU denote the orthogonal projection of C71 onto the 1-dimensional subspace of C71 generated by u. Let Vu(C) = Cw and let (7.2.8)
p(C)=^«107r«o/(Cti),
C€C/.
Then g is holomorphic on U, g(0) = 0 and g'(0) = 1, since / is normalized. Therefore, using the result of Theorem 2.2.10, we deduce that g(U) contains the disc C/i/2- Moreover, by (7.2.8), ^u(g(U}) = (^ u (p(C/))f C 7ru(/(5)). Theorem 7.2.6 now follows from the following elementary result [Gra4], by choosing Lemma 7.2.7. Let fi be a convex set in Cn which contains a neighbourhood o/O. Assume Q has the following property: whenever u e dB, 7r u (fi) contains the disc centered at zero and of radius 1/2 in the subspace Cu. Then fi contains t h e ball B .
7.2. Convex mappings
279
Proof. Let WQ 6 d£l be a point at minimum distance from 0. Let H be a supporting hyperplane through WQ. We may assume that WQ = (r, 0, . . . , 0), where r > 0, because otherwise we can use a unitary transformation to reduce to this situation. The equation of H is therefore Re w\ = r, since H must be perpendicular to WQ. Since £1 is a convex set we have Re w\ < r, for all w e fi. The orthogonal projection of Cn onto the subspace generated by WQ is (ii>i, . . . , wn) !->• (it>i, 0, . . . , 0). Hence all points in the image of fi under this projection satisfy Re w\ < r. In view of the hypothesis we now conclude that r > 1/2. This completes the proof. There is a refinement of this result in the presence of fe-fold symmetry, or of the following weaker notion of symmetry [Gra- Var2] : Definition 7.2.8. Let / : B —> C™ be a normalized holomorphic mapping such that f(B) is a convex domain. We say that / has critical-slice symmetry of order k if there is a point w € d f ( B ) at minimum distance from 0 such that, on setting a = iy/||u;||, the function (£) = {/(Co), a) has symmetry of order k. The following result, obtained by Graham and Varolin [Gra-Var2], is a generalization to higher dimensions of Theorem 2.2.15. Theorem 7.2.9. Let f : B —» C™ have critical- slice symmetry of order k. Then f ( B ] D BTk, where
(7 2 9)
--
rfe
This result is sharp. Proof. Let w e df(B) be a point which satisfies all of the conditions of Definition 7.2.8, and let a = w/\\w\\. If <^>(C) = ( f ( ( a ) , a ) , then by hypothesis 0 is fc-fold symmetric. Let 7ra denote the orthogonal projection of C™ onto the one-dimensional subspace Ca through 0 containing a. By Theorem 2.2.15, <j>(U} 2 Urk and since $(U) C 7T0(/(B)), we deduce that 7T0(/(B)) 2 Urk. Let H be the (unique) supporting hyperplane at w. Then a _L H, and hence 7ra(H) is a line. Since ^a(w) = «;, we must have ||ty|| > r*fc. The sharpness of the radius r* follows from the extension theorem of Roper and Suffridge which will be discussed in Theorem 11.1.2: i f / : U —>• C is a convex function such that /(O) = /'(O) — 1 = 0, then there exists a normalized
280
Growth and distortion results for starlike and convex maps
convex mapping F : B ->• C1 such that F(z\, 0 , . . . , 0) = (/(zi), 0 , . . . , 0). (Also if / is fc-fold symmetric then so is F.) This completes the proof. Prom Theorem 7.2.9 we conclude that the Koebe constant for normalized convex mappings of B with fc-fold symmetry is equal to r^ given by (7.2.9). In particular, for odd convex mappings this constant is 7T/4. We next discuss a theorem of Graham and Varolin [Gra-Varl] about the translations of the image of a convex map of B. It is an analog of a theorem of MacGregor [Mac2,4], which in one variable holds in much greater generality than for convex functions. MacGregor [Mac2] (see also [Mac4]) proved the following: Theorem 7.2.10. Suppose that f e H(U) and that the Taylor series expansion of f has the form f ( z ) = ao + zk + O(|z| fe+1 ). Let £) = f ( U ) . Then any translation of fi through a distance less than ?r/2 has nonempty intersection with fi. In particular the theorem holds for all functions in 5. A conformal map onto an infinite strip of width ?r/2 is extremal. In dimension greater than one the theorem cannot hold for all mappings in 5(J5), because of the existence of disjoint Fatou-Bieberbach domains which are translates of each other [Ros-Rud]. However, it does hold for mappings in K(B). To prove this, we need to recall some properties of convex sets. Let M be a convex set in Rm (not necessarily compact) and let u be a unit vector in R m . There are two natural notions of the width of M in the direction of u. Definition 7.2.11. (i) WM(U) = sup |ci— c%\, where the hyperplanes x-u = d, i = 1,2, have nonempty intersections with M. (ii) dM(u) = sup l(o-X}U n M), where crZ)U = {tx + u : t € M} and I is the x6Rm
Lebesgue measure in Em. We also introduce Definition 7.2.12. WM = inf WM(U), DM = inf C£M(W). ||u||=l
||u||=l
If WM or DM is finite, the corresponding infimum is actually a minimum. When M is a compact convex set we have DM — WM, as shown in [Val, Theorem 12.18]. An elementary argument [Gra-Varl] shows that this remains valid for arbitrary convex sets. Lemma 7.2.13. DM =
7.2. Convex mappings
281
Proof. It suffices to prove this equality for an arbitrary closed convex set M. For 0 < r < oo, let B? = {x € Rm : \\x\\ < r} and let Mr = M n B^. Clearly DMT /* DM and WMT /* WM as r —> oo. Since DMT = WMT for all r, we deduce that DM = WMWe are now able to prove the analog of MacGregor's result [Mac2] as obtained by Graham and Varolin [Gra-Varl]. Theorem 7.2.14. Let f : B -> C™ be a holomorphic mapping such that .D/(0) = / and fi = f ( B ) is a convex domain. Then any translation of $1 through a distance less than ?r/2 meets £1. Proof. We may assume /(O) = 0. Suppose 6 e C1 is such that ft n (ft+6) = 0. Then Wn = Da < \\b\\. Let u e dB be such that wn(u) = WQ. Also let Cu denote the one-dimensional complex subspace of C71 generated by w, and let TTU be the orthogonal projection of C™ onto Cu. Let Ucu denote the unit disc in Cu and let g = TTU o f\uCu. Then g may be regarded as a function of one variable from the unit disc U to C. Moreover, g(Q) = 0 and 7T/2, and thus we conclude that ||6|| > Tr/2. Open Problem 7.2.15. Does the translation theorem (with the constant 7T/2) hold for other classes of maps in C71, n > 2, in particular starlike maps? This problem was formulated by Graham and Varolin [Gra-Varl].
7.2.3
Bounds for coefficients of convex mappings in C1 and complex Hilbert spaces
We now turn to the study of coefficient bounds and distortion theorems for convex mappings in higher dimensions. The known results are somewhat more complete than for starlike mappings, but there still remain a number of unsolved problems. We begin with the unit ball B(p) in C1 with respect to a p-norm, 1 < P < oo, and we give an estimate for the fcth Prechet derivative of a normalized convex mapping at the origin which was proved by Gong and Liu [Gon-Liul]. In the Euclidean case the result was also obtained by Liczberski [Lic2], rediscovered by Kohr [Koh6], and obtained in a slightly stronger
282
Growth and distortion results for starlike and convex maps
form by Pfaltzgraff and SufTridge [Pfa-Su4]. We mention that FitzGerald and Thomas [Fit-Th] gave coefficient bounds for normalized convex mappings of the Euclidean unit ball of Cn that involve combinations of partial derivatives different from those of the homogeneous polynomials Ak(z] which occur in Theorem 7.2.16. Also Liczberski [Licl] obtained sharp bounds for the coefficients of normalized convex mappings of the unit polydisc in C™. Our argument is similar to [Pfa-Su4, Theorem 5.1]. OO
-
Theorem 7.2.16. Let f ( z ) = z+^^ —Dkf(Q)(zk)
be a normalized convex
mapping on the unit ball B(p) with 1 < p < oo. Then (7.2.10)
\\^Dkf(0)(wk)\\
<1,
fc€N,
k>2,
\\w\\p = 1.
Proof. Let Ak(z) = —Dkf(0)(zk) for k > 2. Then Ak(z) is a homogeneous K• polynomial in z of degree k. Now fix k and let gk '. B(p) —> C™ be given by
Since / is a convex mapping on B(p) it follows that gk(B(p}) C f(B(p)) and hence if v(z) = f ~ l ( g k ( z ) ) for z £ -B(p), then v is a holomorphic mapping from B(p) into B(p) such that v(0) = 0. We conclude from the Schwarz lemma (see Lemma 6.1.28) that
M*)||p < ll^llp, z e B(p). Further, using the Cauchy estimates for Schwarz mappings on B(p), we deduce that (7.2.11) ±Dkv(Q)(zk)
71
Next, let z E C , \\z\\p = 1, and consider gk((3z) _
f(v((3z)}
7.2. Convex mappings
283
This function is holomorphic on the unit disc, since f3 = 0 is a removable singularity. Letting ft —>• 0 in the above, we deduce from (7.2.11) that \\Ak(z) II
p
This completes the proof. In the Euclidean case or in a Hilbert space, one may strengthen the conclusion of Theorem 7.2.16 using a result of Hormander on the norm of polynomials defined on Hilbert spaces. We shall briefly consider polynomials and multilinear mappings in complex vector spaces in order to explain Hormander's result. The basic source is [Horl]. For more details about this subject, the reader may consult [Fra-Ve], [Hil-Phi] and [Muj]. Let V be a vector space over C. Definition 7.2.17. A function A : V —> C is called an (abstract) homogeneous polynomial of degree k if for any z, w e V, A(sz + tw) is a homogeneous polynomial of degree k in s, t G C in the algebraic sense. Obviously, if the dimension of V is finite, the abstract and the algebraic definitions of a polynomial coincide. We have [Horl]: Lemma 7.2.18. To each homogeneous polynomial A(z] of degree k in V there exists a unique function A(z^\... ,z^) with values in C, which is defined whenever z® € V, i — 1,..., k, and has the following properties: (i) A(z^\... , z( fc )) is linear in z^l> if the other arguments are fixed, i = !,...,«/ (ii) A(z^,..., z^) is symmetric in its arguments; (Hi) A(z,..., z) = A(z). The function A(z^\..., z^), defined in Lemma 7.2.18, is called the kthpolar form of A(z). We remark that Definition 7.2.17 and Lemma 7.2.18 can be easily extended to the case when the polynomial is defined in V with values in another vector space over C. For further information, see [Horl]. See also [Hil-Phi, Chapter 26]. We can now give the basic result of Hormander [Horl] about the norm of polynomials defined in Hilbert spaces. Theorem 7.2.19. Let X be a complex Hilbert space with inner product
284
Growth and distortion results for starlike and convex maps
{•,•) and the induced norm \\z\\i = \J(z,z}. Also let Y be a complex Banach space with respect to a norm \\ • \\2- If A: X -> Y is a homogeneous polynomial of degree fc, then P(*)»2
Using Hormander's result, Pfaltzgraff and SufFridge [Pfa-Su4] obtained the coefficient bounds in the next theorem for the case of the Euclidean unit ball in C71. Hamada and Kohr [Ham-Koh9] gave the generalization to the Hilbert space case. Theorem 7.2.20. Let X be a complex Hilbert space with the inner product {•, •), and let B denote the unit ball of X with respect to the induced norm || • ||. Let f : B —I X be a normalized convex mapping. Then
k € N,
< 1,
(7.2.12)
k > 2.
These estimates are sharp. Consequently, lUllfc
ze
fc e N.
Proof. For z € B and fc > 2 we define Ak(z), Qk(z), and v(z) analogously to the proof of Theorem 7.2.16. Then Ak(z] is a homogeneous polynomial in z of degree fc. Also v is a Schwarz mapping and (7.2.13)
\\±-Dkv(Q)(wk)
< 1,
w € X, \\w\\ = 1, fc > 2.
1 1 /V •
Since
and
in a neighbourhood of zero, we conclude that
7.2. Convex mappings
285
Prom (7.2.13) and the above equality we deduce that (7.2.14)
T-.Dkf(Q)(wk)\\ Kl
II
< 1, weX,
\\w\\ = 1.
Now, if we apply Theorem 7.2.19 to the multilinear mapping — Dkf(Q), we obtain Kl
k -Dm= sup l ?*/ ( 01 ) ( )« , . . . , z ) = sup Dk f ( 0 ) k( z ) . ife! II ,«)|| =1 llfc! "I ||*||=i " «! II
Consequently, from (7.2.14) and the above equality we conclude (7.2.12), as desired. Next we show that the bounds (7.2.12) are sharp. For this purpose, let u be a unit vector in X and define Fu : B —> X by
Then Fu is a convex mapping, as will be shown in Theorem 11.1.5 (see also Remark 7.2.4). In addition, it is easy to see that
for all w G X, \\w\\ = 1 and k G N, k > 2. Hence for w = u in the above equality, we deduce that
On the other hand, since / is normalized, it has the power series expansion
fc=2
Also since / is bounded on each ball Br, 0 < r < 1, the above series converges uniformly on any such ball (see [Muj, Theorem 7.13]), and we deduce that oo
Df(z)(z)
= z + ^ kAk(z),
z€B.
k=2
(The term-by-term differentiation of power series in infinite dimensional spaces is discussed in [Din2, p. 150-151].)
286
Growth and distortion results for starlike and convex maps Further, it is easy to prove by induction that oo
Dmf(z)(zm}
= £ k(k - 1) • • • (k - m + l)Ak(z], k=m
for all z e B and m > 1. Consequently, from (7.2.12) we obtain
k=m
^
" "'
as asserted. This completes the proof. Some interesting consequences of Theorem 7.2.20 including estimates for individual coefficients have been obtained by Kohr [Koh6] in the case of two complex variables. Also see [Fit-Th] and [Gon4].
7.2.4
Distortion results for convex mappings in C™ and complex Hilbert spaces
In this section we derive upper and lower bounds for ||.D/(2)|| for normalized convex mappings on the unit ball of a complex Hilbert space X. The upper bound is a simple consequence of Theorem 7.2.20 (see [Ham-Koh9]). However, the lower bound is more difficult and its proof uses an ingenious argument of Gong and Liu, who obtained the corresponding result in the finite dimensional case for any norm which satisfies a certain smoothness condition ([Gon-Liu2]; see also [Gon4]). On the Euclidean unit ball in C1, Pfaltzgraff and Suffridge [Pfa-Su4] obtained the upper estimate for ||D/(,z)|| and a lower estimate for ||£)/(2)(z)|| as a consequence of a distortion theorem for linear-invariant families of given norm order. The class of normalized convex mappings of B is a linear-invariant family of norm order 1. When X = Cn, n > 2, the upper bound in (7.2.17) is attained by the Cayley transform, but the lower bound in (7.2.17) is not sharp (see [Lic-St3]). We shall come back to these results in Chapter 10. The generalization of the lower estimate for ||D/(z)|| to complex Hilbert spaces was obtained by Hamada and Kohr [Ham-KohlOj. For the proof of this theorem we need to introduce the infinitesimal Caratheodory and Kobayashi-Royden metrics.
7.2. Convex mappings
287
Definition 7.2.21. Let £) be a domain in a complex Hilbert space X. For z € fi and v € X, the infinitesimal Caratheodory metric is defined by (7.2.15)
EQ(z,v) = sup [\Dg(z)v\ : g € H(fl,U),
g(z) = o}.
The infinitesimal Kobayashi-Royden metric is defined by (7.2.16)
Fn(z,v) = inf ||a| : 3g € H(U,ti) such that
(Strictly speaking, these are pseudometrics, i.e. they may not be positive definite.) A basic property of these metrics is the contractive property under holomorphic mappings. That is, if fi, fi' are domains in X and h: 17 —>• £2' is a holomorphic mapping, then
Ew(h(z),Dh(z)v) < £n(z,u), and similarly for the infinitesimal Kobayashi-Royden metric. It follows that these metrics are preserved by biholomorphic mappings. Using this fact it is not hard to compute these metrics on the unit ball B of X and to show that they coincide on B (cf. [Fra-Ve]). (They also coincide on any convex domain in X by a basic result of Lempert (see [Leml,2]), extended to infinite dimensions by Dineen, Timoney, and Vigue [Din-Tim-Vig].) For further details about invariant metrics, see [Fra-Ve], [Gral], [Jar-Pf], [Kob], [Kran], [Leml], [Pol-Sh]. Lemma 7.2.22. EB(z,v)* = FB(z,v? = J^L + (^\^}2Proof. For any be B, let
where s^ = y^l — ||6||2. We have seen in Chapter 6 that up to multiplication by unitary transformations of X, the biholomorphic automorphisms of B are of the form
288
Growth and distortion results for starlike and convex maps
where
Since &(&) = 0 and the Caratheodory differential metric is a biholomorphic invariant, we deduce that
On the other hand, since EB(O,V) = \\v\\ and sb
we obtain the desired result for EB- A similar argument holds for FB. Theorem 7.2.23. Let f : B —>• X be a normalized convex mapping. Then
on B
-
Proof. First we prove the upper estimate. Since / is normalized, it has the power series expansion
Note that the above series converges uniformly on any ball Br, 0 < r < 1, and
Since — Dkf(Q)(zk~l, •) is a continuous linear operator, as the restriction Ki •
of —Dkf(0) to { ( z , . . . ,z)} x X, we conclude in view of (7.2.12) that fc— 1 times
jjDkf(Q)(zk-\.)\\<
\\z\f~1
on B.
Therefore we obtain 00
k=2
^
" "^
7.2. Convex mappings
289
We now give the proof of the lower estimate in (7.2.17), using a similar method as in [Gon-Liu2] (also see [Gon4]). Fix z € Br, 0 < r < 1. Let t0 = sup{t > 0 : tf(z) € f(Br)}. Prom Theorem 7.2.3 we deduce that 1 < to < oo. Moreover, using similar arguments as in the proof of Theorem 6.3.8, it is not difficult to see that there exists a point z1 € dBr such that f ( z ' ) = t 0 f ( z ) . Let p, = I/to and z(t) = f ~ l ( t f ( z ' } ) for 0 < t < 1. Also let «'»
0<*<1
Since || • ||2 is of class C°° and z(t) is a Cl mapping from [0, 1] to B, it follows that g is of class C1 on (0,1]. It is also clear that g is continuous on [0,1]. We prove that g is decreasing on [0,1] (cf. [Liu-Ren4]). A simple computation yields that
On the other hand, from (7.2.4) we have (1 - \\z\\)\\z\\* < ^ ((Df(z)]-lf(z),
(7.2.18)
\z\\)\\z\\2,
z}
zeB.
Also
t\\Z(t)\r Prom (7.2.18) and (7.2.19) we deduce that g'(t) < 0 on (0,1), and hence g is decreasing on [0,1]. This implies that
and hence
\\z\ 1+r
290
Growth and distortion results for starlike and convex maps
which may be rewritten as
722
( - - o)
i-Ai> r'^L
Now, let G(y) = f ~ l ( ( l - A*)/(y) + A*/ (*'))> for V e B- Since / is a nor' malized convex mapping, G is biholomorphic on B, G(B) C B and G(0) = z. Also, since the Caratheodory differential metric is invariant under biholomorphic mappings, we have EG(B)(ztDG(Q)v) = £B(O,V),
v € X.
Further, since G(B) C 5, the contractive property of the Caratheodory differential metric gives EG(B)(z,DG(0)v) > EB(z,DG(0)v). Thus (7.2.21)
EB(z, DG(0)v) < EB(Q, v).
Using the definition of G, we have
and therefore from (7.2.21) we have (l-n)EB(z,[Df(z)]-lv)<EB(Otv). Setting v = Df(z)£ in this inequality, we obtain
From (7.2.20) and this relation we conclude that r(l + ||2!|
and letting r —> 1 we obtain (7.2.22)
±=1
7.2. Convex mappings
291
Prom Lemma 7.2.22 it can be seen that for fixed z ^ 0 and for \\v\\ = 1, EB(Z,V) is maximized when v = e*ez/||2:||, 6 € R. Using this observation and the inequality (7.2.22), we finally deduce the relation
The estimates (7.2.23) below are immediate consequences of Theorems 7.2.23 and 7.2.26. In the finite-dimensional case these estimates were obtained by Gong and Liu [Gon-Liu2] (see also [Gon4]), on the unit ball with respect to any norm which is Cl except for a lower dimensional manifold. Pfaltzgraff and Suffridge [Pfa-Su4] also obtained these bounds on the Euclidean unit ball in C", and formulated (the finite-dimensional version of) Conjecture 7.2.25. (See also [Lic3] in the case of the maximum norm in C71.) We present these results in the context of complex Hilbert spaces. Corollary 7.2.24. Let f : B —>• X be a normalized convex mapping. Then (7-2.23)
,, ,^1
< \\Df(z)z\\ < ,, H* 1 '
,
z£B.
Proof. The upper bound follows directly from Theorem 7.2.23. The lower bound follows from Theorem 7.2.26 for £ = z/||z||, z € B \ {0}. Conjecture 7.2.25. If /: B —t X is a normalized convex mapping of the unit ball B in a complex Hilbert space X then
\\Df(z)v\\>
a.
,
.
V 1 ~T~ I I Z I I /
One may also formulate a distortion theorem which takes account of the behaviour of Df(z) in different directions at z by using the infinitesimal Caratheodory or Kobayashi-Royden metric. In the finite-dimensional case this result was obtained by Gong and Liu [Gon-Liu2] for convex mappings on bounded convex circular domains in Cn. The generalization to Hilbert spaces is again due to Hamada and Kohr [Ham-KohlO]. We omit the proof and we leave it for the reader. Theorem 7.2.26. Let f : B -> X be a normalized convex mapping. Then
292
Growth and distortion results for starlike and convex maps
for all z e B and £ e X. For the case of the Euclidean unit ball in C1 , there is an earlier reformulation of this theorem due to Gong, Wang, and Yu [Gon-Wa-Yu3] (see also [Gon4]). Again we leave the proof for the reader. Theorem 7.2.27. Let f : B —> C71 be a normalized convex mapping. Then the following estimates hold for all z = (zi, . . . , zn) e B:
Here G is the matrix of the Bergman metric of the unit ball B (up to a constant factor), i.e.
( 1
V
_ | Uz| | 2 \ 2 \\ \\ )
and 6j}k has the value 1 when j = k and is 0 when j ^ k. To see that this result is indeed a reformulation of Theorem 7.2.26 in the finite-dimensional case, it suffices to observe that the square of the infinitesimal Caratheodory or Kobayashi-Royden metric gives the metric with matrix G(z). The extent to which this result is sharp is discussed in [Gon-Wa-Yu3] and [Gon4]. Gong and Liu also gave estimates for the eigenvalues of Df(z) in the finite-dimensional case ([Gon-Liu2], [Gon4]). We conclude this section with a two-point distortion theorem that provides a necessary and sufficient condition for univalence on the Euclidean unit ball of C71, using properties of the Caratheodory distance (see [Koh8]). For other results of this type, the reader may consult [Gon4, Section 7.5]. If fi is a domain in Cn , then the Caratheodory distance is defined by =
sup dh(g(z),g(w)),
z,weft,
where dh is the hyperbolic distance on U. For further information about the Caratheodory distance, the reader may consult [Jar-Pf], [Kob], [Kran], [PolSh], Theorem 7.2.28. Let f : B —>• C" be a convex mapping. Also let
=
(l-\\z\\2)\\[Df(z)}-ll\\-1
7.2. Convex mappings
293
Then (7-2.24)
ll/(a)-/(6)||
for all a, 6 e B. Conversely, letf:B—*Cnbea locally biholomorphic mapping which satisfies (7.2.24)- Then f is univalent on B. Proof. Fix 6 € B and define F : B -> C1 by
where & is given by (6.1.9), i.e.
Since F is normalized convex, Theorem 7.2.2 gives
and since ~ 11*1
we obtain i[l -exp(-2Ck(z,0))] < ||F(z)|| < For 2; = (0_b)~ 1 (a) = 06(a), we obtain - exp
-
< \\[D
- f(b))\\
< i [exp (2Cs((0_6)-1(a), (^-ty))) - l] . On the other hand, since [^^-^(O)]"1 = D6(6) and the Caratheodory distance is invariant under biholomorphic mappings, we obtain from the above that i(l - exp(-2CB(a,&))) < \\[D
- f(b))\\
294
Growth and distortion results for starlike and convex maps
Moreover, it is not difficult to check that IKIty.^O)]"1!! = 1/(1 - ||6||2) (see (10.2.7)), and since
the relation (7.2.24) easily follows, as desired. Conversely, assume that / is locally biholomorphic on B and satisfies (7.2.24). Let a, b € B such that /(a) = /(&). In view of (7.2.24) we deduce that exp(2Cs(a, &)) — 1 or V\f(a) — T>if(b) — 0. However, the latter condition cannot hold since / is locally biholomorphic. Thus we must have a = b. This completes the proof. Problems
7.2.1. Prove Theorem 7.2.26. 7.2.2. Prove Theorem 7.2.27.
Chapter 8
Loewner chains in several complex variables 8.1
8.1.1
Loewner chains and the Loewner differential equation in several complex variables The Loewner differential equation in C1
In this chapter we shall consider the generalization of the Loewner differential equation to n dimensions. This subject was first studied by Pfaltzgraff [Pfal]; later contributions, refining Pfaltzgraff's results in C" and permitting generalizations to the unit ball of a Banach space, were made by Poreda [Por3]. We also include recent improvements in the existence theorems for the Loewner equation which depend on the fact that the class .M is compact [Gra-HamKohj. We give various applications, including univalence criteria and characterizations of subclasses of S(B), such as spirallike and close-to-starlike maps. Finally, we shall obtain some quasiconformal extension results to C" of univalent mappings which can be embedded in Loewner chains. We shall see that in several variables the class S°(B) of mappings which admit a parametric representation remains well-behaved, and satisfies growth theorems and coefficient estimates. For the case of the polydisc these mappings were studied by Poreda [Porl-2]; on the Euclidean ball they were studied 295
296
Loewner chains in several complex variables
extensively by Kohr [Koh9]; and they have recently been studied on the unit ball with respect to an arbitrary norm by Graham, Hamada, and Kohr [GraHam-Koh]. Somewhat surprisingly, we shall see that in higher dimensions there exist univalent mappings which can be embedded in Loewner chains, but which do not admit a parametric representation. We shall work on the Euclidean unit ball B of Cn, and we begin this section by giving some basic facts about subordination and Loewner chains in this context. As usual, by {•, •) we denote the inner product on C71 and by || • || the induced norm. Also let H(B) be the set of holomorphic mappings from B into C1. Definition 8.1.1. Let /, g e H(B). We say that / is subordinate to g (f -< g) if there exists a Schwarz mapping v on B such that f ( z ) — g(v(z}}, z e B. As in the case of one variable, if / -< g then /(O) = e/(0) and f(B] C g(B}. Also if g is univalent on B, then / -< g if and only if /(O) = g(Q) and f ( B ) C g(B). Furthermore, if / -< g then f ( B r ) C g(Br) for each r, 0 < r < 1, and ||£>/(0)|| < \\Dg(Q)\\. In this chapter, if g is a mapping which depends holomorphically on z € B and also depends on other (real) variables, we shall write Dg(z, •) for the differential of g in the z variable. Definition 8.1.2. A mapping / : B x [0,00} —>• Cn is called a subordination chain if it satisfies the following conditions: (i) /(-,«) 6 H(B) and Df(Q,t) = y?(*)I, t > 0, where
8.1. The Loewner differential equation
297
somewhat restrictive, but without it the chain cannot necessarily be normalized. Also as in one variable, we shall see that a normalized Loewner chain satisfies a strong regularity condition in t: it is locally Lipschitz in t locally uniformly with respect to z. This normalization forces the normalization £)v(0, s, t) = es~*J, 0 < s < t < oo, on the Schwarz mapping v(z, s,i). Moreover, using the assumption (ii) of Definition 8.1.2, it is not difficult to see that if f ( z , t ) is a Loewner chain, then the mapping v(z,s,t), given by the equality f ( z , s ) = /(v(z,s, £),£), satisfies the semigroup property v(z, s, n) = v(v(z, s, £), £, u),
z e £,
0 < s < t < u < oo.
This property combined with the fact that ||v(z,s,t)|| < ||z|| implies that ||v(z,s,u)|| < ||«(*,a,t)||,
zeB, 0<s
and thus \\v(z, s, -)|| is a decreasing function on [s, oo). Moreover, the univalence of /(•,£) implies that v(-,s,t) is uniquely determined and is also univalent on B. Recall the class M given by M = {h e H(B) : /i(0) = 0, Dh(fy = I, Re (h(z), z) > 0, z 6 B \ {0}}, which is related in higher dimensions to the Caratheodory class of functions with positive real part in the unit disc. This class will play a fundamental role throughout this chapter. As we have seen in Theorem 6.1.39, the class M. is 4?" compact, and for each r € (0,1) there is a number M = M(r) < — ^ such that \\h(z)\\ < M(r) for \\z\\ < r and h e M, by the proof of Theorem 7.1.7. The basic existence theorem for the Loewner differential equation on B is due to Pfaltzgraff [Pfal, Theorem 2.1]. His assumption concerning the boundedness of h(z, t) can be omitted since it follows automatically from the compactness of M. (Pfaltzgraff assumed that for each r € (0,1) and T > 0 there exists a number K = K(r,T) > 0 such that ||/i(M)ll < K(r,T} for \\z\\ < r and 0 < t < T.)
298
Loewner chains in several complex variables
Later Poreda [Por3] studied the initial value problem (8.1.1) on the unit ball of a Banach space under the assumption that the mapping h is bounded on Br x [0, oo) for each r e (0,1). This condition is at first sight more restrictive than Pfaltzgraff's, but again it is automatically satisfied in view of Theorem 6.1.39 or Theorem 7.1.7. The reader may compare the result below, due to Pfaltzgraff [Pfal], with Theorem 3.1.10. Theorem 8.1.3. Let h : B x [0, oo) —>• C1 satisfy the following conditions: (i) h ( - , t} e M for each t > 0; (ii) for each z € B, h(z, t) is a measurable function o f t e [0, oo). Then for each s > 0 and z G B, the initial value problem
(8.1.1)
dv — = — h(v,t), c/1
a.e. t > s,
v(s) = z,
has a unique locally absolutely continuous solution v(t] = v(z, s, t) = es *z + .... Furthermore, for fixed s and t, 0 < s < t < oo, vsj(z} — v(z,s,t) is a univalent Schwarz mapping, and for fixed s > 0 and z £ B it is a Lipschitz function oft>s locally uniformly with respect to z. Proof. As in the proof of Theorem 3.1.10 we shall apply the classical method of successive approximations to construct the solution. Fix s > 0 and r 6 (0,1). We shall show that if z € Br, then (8.1.1) has a unique solution on any interval [s, T] with T > s. Estimating with Cauchy's integral formula in several variables and using the fact that for each p <E (0,1) there is K = K(p) > 0 such that ||Mz,£)|| < K(p), z e Bp, t > s, we deduce the existence of a constant M = M(r) > 0 such that (8.1.2)
\\h(w,t)-h(z,t)\\<M(r)\\z-w\\
for z, w E Br and t > s. Let R = (1 + r)/2, and let
8.1. The Loewner differential equation
299
Fix z € Br and consider the following successive approximations vo(t) = v0(z,t) = z,
,
(8-1-3) vm(t) = vm(z,t) = z- I
Js
h(vm-i(z,r),
on s < t < s + c. If we assume that vm-i(z, t) 6 BR for s < t < s + c, then we obtain IMM) - z\\ < K(R)(t -s)< K(R)c < (1 - r)/2, and therefore ||vm(^,t)|| < R < 1 (m = 0, 1, . . .). Since vo(z,t) = z € Br C BR, we see by induction that the successive approximations are well defined. Using (8.1.2) and the locally uniform boundedness of A4, we obtain by induction that \\vm(z,t) - i;m_iGM)|| when s < t < s + c and ||^|| < r. Hence the mapping (8.1.4)
v(t) = v(z,t)= lim vm(z,t]
is well denned and continuous on s < t < s + c. Also since the convergence is uniform on Br for fixed t, v(z, t) is holomorphic as a function of z. Clearly v(z,t) 6 BR when z € Br and s < t < s + c. Moreover, by using (8.1.2), (8.1.3), and (8.1.4) we deduce that v(z,i) satisfies the integral equation ft v(z,t) = z— I h(v(z,T),
(8.1.5)
Js
on s < t < s + c. This equation, the hypothesis (ii) and the boundedness of ||/i(2,t)|| on BR x [s,oo) show that v(z,t) is a Lipschitz continuous (hence absolutely continuous) function of t € [s, s + c] uniformly with respect to z € J3r, and that r\
(8.1.6)
-£(z,t) = -h(v(z,t},t], ot
a.e. te[s,s + c], v(z,s) = z.
We shall prove that ||u(z, t)\\ < \\z\\ for s < t < s+c. Clearly, if we establish this relation on [s, s + c] then the solution of the initial value problem (8.1.6)
300
Loewner chains in several complex variables
can be continued to s+c < t < s+2c with the initial condition v(t) = v(z, s+c) when t = s + c, by applying the method of successive approximations with the same constants r, R, c, K(R), M(R). By iterating this procedure we may extend the solution to the full interval [s, T] in a finite number of steps. Since T is arbitrary, the above arguments show the existence of a solution to the initial value problem (8.1.1) on s < t < oo. This solution will be a Lipschitz continuous function of t locally uniformly with respect to z e B, and holomorphic in z G B. Thus our objective is to prove that (8.1.7)
HM)|| < INI on [s,s + c].
Since v(z, •) is a Lipschitz continuous function of t € [s, s + c], the same is true of ||i>(2, -)||. Therefore d||i>(z,t)||/c?£ exists a.e., is an integrable function and
(8.1.8)
|Wz,t)||= ||z|| + y"5tfclldT,
s
+ c.
On the other hand, because
we may substitute from (8.1.6) and use the fact that /i(-, t) € M. for t > 0, to deduce that =
_ Re kvz,t,t,vz,t
< 0.
Therefore, d\\v(z,t)\\/dt < 0 a.e. on [s,s + c], and from (8.1.8) we deduce that HZ, t)|| < ||2|| on [s,s + c]. Let s and t be fixed, 0 < s < t. We have already noted that the solution v(z,s,t) of (8.1.6) is holomorphic in z, and hence v(z, s,t) is a Schwarz mapping. We next prove that the solution v(z, s,t) of (8.1.1) is unique. Assume u(z,s,t) is another locally absolutely continuous solution of (8.1.1) such that w(z, s, s) = z. Fix T > s. Then for t € [s, T], we have ft v(t) = v(z,s,t) = v(z,s, s) — I /i(i;(2,s,r),r)dr Js
8.1. The Loewner differential equation
301
and
u(t) = u(z, s, t) = u(z, s, s) — If* h(u(z,s,r),r)dT. J8
Also, for ||z|| < r < 1 and s < t < T, we have \\v(z, s,t)\\ < r and ||w(z, s,£)ll ^ r - Hence from (8.1.2) we deduce that ft I
\\v(z,a,t)-u(z,8,t)\\<
\\h(v(z,s,T),r)-h(u(z,s,T),r)\\dr
Js
ft
< M(r) I \\v(z, s,r) - u(z,s,r)\\dT. Js
Let L = sup \\u(i) — v(i)\\. Using the previous estimate, one may prove s
inductively that (+ _ o\m r L - ,
t € [a, 21,
for all m 6 N and \\z\\ < r. This implies that u(t) and v(i) coincide on [s,T] for each T > s. Hence the solution of (8.1.1) is unique. We next show that v(-,s,t) is univalent on B whenever t > s. Since v(z, s, s) = z is univalent, it suffices to prove this assertion for t > s. For this purpose, let £Q > s be fixed and let z', z" be such that \\z'\\ < r, ||z"|| < r < 1, and v(z',s,to) = v(z",s,to). Also let w(t) = vi(i) — v^(t] for t > s, where vi(t) = v(z',s,t) and vi(t} = v(z",s,t). Then w(to) = 0. Moreover, from (8.1.2) we obtain dw(t)\\
at
h(vi(t),t) - h(v2(t),t)\\ < M(r)\\w(t)\\,
a.e. t € [«,*,].
Let : s < t
Then for t € [s, to] we obtain the estimate \Ht}\\ =
II fto dwM II Cio / ^^-drll < / M(r)Adr = M(r)A(t0 - t), \\Jt or || Jt
and using an inductive argument, we deduce that \\w(t)\\<
ml
A(t0-t)m,
302
Loewner chains in several complex variables
Consequently, we must have w(t) = 0 for t € [s, to]. But the condition w(s) = 0 implies that z' = z". It remains to show that Dv(0, s, t) = e3^!. For this purpose, let s > 0 be fixed and let V(t) = Dv(Q,s,t) for t > s. From (8.1.5) and the fact that M is locally uniformly bounded, we deduce that v(z,s,t) satisfies the Lipschitz condition (8.1.9)
h(v(z,s,r),r)dr
s
\\z\\
Now Cauchy's integral formula for vector-valued holomorphic functions of a single variable gives 1 r v((,w,s,t) 2m 7 C2 ICI=P
'
for all w € C n , ||iu|| = 1. Together with (8.1.9) this gives (taking the supremum over w)
\\V(ti) - V(t2)\\ < c\ti - «2|,
*i,*z e [s,oo),
for some constant C = C(r). Hence V is a Lipschitz continuous function of t € [s, oo), so that dV(t)/dt exists a.e. t > s. Also since
3 —Dv(Q,s,t) = —Dh(v(Q,s,t),t)Dv(Q,s,t), C/6
a.e. t>s,
we obtain, using Dh(Q, t} = I and v(0, s, t) = 0, that (8.1.10)
—-(t) = ~V(t],
a.e. t>s,
V(s) = I.
Solving the initial value problem (8.1.10), we obtain the unique solution V(t) = es~il. Hence Dt>(0, s,£) — es~ll, as desired. This completes the proof. The following growth result is due to Pfaltzgraff [Pfal]: Lemma 8.1.4. Suppose that h(z,t) satisfies the hypotheses of Theorem 8.1.3, and let v(z,s,t) be the solution of the initial value problem (8.1.1). Then -\\v(z,s,tW-(l- \\z\
8.1. The Loewner differential equation
303
and
/or all z e B and 0 <s
(8.1.13)
=-
B
e (h(v(t),t),v(t))
i + IK*)ll
'
making use of (6.1.27) and the fact that
a.e. t>s. Hence \\v(t)\\ is strictly decreasing from \\z\\ = \\v(s)\\ to 0 as t increases from s to oo. Also we know from the proof of Theorem 8.1.3 that ||v(£)|| is Lipschitz (and hence absolutely continuous) in t for t > s. Thus we may integrate in (8.1.13) and change variables to deduce that _ /"HI fl + x\ dx=_[t (l + \\v(r}\\\ I d\\v(r}\\ 7,1,11 \l-x) x Ja \l-\\v(r)\\J\\v(r)\\ dr
> I dr = t-s. Js
From this we obtain
\\z\\
1(1 -INI) 2
(i-IK^ H*,a,t)||
which is equivalent to (8.1.11). Similar reasoning gives the estimate (8.1.12). This completes the proof. Next we show that if v(t) = v(z,s,t) is the unique solution of the initial value problem (8.1.1), then lim etv(z,s,t) exists and gives rise to a Loewner t—too chain. Our proof expands on ideas of Poreda [Por3, Theorems 2 and 3]. Poreda obtained this result on the unit ball of a Banach space, with slightly stronger regularity assumptions in t. (The use of normal families in the first step of the
304
Loewner chains in several complex variables
argument below is not essential, and the univalence of /(•, t) can be established by another argument in infinite dimensions [Por3, Theorem 4].) Theorem 8.1.5. Let h : B x [0, oo) —>• C1 satisfy the conditions (i)-(ii) of Theorem 8.1.3. Let v(t) = v(z, s,t) be the solution of the initial value problem (8.1.1). Then for each s > 0, the limit (8.1.14)
lim e*v(z, s, t) = f ( z , s),
t—>-oo
s > 0,
exists locally uniformly on B. Moreover, /(•, s) is univalent on B and f ( z , s ) = f ( v ( z , s , t ) , t ) for all z € B and 0 < s < t < oo. Hence f ( z , t ) is a Loewner chain, and this Loewner chain has the property that {e~*f(z, t)}t>o is a normal family on B. In addition, /(z, •) is a locally Lipschitz function on [0, oo) locally uniformly with respect to z £ B. Finally, for a.e. t> 0, j£(z,t) = D f ( z , t ) h ( z t t ) ,
VzeB.
Proof. Fix s > 0 and let tp(z, t) = etv(z, s, t) for t > s. Then from (8.1.11) we have
and hence the mappings {
m—too
exists locally uniformly on B. Moreover, since ip(z, tm) is a univalent mapping and D(p(Q,tm) = esl for each tm, the Jacobian of f ( z , s) cannot be identically zero. Consequently, /(•, s) is a univalent mapping on B such that /(O, s) = 0 and Df(Q, s) = esl. However, we shall establish the existence of the limit (8.1.14) independently of this sequence. Since v(t} = v(z, s, t) is the solution of (8.1.1), we have -^(z,t) = v(z,t)-eth(e-t
a.e. t > s,
for all z € B. Let g(z, t) = h(z, t} - z for z 6 B and t > 0. Then g(-, t) € H(B), , t) = 0, and D 0. Moreover, we have (8.1.15)
8.1. The Loewner differential equation
305
for all z € B. Since for each r G (0, 1) there exists M = M(r) > 0 such that \\h(z, t)\\ < M(r), ||z|| < r, t > 0, we deduce that (8.1.16)
\\g(ztt)\\
\\z\\ < r,
t > 0.
Let r0 € (0, 1) be fixed. Since 0(0, £) = 0 and Dg(Q,t) = 0, it follows from Lemma 6.1.28 and the relation (8.1.16) that
(8.1.17)
||0(JM)||
Next, using Lemma 8.1.4, we obtain (8.1.18)
\\v(z,8,t)\\<e>-*
r
° U - ro)
N|
*>
Hence, from (8.1.17) and (8.1.18) we deduce that (8.1.19)
||0(e-V(*. *),*)!! < ^(ro
(1-ro)
for ||z|| < TO and t > s, where we have used the fact that \\e t
< K(ro)e~2tt2,
\\z\\ < r0,
t > to.
Further, using this inequality and the relation (8.1.15), one concludes by integration that
POO
for h > ti > to and ||z|| < ro. The convergence of the integral / t2e~tdt Jo implies that for any e > 0 there exists TO > 0 such that for ti,t2 > TO,
Hence the limit (8.1.14) exists locally uniformly on B, and by Weierstrass' theorem gives a holomorphic mapping on B. We now prove that the mapping / : B x [0, oo) —>• C" , constructed in this way, is a Loewner chain. As in one variable, the semigroup property v(z, s, t) = v(v(z, s,r),r,t),
0 < s < r < t < oo,
306
Loewner chains in several complex variables
follows from the uniqueness of solutions to initial value problems of the form (8.1.1). Using it, we deduce that
f ( z , s ) = lim etv(z, s, t] £->oo
= lim e*v(t;(z,s,T),T,t) = /(V(Z,S,T),T), t—>oo
for z e B and 0 < s < r < oo. We also have /(0,t) = 0 and £>/(0,t) = e*I for i > 0, and hence by Hurwitz's theorem /(-,£) is univalent for each t > 0. Hence f ( z , t) satisfies all of the conditions needed to be a Loewner chain. Moreover, f ( z , t ) is locally Lipschitz in t locally uniformly in z. Indeed, in view of Lemma 8.1.4 and the definition of f ( z , t), we have
and using the Cauchy integral formula (6.1.18) and the mean value theorem (applied to the real and imaginary parts of the components of /), it is easy to deduce that for all r e (0, 1) and T > 0 there is a constant L = L(r, T) such that \\f(z,t)-f(w,t)\\
*o,*,t) - ZQ\\ < L(\\z0\\,T)M(\\zQ\\)(t - s}, for 0 < 5 < t < T and ZQ e B, where we have used the fact that \v(zo,s,t)-ZQ\\ =
h(v(zQ,s,r),r)dr
<M(\\zo\\)(t-s).
In particular, f ( z , t ) is locally absolutely continuous in t locally uniformly with respect to z. In addition, (8.1.14) and (8.1.11) imply that {e~tf(z,t)}t>o is a normal family on B. It remains to prove that for almost alH > 0, (8.1.20)
2;(z,t) = Df(ztt)h(ztt)1
z € B.
8.1. The Loewner differential equation
307
By using the subordination property and applying the mean value theorem to the real and imaginary parts of the components of /, we obtain (8-1.21) = - [/(z, t + TJ) - f ( v ( z , t, t + rj) , t + 77)] 2 € B , t>Q, r ? > 0 , where A(z, t, 77) is a real-linear operator which tends to the invertible complexlinear operator D f ( z , t ) as 77 —» 0+. In view of this we deduce that the difference quotient in the first member of (8.1.21) has a limit as 77 —>• 0+ if and only if the same is true of the difference quotient in the last member of (8.1.21). But as just remarked, f ( z , t ) is locally Lipschitz in t locally uniformly in z. Using this fact and Vitali's theorem (Theorem 6.1.16) we deduce that, except for a set of measure 0 in t, the two sided-limit of the left-hand side of (8.1.21) exists for all z €E B as 77 —>• 0. Hence we obtain for a.e. t > 0,
where h(-,t) € M. in view of Lemmas 6.1.30 and 6.1.33 and the normalization of the transition mappings. To complete the proof, we apply Corollary 8.1.10 to conclude that h(-,t) = h(-,i) for a.e. t > 0. Next we show that solutions of the Loewner differential equation in n dimensions which satisfy a growth condition in t give Loewner chains. This is one of the basic results in the theory of univalence in several complex variables. It is originally due to Pfaltzgraff [Pfal, Theorem 2.3], and is a generalization to higher dimensions of Theorem 3.1.11. We have incorporated improvements in the assumptions on h resulting from the compactness of the class M. [GraHam-Koh], and we have added some details/variations in other parts of the argument. We also combine Pfaltzgraff 's result with [Por3, Theorem 6] to conclude that the mapping f ( z , t ) which solves the differential equation (8.1.22) coincides with the mapping defined by (8.1.14). This situation corresponds to the one variable case given in Theorem 3.1.12. Poreda [Por3] established the analogous result in the context of complex Banach spaces.
308
Loewner chains in several complex variables
Theorem 8.1.6. Suppose that r € (0,1] and that ft(z) = f ( z , t) = etz+... is a mapping from Br x [0,oo) into C71 such that ft £ H(Br) and f ( z , t ) is a locally absolutely continuous function of t € [0, oo) locally uniformly with respect to z € Br. Suppose h : B x [0, oo) -> Cn satisfies the conditions (i) and (ii) of Theorem 8.1.3, and that for all z €. Br, (8.1.22)
( z , t ) = Df(z,t)h(z,t),
a.e. t>Q.
Further, suppose there exists a mapping F G H(Br) and a sequence tm > 0, increasing to oo, such that (8.1.23)
lim
locally uniformly on Br. Then f ( z , t ) extends to a Loewner chain g : B x [0, oo) ->• C1 and lim e*v(z, s, t) = g(z, s) t—><x>
locally uniformly on B for each s > 0, where v(t) = v(z,s,t) is the solution of the initial value problem (8.1.24)
dv — = -h(v,t), ot
a.e. t > s,
v(s) = z,
for all z e B. Proof. First we show that f ( z , t) is a locally Lipschitz continuous function of t e [0,oo) locally uniformly with respect to z e Br. Indeed, since f ( z , t ) is a locally absolutely continuous function of t € [0, oo) locally uniformly with respect to z € Br, it is obvious that f ( z , t ) is continuous on Br x [0, oo). Therefore, for each p € (0, r) and T > 0 there exists K = K(p, T) > 0 such that \\f(z,t)\\
\\Df(z,t)\\
\\z\\
0 < t < T.
8.1. The Loewner differential equation
309
for \\u\\ — 1 and S € (0,r — p), we deduce for 6 = (r — p)/2 that
for all z,u with ||z|| < p < r, ||u|| = 1, and £ e [0,T], as claimed. Hence if \\z\\ < p, ||w|| < p and* <E [0,T], then by (8.1.25) we obtain *,t)-/(tM)|| =
— T)Z + rw, i)(w — z)dr
On the other hand, taking into account the relations (8.1.22) and (8.1.25), and the local uniform boundedness of M., we easily deduce that for each p e (0, r) and T > 0, there exists N = N(p, T) > 0 such that
\\z\\
a.e. te[0,T\.
Together with the absolute continuity hypothesis, this implies that for 0 < ti < t2 < T and \\z\\ < p < r,
Next, fix s > 0. We prove that /(u(z,s,£),£) = /(z,s) for all z G Br and t > 5, where v = v(z,s,t) (t > s, z e B) is the univalent Schwarz mapping that gives the solution of the initial value problem (8.1.24). For this purpose, let f ( z , s , t ) = f ( v ( z , s , t ) , t ) , fort>s and z € Br. In the proof of Theorem 8.1.3 we have seen that v(z, s,t) is a Lipschitz continuous function of t locally uniformly with respect to z. Since /(z, t) is also locally Lipschitz continuous in t > 0 locally uniformly with respect to z € Br, it is not difficult to verify that the same is true of /(z,s,£). Indeed, for all p G (0,r) and T > 5, we obtain the relations - f ( z , s, ta) j| = \\f(v(z, s,
- f ( v ( z , s, , t2), *2)||
310
Loewner chains in several complex variables < N(p,T)(t2 - ti) < N(p,T)(t2 - ti) + M(p,T)(t 2 - *i), 5 < ^ < t2 < T, ||z|| < p,
where for the last inequality we have used the Lipschitz continuity of v(z, s, •) on [s, oo) locally uniformly with respect to z. O f
Hence -~-(z, s, t) exists a.e. t > s and for all z € Br. For 77 > 0 we have
= - [f(v(z, s, £ + 77), £ + 77) - f(v(z, 5, t), t + r?)] + - [/(v(z, s, t), i + r/) - /K-z, 5, t), *)]. 77 L
J
The difference quotient on the left of the preceding equation has the limit — (z, s,£) a.e. t > s as 77 —> 0 + , and the second difference quotient on the c/c f\ f right has the limit — (i;(z,s,£),t) a.e. t > s as 77 —> 0+. Applying the mean C/L value theorem to the real and imaginary parts of the components of /, we may replace the first difference quotient on the right by A(z,s,t,77)( - \v(z,5,t + 77) - v(z,s,t)\}, \77 L J/ where A(z, 5,^,77) is a real-linear operator which tends to the complex-linear operator Df(v(z,s,t},t) as 77 —>• 0+. The difference quotient involving v has 5v the limit -zr(z, s, t} a.e. i > s as 77 —)• 0+. Hence for almost all t > s, we obtain ot
-(z,s,t) + h(v(z,s,t),t) =0, ac / making use of relations (8.1.22) and (8.1.24). It follows that f(z,s,t) is a constant function with respect to t, and thus f(v(z,8,t),i)
as claimed.
= f(v(z,S,s),s)
= f(z,s),
Ze B r , t > S ,
8.1. The Loewner differential equation
311
Next, let if)(z, t) = etv(z, s, t) = esz + ..., t > s, z 6 B. The limit lim etv(z, s,£) = lim tjj(z,i) = g(z,s) exists locally uniformly on B and the mapping g : B x [0,oo) —>• C™ is a Loewner chain, by Theorem 8.1.5. Moreover, we can show that g(z,s) gives the desired holomorphic extension of f ( z , s ) . Indeed, it follows from (8.1.23) and the Schwarz lemma (Lemma 6.1.28) that for each p € (0, r), there exists KQ = KQ(P) > 0 such that \\e-tmf(z,tm)\\ < K0(p), \\z\\ < p, m € N, and thus \\e-tmf(z, tm) - z\\ < (K0(p) + p) W-, \\z\\ < p, m 6 N. Hence, by Lemma 8.1.4 and the equality f ( z , s ) = f(v(z,s,t),i), z € Br, 0 < s < t < oo, we deduce that
and therefore lim etmv(z, s, tm) = f ( z , s)
in—too
uniformly on compact subsets of Br. Combining the above observations, we deduce that f ( z , s ) = g(z,s), zeBr, s>Q. This completes the proof. Remark 8.1.7. (i) The assumption (8.1.23) in Theorem 8.1.6 can be replaced, when r e (0,1), by the assumption that ||e~ t /(- 2; >*)ll < C on Br for t > 0. A priori the exceptional null set in (8.1.22) may depend on z, but the local Lipschitz property of f ( z , t) and Theorem 8.1.9 show that there is no loss of generality in assuming that it is independent of z (see [Gra-Ham-Koh]). (ii) In higher dimensions, univalent solutions of the generalized Loewner equation (8.1.22) need not be unique. For if f ( z , t ) is a Loewner chain which satisfies the differential equation (8.1.22) for a.e. t > 0 and all z € B, and if
312
Loewner chains in several complex variables
<£ : Cn —»• C™ is a normalized entire biholomorphic mapping, not the identity, then g(z, t) = <E»(/(z, i)) is another Loewner chain satisfying (8.1.22) for almost alH > 0 and all z <E B ([Gra-Ham-Koh]; compare with [Bec3-4], [Pom3]). (iii) However, if h(z, t) is a mapping which satisfies the conditions (i) and (ii) of Theorem 8.1.3, then there is a unique Loewner chain f ( z , t ) which satisfies the generalized Loewner differential equation (8.1.22) for almost all t > 0 and all z 6 B, and which has the property that {e~tf(z,t}}t>o is a normal family on B. (Compare with Theorem 3.1.13.) Proof. The existence of such a Loewner chain follows from Theorem 8.1.5. To show uniqueness it suffices to use Corollaries 8.1.10 and 8.1.11.
8.1.2
Transition mappings associated to Loewner chains on the unit ball of Cn
In this section we study properties of transition mappings associated to Loewner chains and deduce some consequences. We recall that all Loewner chains are assumed to be normalized. We start with a Lipschitz regularity result (see [Cu-Kohl,2], [Gra-Koh-Koh2]); we remark that part (ii) is proved under weaker assumptions than in the source papers. Theorem 8.1.8. Let f ( z , t ) be a Loewner chain and let v(z,s,t] be the transition mapping associated to f ( z , t ) . Then the following statements hold: (i) For each r e (0, 1) there exists M = M(r} < 4r/(l — r) 2 such that \\v(z,s,t)-v(z,s,u)\\
< M(r)(l - e*~u), ||z|| < r, Q < s
Thus for each s > 0, v(z, s, t) is a Lipschitz continuous function of t > s locally uniformly with respect to z E B. (ii} For each r G (0, 1) and T > 0 there exists N = N(r, T) > 0 such that ||/(*, a) - /(*,t)|| < AT(r,T)(l - e'-'), ||*|| < r, 0 < s < t < T. Thus f ( z , t ) is a locally Lipschitz continuous function oft£ [0, oo) locally uniformly with respect to z G B. Proof. First we show (i). For this purpose, we show that for each r e (0, 1) there exists M = M(r) < 4r/(l — r) 2 (independent of v] such that (8.1.26)
\\z-v(z,s,t)\\ <M(r)(l-e s ~*), \\z\\ < r, 0 < s < t < oo.
8.1. The Loewner differential equation
313
In fact we can take M(r) = 4r/(l — r)2 — 3r, 0 < r < 1. (This is an increasing function of r € (0, 1).) To see this, fix s, t with 0 < s < t < oo, and let z — v(z,s.t) _ Then it is obvious that ga>t € M. Indeed, gSjt € H(B], gs,t(Q) = 0, DgStt(0) = I and
Re (gs,t(z), z) = -^-^ [\\z\\2 - Re (v(z, s, t), z)]
z e B. From Lemma 6.1.30, one deduces that gsj € M.. Also using the proof of Theorem 7.1.7 we deduce that for each r € (0, 1) there exists M = M(r) > 0 given by M(r) = 4r/(l — r)2 — 3r, such that ||<7«,tOz)|| < M(r), ||z|| < r. Hence the relation (8.1.26) follows. Next, using the semigroup property of the transition mapping v(z, s, t), we obtain for 0 < s < t < w < o o and ||^|| < r that \\v(z, s, t) - v(z, s, u)\\ = \\v(z, s, t) - v(v(z, s, t),t, u) || < Af (r)(l - e*~u) < M(r)(u - t), where we have used the fact that ||v(2,s,£)|| < \\z\\ < r. Therefore v(z, s,t) is Lipschitz continuous in t for t > s, locally uniformly with respect to z 6 B. We now prove the second statement. To this end, we remark that a similar argument to the above yields that for each r € (0, 1) there exists R = R(r) > 0 such that ||v(z, s, t) - v(z, T, t)\\ < R(r)(l - e s ~ T ), ||z|| < r, 0 < s < r < £ < o o . (See also Problem 8.1.4.) Next, fix T > 0 and r € (0, 1). Since f ( - , T ) is holomorphic on B, for each p 6 (0, 1) there exists L = L(p,T) > 0 such that ||/(*, T)||
\\z\\ < p.
Applying the Cauchy integral formula, it is not difficult to deduce that there exists K = K(r, T) > 0 such that \\Df(z,T)\\
\\z\\ < r.
314
Loewner chains in several complex variables
Now, let 0 < s < t < T and z <E ~Br- Since f ( z , s ) = /(t;(z,s,T),T) and f ( z , t ) = f(v(z,t,T),T), we obtain \\f(z, s) - f ( z , t) || = \\f(v(z, 8, T), r) - f ( v ( z , t, r o Combining the above arguments, we conclude that
\\f(z,s] - f ( z , t ) \ \ < K(r,T)R(r)(l
- e-') = N(r,T)(l - e*-*},
as desired. This completes the proof. We are now able to prove that any Loewner chain on the Euclidean unit ball in C" satisfies the generalized Loewner differential equation (8.1.27) ([GraHam-Koh]; compare with [Cu-Koh2] and Theorem 3.1.12). Theorem 8.1.9. Let f ( z , t ) be a Loewner chain. Then there is a mapping h = h(z,t] such that h ( - , t ) e M. for each t > 0, h(z,t) is measurable in t for each z £ B, and for a.e. t > 0, (8.1.27)
?i(z,t) = Df(z,t)h(z,t),
VzeB. r\ f
(That is, there is a null set E C [0, oo) such that for all t € [0, oo)\E, -«T(-, t) C/if exists and is holomorphic on B. Also, for t G [0, oo) \ E and z € B, (8.1.27) holds. We interpret the left-hand side of (8.1.27) as a right-hand derivative when t = 0.) Moreover, if there exists a sequence {tm}meN such that tm > 0, tm —t oo, and (8.1.28) lim e-tmf(z,tm) = F(z) m—too
locally uniformly on B, then /(z, s) = lim etw(z, s, t) locally uniformly on t—¥OO B for each s > 0, where w(t) = w(z,s,t) is the solution of the initial value problem r\ -7— = — /i(iy,t), at
a.e. t > 5,
w(s) = z,
for all z e B. Proof. Let v — v(z,s,t) be the transition mapping defined by the chain /(z, t), i.e. /(z, s) = f ( v ( z , s, t), t), z 6 B, 0 < s < t < oo. Taking into account
8.1. The Loewner differential equation
315
the normalization of f ( z , t ) , we deduce that Du(0, s,t) = e8**! for t > s > 0. By using the subordination property and applying the mean value theorem to the real and imaginary parts of the components of /, we obtain (8.1.29) = i [/(«, t + r) - f(v(z, t,t + r),t + r)] = A(z,t,r)(-\z-v(z,t,t
+ r)] j ,
z € B,
t>0, r > 0,
where A(z, i, r) is a real-linear operator which tends to the invertible complex linear operator Df(z, t) as r —>• 0+. In view of this we deduce that the difference quotient in the first member of (8.1.29) has a limit as r —> 0+ if and only if the same is true of the difference quotient in the last member of (8.1.29). Since f(z,i) is locally Lipschitz in t locally uniformly with respect to z 6 B by Theorem 8.1.8, the difference quotients on the left-hand side of (8.1.29) are a family of locally uniformly bounded holomorphic functions of z € B. Let Q be a countable set of uniqueness for the holomorphic functions on B. For each z € Q the limit as r —» 0 of the difference quotients on the left side of (8.1.29) exists except when t € Ez, where Ez is a subset of [0, oo) of measure 0. The set E — \J{EZ : z € Q} also has measure 0, and Vitali's theorem (Theorem 6.1.16) implies that for t £ E, the difference quotient on the left-hand side of /•) / (8.1.29) has a limit as r —>• 0 which is holomorphic in z. Therefore, •^r('>0 is holomorphic on B for each t G [0, oo) \ E. Moreover, since v(z, s, t) is a Schwarz mapping and Dv(Q, s, t) = es~ll, the difference quotient on the right has a limit h(z,t) in M as r ->• 0+ for t £ E, by Lemmas 6.1.30 and 6.1.33. For t € E and z € B, we let h(z,t) = z. Then it is clear that h(-,t) € M. for t>0. The mapping h(z,t) is measurable in t € [0,oo) for each z € B, since Q r
— ( z , t ) and [Df(z,t)]~l are measurable in t. To deduce the measurability of [Df(z, t)]~ , it suffices to use the Cauchy integral formula and the local Lipschitz continuity of f ( z , t } in t, and to prove that D f ( z , t ) is also locally Lipschitz continuous in t locally uniformly with respect to z G B.
316
Loewner chains in several complex variables
Finally it suffices to note that if (8.1.28) holds locally uniformly on B, then the final statement follows from Theorem 8.1.6. This completes the proof. Next we give some consequences of the above result. (Compare with Theorem 3.1.12 and [Gra-Koh-Koh2] . Another proof of (8.1.30) is given in [CuKoh2].) Corollary 8.1.10. Let f ( z , t ) be a Loewner chain and let v(t} = v(z,s,t) be the transition mapping associated to f(z,t). Also let h(z,t) be the mapping given by Theorem 8.1.9. Then for each s > 0 and z G B, v = v(z, s, t) satisfies the initial value problem dv — = — /i(v,t),
(8.1.30)
a.e.
t>s,
v(z,s,s) = z.
\Jl/
Moreover, on any interval (0,£], v = v(z,s,t) satisfies the initial value problem dv (8.1.31) — (z, s, t) = Dv(z, s, t)h(z, s), C/S
a.e. s <E (0, t], v(z, t, t) = z.
Proof. Fix s > 0 and let w(t] = w(z, s,t) be the unique locally absolutely continuous solution of the initial value problem r\
(8.1.32)
-^- = -h(w,t),
a.e. t > s,
w(s) = z,
\Jl/
for all z e B. Also let T > s and let g(z,s,t) — f(w(z,s,t),t) for t 6 [s,r] and z e B. Since both mappings f ( z , t ) and w(z, 5, t) are locally Lipschitz continuous in t locally uniformly with respect to z 6 B, it follows by a similar argument as in the proof of Theorem 8.1.6 that the same is true for g(z, s,t). Then this mapping is locally absolutely continuous in t, so is differentiable with respect to t for almost all t e [S,T]. Using (8.1.32) and (8.1.27), we obtain for almost all t € [s,r] that
= Df(w(z, s, t),t)h(w(z, s, t), t) - Df(w(z, s, t), t)h(w(z, s, t), t) = 0. Hence g(z, s,t) — g(z,s,s) — f ( z , s ) for t e [s,r], that is f(w(z,s,i),t) = f ( z , s}. Since r is arbitrary, we deduce that this equality holds for all s, t with 0 < s < t < oo, and z G B.
8.1. The Loewner differential equation
317
On the other hand, since v = v(z, s, t) is the transition mapping associated to f ( z , t ) , we have f(v(z,s,t),t)
= f(z,s),
t>s,
zeB.
Combining the above arguments, we obtain /(u(z,a,t),t) = /(w(2,s,t),t),
z e B,
0<s
Since /(•, t) is univalent on B, we must have w(z, s, t) = v(z, s, t) for all z € B and t > s. In order to prove (8.1.31), we use the equality /(z,s) = f(v(z,s,t),t). Fix t > 0. If we differentiate both sides of this equality with respect to s G (0, t] (this is possible since v(z, s,£) is Lipschitz continuous in s for s € [0, t] locally uniformly with respect to z e B (see Problem 8.1.4)), we obtain - ; ,
, , , - , , ,
a.e.
On the other hand, in view of (8.1.27) and the above equality, we deduce that
dv Df(z, s)h(z, s) = Df(v(z, s, t), t)-^(z, s, t),
a.e. s 6 (0, t},
and since Df(z, s} = Df(v(z, s, t), t)Dv(z, s, t), we obtain dv Df(v(z, s, t), t)Dv(z, s, t)h(z, s) = Df(v(z, s, t), t)^(z, s, t). Since /(-,*) is univalent on B, Df(v(z,s,t),t) is a nonsingular matrix, and therefore the above equality implies (8.1.31), as desired. This completes the proof. Another consequence of Theorem 8.1.9 is given below. (Compare with Theorem 3.1.12 and [Gra-Koh-Koh2]. Another proof of Corollary 8.1.11 is given in [Cu-Kohl].) Corollary 8.1.11. Let f ( z , t ) be a Loewner chain such that the family t {e~ f(z,t}}t>o is a normal family on B. Then for each s > 0, the limit f ( z , s ) = lim etv(z, s,t) t—too
318
Loewner chains in several complex variables
exists locally uniformly on B, where v = v(z,s,t) is the transition mapping associated to f ( z , t ) . Proof. It suffices to apply Theorem 8.1.9 and Corollary 8.1.10. We next give the growth result for Loewner chains which satisfy the conditions of Corollary 8.1.11 (see [Gra-Ham-Koh], [Cu-Kohl], [Gra-Koh-Koh2]). Corollary 8.1.12. Let /(z, t) be a Loewner chain such that {e~*f(z, £)}t>o is a normal family on B. Then (8-L33) 7^TTTTv> < IK'/M)!! < „ ^ ' a » * € B, 0 < t < oo. In particular, if f ( z ) = /(z,0) then
Proof. It suffices to apply Corollaries 8.1.10, 8.1.11 and Lemma 8.1.4, to obtain (8.1.33). Remark 8.1.13. In several complex variables there exist Loewner chains f ( z , t ) such that f ( z , 0 ) does not satisfy the 1/4-growth result above (see Example 8.3.12). We conclude this section with a compactness result for the set of Loewner chains (see [Gra-Koh-Koh2] and compare with Theorem 3.1.9). Theorem 8.1.14. Every sequence of Loewner chains {fk(z,t)}keN> such that {e~tfk(z^t)}t>o is a normal family on B for each k € N, contains a subsequence that converges locally uniformly on B for each fixed t > 0 to a Loewner chain f ( z , t ) , such that {e~tf(z,t)}t>o is a normal family on B. Proof. Using the upper bound in (8.1.33) and the argument in part (ii) of Theorem 8.1.8, we deduce that for each r € (0,1) and T > 0 there exists AT = 7V(r,T) > 0 such that (8.1.34) ||/ fc (z,t) - fk(z,8)\\
< N(r,T)(t - a), \\z\\
for all k E K The upper bound in (8.1.33) also implies that for 0 < r < 1 and T > 0 there exists a constant L — L(r, T) > 0 such that for each k e N, (8.1.35)
||AOM)-/*(tM)||
8.1. The Loewner differential equation
319
for ||z|| < r, \\w\\ < r, 0 < t < T and k € N. The estimates (8.1.34) and (8.1.35) together imply that the mappings /fc(z,£), k €. N, are equicontinuous on {(z,t) : ||z|| < 1 — 1/m, 0 < t < m}, m = 2,3, ---- The Arzela-Ascoli theorem (see e.g. [Roy, p.167-169]) implies that for m fixed, there is a subsequence {fkp(z,i)}p£w which converges pointwise on -Bi_i/m x [0, m] to a limit fm(z,t), and furthermore the convergence is uniform on -Bi_i/m x [0, m]. A diagonal sequence argument then shows that there exists a subsequence, which we denote again by {/fcp(2,£)}peN5 which converges pointwise on B x [0,oo) to a limit f ( z , t ) , and furthermore the convergence is uniform on each compact subset of B x [0, oo). In particular f k p ( z , t ) —>• f ( z , t ) locally uniformly on B for each t > 0, and f ( z , t ) is holomorphic in z. Since /fcp(0, t) = 0 and Dfkp(Q,t) = e*!, p € N, it follows that the same is true for /(z,£), and hence this limit must be univalent on B. Furthermore, since f k p ( z , s ) -< f k p ( z , t ) , 0 < s < t < oo, z 6 B, it follows that there exist univalent Schwarz mappings Vkp = Vkp(z,s,t) such that Q,s,t = ea~tl and
(8.1.36)
fkp(z,8) = fkp(vkp(z,s,t),t),
p=l,2,....
Since ||ufc p (z,s,i)|| < \\z\\, p = 1,2, ..., we deduce that {vfc p (z,s,£)}peN is a normal family. Therefore there exists a subsequence {k'p}p^ of {fcp}peN such that Vk> (z,s,t) —> v(z,s,i) locally uniformly on B. This limit is univalent on B and satisfies the following conditions: u(0,s,£) = 0,
Dv(Q,s,t) = et>-tI,
\\v(z,s,t)\\ < \\z\\.
Taking the limit in (8.1.36) through the subsequence {k'p}p£N, one deduces that f ( z , s) = f ( v ( z , a, t), t), z e B, 0 < s < t < oo. We have therefore proved that f ( z , t ) is a Loewner chain. Moreover, in view of (8.1.33), we have l|e-*/q(*.*)ll<
(1 _||i||)2'
zeB
>
0
*£B,
0
for p = 1, 2, . . . , and thus H
e
" * / ( g . * ) H < .
320
Loewner chains in several complex variables
Hence {e~tf(z, i)}t>o is a normal family, as desired. This completes the proof. Notes. As noted by Pfaltzgraff, Theorem 8.1.6 can be generalized to the unit ball with respect to an arbitrary norm in C71 , if one introduces the appropriate generalization of the class M. (see Section 6.1.6). However, the proof does not generalize to the unit ball of an infinite-dimensional Banach space because of the reliance on Montel's theorem. Poreda [Por3] later showed that one could get around this difficulty by introducing slightly stronger assumptions. Also Poreda studied univalent mappings on the poly disc which admit parametric representations, and which therefore can be embedded in Loewner chains [Porl,2]. The study of the Loewner differential equation has recently been extended to bounded balanced pseudoconvex domains whose Minkowski function is of class Cl in Cn \ {0} by Hamada [Ham5] and Hamada and Kohr [Ham-Kohl, 4]. Loewner chains on the unit ball are also studied in [Cu3]. At an earlier stage, Pfaltzgraff [Pfa2] used subordination chains to study the quasiconformal extension of holomorphic maps in several complex variables. More recently, Chuaqui [Chu2] studied a subclass of the normalized star like mappings of B, called strongly star like mappings and showed that quasiconformal extensions to C71 exist for such mappings. Hamada [Ham4] extended the above results to strongly starlike mappings on bounded balanced pseudoconvex domains fi whose Minkowski function is of class Cl on Cn\{0}, and Hamada and Kohr [Ham-Koh6] obtained some similar results in the case of strongly spirallike mappings of type a on fi. We shall discuss these results in Section 8.5.
Problems 8.1.1. In Theorem 8.1.3, prove that for each t > 0, there is a null set dv Et C (0, t] such that for all s € (0, t] \ Et, ~^-(z, s > *) exists for all z € B and is c/ s holomorphic on B. Also prove that v(z, s,t) satisfies the initial value problem
dv OS
, ,
, , , s ) ,
a.e. s E ( 0 , £ ] ,
v(z,t,f) =z .
Hint. Use similar arguments as in the proofs of Theorem 8.1.9 and Corollary 8.1.10.
8.1. The Loewner differential equation
321
8.1.2. Let / = f(z,t) : B x [0, oo) -> C1 be such that /(0,t) = 0 and -D/(0, t) = ai(£)7, t > 0, where ai(t) ^ 0, t > 0. Assume that f ( z , t) is a locally absolutely continuous function of t > 0 locally uniformly with respect to z € B, and /(•, t) is holomorphic on B for each t > 0. Let /i = h(z, t) : B x [0, oo) ->• C71 be such that (i) h(-,t) e H(B), /i(0,£) = 0 and Re (h(z,t),z) > 0 for z e 5 \ {0} and *>0; (ii) /i(a:, •) is a measurable function on [0, oo) for each z € B. Also assume that -j-(z,t} = Df(z,t)h(z,t}, at
a.e. t > 0,
Vz 6 5.
Further, if oi(-) 6 C^QO, oo)), |ai(-)| is strictly increasing on [0, oo), |ai(£)| —>• oo as t —> oo, and if there is a sequence {£m}meN> tm > 0, tm —t oo, such that
= F(z) locally uniformly on B, then prove that /(z, t) is a (non-normalized) Loewner chain. (Curt, 1994 [Cul]; see also Chen and Ren, 1994 [Che-Ren].) 8.1.3. Let f ( z , t ) be a (normalized) Loewner chain and $ be an entire normalized biholomorphic mapping. Also let g(z,t) = 3>(f(z,t)), z € B, t > 0. Show that g (2, t) is a Loewner chain. 8.1.4. Let f ( z , t ) be a Loewner chain and v = v(z,s,t) be the transition mapping associated to f ( z , t ) . Prove that for each r e (0,1), there exists R = R(r) > 0 such that \\v(z,s,t)-v(z,r,t)\\
322
Loewner chains in several complex variables
for \\z\\ < r, \\w\\ < r, 0 < r < t < oo, we obtain in view of the Cauchy integral formula, \\v(z,T,t)-v(w,T,t}\\
8.2 8.2.1
Close-to-starlike and spirallike mappings of type alpha on the unit ball of Cn An alternative characterization of spirallikeness of type alpha in terms of Loewner chains
In this section we shall give a characterization of normalized spirallike mappings of type a on the Euclidean unit ball of C" in terms of Loewner chains. The results of this section can be generalized to the case of an arbitrary norm in C". Let a G R be such that |a| < Tr/2. Recall that a normalized locally biholomorphic mapping / : B —>• Cn is called spirallike of type a if (8.2.1) Let (8.2.2)
Re [e-ia([Df(z)rlf(z),
z}\ > 0,
f ( z , t ) = e(1-ia»f(eiatz),
z € B \ {0}.
z e B,
t>0,
where a = tana. The following theorem of Hamada and Kohr [Ham-Koh4] is the ndimensional version of Corollary 3.2.9, and shows that spirallike mappings of type a can be embedded in Loewner chains. Theorem 8.2.1. Let f : B —>• U1 be a normalized locally biholomorphic mapping and let a 6 IR, |o;| < 7T/2. Then f is a spirallike mapping of type a iff f ( z , t ) given by (8.2.2) is a Loewner chain. Proof. First, assume that / is spirallike of type a. Then it is easy to check that ft(z) = f ( z , t ) e H(B), ft(Q) = 0, and Dft(0) = e*I, t > 0. We note that f ( z , t ) is a C°° mapping on B x [0, oo).
8.2. Close-to-starlike and spirallike mappings
323
A short computation yields the equality (8.2.3)
|£(z, t) = D f ( z , t)h(z, t),
zeB, t> 0,
where (8.2.4)
h(z, t) = iaz + (I -
ia)e-iat[Df(eiatz)]-lf(eiatz).
It is clear that h(z, t) is a measurable function of t € [0, oo) for each z G B, h(0, t) = 0 and Dh(Q, t) = I. Moreover, f P~ia (8.2.5) Re (M*, *),*)= Re -[cosa
1 ([Df(eiatz)}-1f(eiatz),eiatz}\>Q, J
for all z € B \ {0} and t > 0, by (8.2.1). Hence h ( - t t ) € M. Let tm = m if a = 0 and tm = 2-Trm/a if a ^ 0. Then e~tmf(z,tm) — f ( z ) holds for any m € Z. Therefore, using the result of Theorem 8.1.6, we conclude that f ( z , t ) is a Loewner chain. Conversely, assume that f ( z , t ) is a Loewner chain. Since f ( z , t ) is of class C°° on B x [0,oo), the mapping h = h(z,t) £ M. given by Theorem 8.1.9 is also of class C°° on B x [0, oo), and (8.2.3) holds for all z e B and t > 0. It is obvious that h(z, t) is given by (8.2.4). Using the fact that Re(h(z,t),z)>0,
zeB\{0},
t > 0,
we obtain from (8.2.5) that Re [ e - i a ( ( D f ( z ) ] - l f ( z } ,
z)] > 0,
z 6 B \ {0}.
Thus / is spirallike of type a, as desired. Theorem 8.2.1 yields the following geometric interpretation of the notion of spirallikeness of type a, as in the case of one complex variable (see [Ham-Koh4] and also Section 2.4.2). Corollary 8.2.2. Let f : B —)• C" be a normalized locally biholomorphic mapping and let a e R with \a\ < n/2. Then f is spirallike of type a if and only if f is biholomorphic on B and the spiral exp(— e~tar)f(z) (r > 0) is contained in f ( B ) for any z £ B.
324
Loewner chains in several complex variables
Proof. Prom Theorem 8.2.1, we know that / is spirallike of type a if and only if
is a Loewner chain. In this case / is biholomorphic on B and /(*H /CMH e^-^/Oz),
*eJ3,
t>0,
which implies that (8.2.6)
exp ( -e~ia —— ) f ( z ) e /(£), \ cos a /
z £ B,
t > 0.
Conversely, if / € S(B) and each spiral exp (-e~iar] f ( z ) , r > 0, is contained in /(B), then (8.2.6) holds and it is easy to see that (8.2.2) defines a Loewner chain. This completes the proof. The case a = 0 in Theorem 8.2.1 yields the familiar characterization of starlikeness in terms of Loewner chains, due to Pfaltzgraff and Suffridge [PfaSul]. Corollary 8.2.3. Let f : B —> Cn be a normalized locally biholomorphic mapping. Then f is starlike if and only if f ( z , t ) = e t f ( z ) is a Loewner chain. 8.2.2
Close-tost arlike mappings on the unit ball of Cn
In Section 2.4 we studied certain properties of close-to-convex functions in the unit disc, and we saw that these functions appear as a natural generalization of starlike functions. The extension to higher dimensions of this class of univalent functions was considered by Pfaltzgraff and Suffridge [Pfa-Sul] and by Suffridge [Su4j. In fact, because of the failure of Alexander's theorem in several variables, there is more than one possible generalization of close-toconvexity. We shall restrict our attention to the Euclidean case in discussing these notions. Suffridge [Su4] defined close-to-convexity in higher dimensions as follows: Definition 8.2.4. Let / : B —>• Cn be a holomorphic mapping. We say that / is close-to-convex if there exists a convex mapping g G H(B) such that (8.2.7)
Re ( D f ( z ) [ D g ( z ) ] - l u , u )
> 0, z € B \ {0}, u e Cn \ {0}.
8.2. Close-to-starlike and spirallike mappings
325
Using a version of the Noshiro-Warschawski- Wolff theorem (see Lemma 2.4.1) in higher dimensions, Suffridge [Su4] showed that a close-to-convex mapping of B is univalent. However, examples show that in dimension n > 2, the close-to-convex mappings do not satisfy a growth theorem and do not form a normal family. The notion of close-to-starlikeness, introduced by Pfaltzgraff and Suffridge [Pfa-Sul], is more closely connected with Loewner chains. Again we shall just give the definition in the Euclidean case, although Pfaltzgraff and Suffridge worked with an arbitrary norm in Cn . Definition 8.2.5. Let / : B —> C™ be a normalized holomorphic mapping. We say that / is close-to-starlike if there exist h 6 M. and a starlike mapping g on B such that (8.2.8) Df(z)h(z) = g ( z ) , zeB. Note that since / and h are normalized, it follows that g must also be normalized. It is also clear that any normalized starlike mapping on B is close-to-starlike (with respect to itself). When n = 1 the above definition is equivalent to the usual definition of close-to-convexity (see Definition 2.4.3), because of Alexander's theorem. Indeed, since h € M., we may write h(z) = zp(z), z € U, where p € P. The relation (8.2.8) is therefore equivalent to / \
9(z)
—
-/
\>
P(z)
<
,
and we obtain Re
—^—r- > 0 on U. Moreover, if (p(z) = I —— dt, z £ U, Jo * then (f> £ K in view of Alexander's theorem, and thus we obtain the relation L 9\z] J
Re
£& > 0,
zeC7,
which gives the usual definition of close-to-convexity. Pfaltzgraff and Suffridge [Pfa-Sul] studied generalizations of Theorem 3.2.10 to higher dimensions, and showed that the condition of close-tostarlikeness can be used to give characterizations in terms of Loewner chains. In one direction, they proved the result below.
326
Loewner chains in several complex variables
Theorem 8.2.6. Let /,# : .B —>• C™ be normalized holomorphic mappings such that g is starlike. If F(z, t) = /(z) + (e* - l}g(z),
z G B,
t > 0,
is a Loewner chain, then f is close-to-starlike relative to g. Proof. Since F(z, t) is a Loewner chain, f(z) = /(z, 0) is univalent on B. In view of Theorem 8.1.9 we deduce that there exists a mapping h — h(z,t) such that h(-,t) G M for each t > 0, h(z, •) is measurable on [0, oo) for all z G B, and OF — (z, t) = DF(z,t)h(z, t), z G 5 , t > 0 . It is easily seen this equation holds for all t > 0 since the specific chain F(z,£) is a C°° mapping with respect to (z,£) G B x [0, oo). Setting t = 0 in the above, we obtain g(z) = Df(z)h(z), z G B, where h(z) = /i(z,0) G M. Thus / is close-to-starlike relative to g, as desired. This completes the proof. Pfaltzgraff and Suffridge [Pfa-Sul] also obtained the converse of this result. Since the proof is rather long, we leave it for the reader. However, we mention that it is interesting. Here we present only a special case. Theorem 8.2.7. Let (f),ip G H(B,C) be such that 0(0) = ip(0) = 1 and (j)(z) ^ 0, z G B \ {0}. Also let f , g e H(B) be given by f(z] = z^(z), g(z) = zip(z), z e B, and assume that f is locally biholomorphic on B and close-to-starlike relative to g € S*(B). Then F(z, t) = f(z) + (e* - l)g(z),
zeB,t>0,
is a Loewner chain, and hence for any r G (0,1) the complement of f ( B r ) is a union of nonintersecting rays. Proof. In view of Theorem 7.1.14 we know that the assumption that g be starlike is equivalent to Re { I±-L^^L- \ > o, z G B.
A computation similar to the proof of Theorem 7.1.14 shows that the assumption that / be close-to-starlike relative to g is equivalent to Re { " v
y
'~^~,~ ,
>
Q^
z eR
8.2. Close-to-starlike and spirallike mappings
327
Taking into account the above inequalities, we obtain - + (l-e-*) V>(z) +,y* ) *l>0, for all z e B and t > 0. Now let h : B x [0, oo) -» C" be given by 1
h(z,t) = z< e~ I
-T-T z
v>( )
H (1 — e~ )
-r—?
Y(Z)
>
j
Then h(-,t) e M, t > 0, h(z, •) is measurable on [0, oo), z € B, and ^(z,t) = DF(z,t)/iCM),
*eB' *^°-
On the other hand, it is clear that lim e~*F(z, t} = g(z) locally uniformly t—¥<x> on B. Hence in view of Theorem 8.1.6 we conclude that F(z,t) is a Loewner chain. As in the case of one variable (see Corollary 3.2.11), the rays L(z, r) = {/(z) + tg(z) : t > 0},
||z|| = r < 1,
are disjoint rays which fill up the complement of f(Br). This completes the proof. Poreda [Por4] obtained an integral representation for close-to-starlike mappings in higher dimensions, and in addition showed that if h € Ai and g € S*(B) are given, there is a unique normalized holomorphic mapping / which satisfies (8.2.8). We have Theorem 8.2.8. Let h G Ai and let g € S*(B). Then there exists a unique normalized holomorphic solution f of (8.2.8), and it is given by (8.2.9)
/(z) = f°° g(v(z, t))dt, Jo
z£B,
where v = v(z,t) is the solution of the differential equation
dv -£-(z,t) = -h(v(z,t)), (z,t) € B x [0,oo), v(z,0) = z. Proof. First we show that any normalized holomorphic solution of (8.2.8) must have the form (8.2.9).
328
Loewner chains in several complex variables For fixed z G J3, we have
and integrating with respect to t gives
r Jo
By Lemma 8.1.4 and the growth theorem for starlike mappings on B (see Theorem 7.1.1), we have (8.2.11)
\\g(v(.
The improper integral
-(i-NI) 2 '
/>oo / g(v(z,t)}dt
(8.2.10) yields
-
therefore converges absolutely, and
poo
f(z) — I 9(v(z,t\)dt, Jo as desired. Here we have used the fact that lim v(z, t) = 0 by Lemma 8.1.4. t-too
Conversely, let us show that the mapping / defined by (8.2.9) satisfies (8.2.8). The estimate (8.2.11) implies that the integral in (8.2.9) exists and that / is holomorphic on B. Using the semigroup property v(v(z, £), T) = v(z, t + r), z € B, t,r > 0, which follows from the uniqueness of solutions of the initial value problem dw/dr = — h(w(z,r)), w(z,0) = v(z,t), we obtain
oo
/
g(v(z,t))dt,z£B,T>Q.
Differentiating with respect to r gives
Setting r = 0 gives the desired result and completes the proof. Poreda [Por4] showed that it is possible to deduce the growth theorem for close-to-starlike mappings on B from the integral representation (8.2.9) (compare with Corollary 8.3.9).
8.2. Close-to-starlike and spirallike mappings
329
Corollary 8.2.9. // / : B —>• Cn is a close-to-starlike mapping then II/COII <
(i-INI)2'
z € B.
We finish this section with some examples of close-to-starlike mappings on the Euclidean unit ball B of C" (see [Pfa-Sul], [Su4]). Example 8.2.10. Let /j, QJ be normalized holomorphic functions on U such that gj 6 5*, j = 1,... ,ra. Also let hj(z) = I dt, j = 1,... ,n. Jo t Assume fj is close-to-convex with respect to hj for j = 1,..., n. Then
Re
= Re
> 0, C e U.
Also let /(z) = (/i(zi),...,/ n (z n )), 2 = (zi,...,z n ) e 5. Then / is a normalized biholomorphic mapping on B. In view of Problem 6.2.5, we deduce that g(z) = (fli(zi), . . . ,gn(zn}} is a starlike mapping on B. Moreover, a short computation yields that
Re
= Re
>0
for z = (zi,... zn) G B \ {0}. Consequently, / is close-to-starlike relative to g. Example 8.2.11. Let /, g be normalized holomorphic functions on U such that g e S*. Assume that
Re "C/'CO" 0(0
Hence / is close-to-convex. Also let u e C" with ||w|| = 1 and define =
(z,u)
(z,u)
Then F is normalized holomorphic on B and it is not difficult to check that G is normalized starlike on B (see Problem 6.2.4). Moreover, letting
n(z) =
\'(, u z, {z,w)/'((z,w»
z€B,
we easily deduce that h € M. and DF(z)h(z) — G(z), z e B. Therefore F is close-to-starlike relative to G.
330
Loewner chains in several complex variables
Problems
8.2.1. Give an example of a close-to-starlike map on the Euclidean unit ball B of C™ which is not spirallike on B. 8.2.2. Let B be the Euclidean unit ball of C2 and let / : B ->• C2 be given by f(z] = (z\ 4- az\, z z ) , z = (zi,Z2) £ B. Find the values of a e C for which / is close-to-convex.
8.3
Univalent mappings which admit a parametric representation
8.3.1
Examples of mappings which admit parametric representation on the unit ball of C"
Univalent mappings of the unit ball in C1 which arise as in Theorem 8.1.5 (specifically, the mappings f ( z ) = f ( z , 0), z € B) are said to admit a parametric representation. The purpose of this section is to derive properties of such mappings on the Euclidean unit ball. Growth, covering, and distortion theorems will be given, as well as coefficient estimates and conjectured coefficient estimates. Also we shall show that in higher dimensions, there exist normalized univalent mappings on the unit ball which can be embedded in Loewner chains, but which do not have parametric representation. In fact, the class of mappings which admit a parametric representation is compact, whereas the class of all mappings which can be embedded as the first element of a Loewner chain is not a normal family. In one variable, as we have seen in Chapter 3, any function / £ S has parametric representation. Thus we have another difference between the onevariable theory and the several-variables theory of univalent mappings. Poreda [For 1,2] studied univalent mappings of the polydisc which admit parametric representation. Also Kubicka and Poreda [Kub-Por] considered parametric representations of starlike mappings of the Euclidean unit ball of C n . Most of the theorems presented in this section are due to Kohr [Koh9] in the Euclidean case; they were generalized to the case of an arbitrary norm in
8.3. Parametric representation
331
C71 by Graham, Hamada, and Kohr [Gra-Ham-Kohj. Some of them may also be found, along with further results, in [Koh-Lic2]. We also mention that some of the theorems have recently been extended by Hamada and Kohr [Ham-Koh5] to the case of bounded balanced pseudoconvex domains whose Minkowski function is of class Cl on Cn\{0}. However, we shall restrict our discussion to the case of the Euclidean unit ball. Definition 8.3.1. Let g € H(U] be a univalent function such that p(0) = 1, <j(£) = <j(£) for £ 6 U (i.e. g has real coefficients), Re 0 on {/, and assume g satisfies the following conditions for r € (0,1): minRe p(C) = min{0(r),p(-r)} (8.3.1) max Re #(C) = max{p(r),p(-r)}. We define M.g to be the class of mappings given by Mg =
Note that (I h(z), . z*||2 \) is understood to have the value 1 (its limiting \ \\ \\ I value) when z = 0. Clearly, if #(C) = (1 + C)/(l - Oi C € U, then Mg becomes the class M, which plays the role of the Caratheodory class in several complex variables. Taking into account Theorem 8.1.5, we introduce the following class of univalent mappings of B (see [Koh9]; cf. [Porl]): Definition 8.3.2. Let g : U —I C be a univalent function satisfying the assumptions of Definition 8.3.1. Also let / e H(B}. We say that / € Sj(B) if there exists a mapping h : B x [0, oo) —> C" which satisfies the conditions (i) for each t > 0, h(; t) e Mg\ (ii) for each z e B, h(z, t) is a measurable function of t € [0, oo); (iii) lim etv(z, t) = f ( z ) locally uniformly on 5, where v = v(z,t) is the t —too solution of the initial value problem
332
Loewner chains in several complex variables
for all z € B. The class S®(B) is called the class of mappings which have g -parametric representation on B. If g((,) — (I + £)/(! — C), the class Sg(B) is denoted by 5° (B], and is called the class of mappings which have parametric representation on B (cf. [Porl]). Remark 8.3.3. Theorems 8.1.5 and 8.1.9 imply that / e S°(B) if and only if there is a Loewner chain /(z, t) such that [e~tf(z, t)}t>o is a normal family on B and / can be embedded as the first element of this chain, i.e. f ( z ) = /O2, 0), z £ B (compare with [Porl]). It is clear that another class of interest is the subclass Sl(B) of S(B) consisting of maps / which can be embedded as the initial element of a Loewner chain f ( z , t ] (without the requirement that {e~tf(z,t}}t>o be a normal family), i.e.
S1(B) = {/ e S(B) : 3 f ( z , t ) Loewner chain, f ( z ) = /(*,0), z e B}. We have
5j(5) C S°(B) C Sl(B) C S(B). When n = 1 we have 5° (17) = S, by Theorems 3.1.8 and 3.1.12. In this chapter we shall see that in higher dimensions, S°(B) g S(B) and S°(B) § Sl(B). Theorem 8.1.6 can be used to generate a large number of examples of mappings in S^(B) if the mapping h = h(z,t) satisfies conditions (i) and (ii) of Definition 8.3.2. In connection with this result, we say that a mapping / : B x [0,oo) -> C" is a g-Loewner chain if f ( z , t ) satisfies the assumptions of Theorem 8.1.6 for r = 1, and the mapping h = h(z,t), for which (8.1.22) holds a.e. t > 0 and for all z e B, satisfies conditions (i) and (ii) of Definition 8.3.2 ([Koh9], see also [Gra-Ham-Koh]). Combining Theorems 8.1.6 and 8.1.9 and Corollary 8.1.10, we deduce that f ( z , t ) is a g-Loewner chain if and only if f ( z , t) is a Loewner chain such that {e~*f(z, £)}t>o is a normal family on B, and the mapping h — h(z, t) which occurs in the Loewner differential equation
,
,
,
,
a.e. t>0,
satisfies /&(-,£) G M.g for a.e. t > 0. Obviously, if f ( z , t ) is a g- Loewner chain and f ( z ) = f(z,0), z € B, then / € 3j(B) by Theorem 8.1.6, and conversely, if / 6 5°(B) there exists a g-
8.3. Parametric representation
333
Loewner chain f ( z , t ) such that f ( z ) = f ( z , 0), z € B, in view of Theorem 8.1.5 and the above arguments. Example 8.3.4. Let / € H(B) be a normalized starlike mapping. Then f ( z , t) = etf(z) is a Loewner chain with transition mapping v(z,s,t) = /"V'VOO),
z € B,
t>s>0.
It is easy to see that dv — = -h(v,t),
where (8.3.2)
h(z,t) =
t>s, z&B, [Df(z)]-1f(z).
Moreover, f-1(e-tf(z))
= e-tf(z) + 0(e-2t)
as t -> oo
locally uniformly in 2, and hence lim etv(z,Q,t) = f ( z ) locally uniformly on t—>-oo
B. Since / is starlike, h(-,t] € M and thus / € S°(B) (see also [Kub-Por]). This result can also be deduced from Theorem 8.1.6, using the fact that the Loewner chains for starlike mappings are 0-Loewner chains with #(£) = (1 + C)/(l — 0- We note that for the case of a starlike map, the mapping h(z, t) in Loewner's equation
^(*,t) = Df(z,t)h(z,t)
t > 0, z € B,
is given by (8.3.2). Remark 8.3.5. We may use Theorems 8.1.6, 8.1.9 and 8.2.1 to conclude that the class of spirallike maps of type a, |a| < ?r/2, is also a subclass of S°(B). The Loewner chains which characterize the above mappings are gLoewner chains with g(Q = (1 + C)/(l - C)> C € U. A case of particular interest in our study is the set Sg(B) with g(£) = l+£, £ € U. In this case g(U) is the open disc centered at 1 and of radius 1. Therefore Mg =
l*ll:
334
Loewner chains in several complex variables
Let u G Cn with ||u|| = 1. For a normalized locally biholomorphic mapping / on B, let
,u
a-
In Section 6.3, we have defined the set Q of quasi-convex mappings of type A, as the set of normalized locally biholomorphic mappings / : B —>• C71 such that Re Gf(a,(3] > 0 for all a,/3 e C, \/3\ < \a\ < 1, and any u € Cn, ||u|| = 1. We shall refer in this section to the set Q as the set of quasi-convex mappings. We also proved that K(B) cQc S*(B}. In fact we proved in Theorem 6.3.19 that if / £ Q then / is star like of order 1/2, that is / satisfies the relation
which is equivalent to (8.3.3) If we set h(z) = [Df(z)] 1f(z) and use the above inequality, we see that h € Mg. Now, let f ( z , t ) = e t f ( z ) . Since / € £, it follows that / € S*(5), and thus f ( z , t ) is a Loewner chain. Using similar reasoning as in Example 8.3.4, we deduce that f ( z , t) is a p-Loewner chain and / € Sg(B). Therefore we have proved that
K(B)cGcS°g(B)
with 0(C) = i + C, C e ^ -
These inclusion relations between K(B), Q and Sg(B) with ^(C) = 1 + C were one of the motivations for studying the set Sg(B}. Thus some important subclasses of S(B} are in fact subclasses of Sg(B) for certain choices of g.
8.3.2
Growth results and coefficient bounds for mappings in
One of the main results of this section is the following growth theorem for the class 5°(B) ([Koh9], [Gra-Ham-Koh]):
8.3. Parametric representation
335
Theorem 8.3.6. Let g : U —> C satisfy the assumptions of Definition 8.3.1 andlet f £S°(B). Then
„
i"11*11 r
i
1 dx
• r / x /-v T " J1 x~> lmin{s(x),0(-z)} Proof. First we note that it is not difficult to prove the convergence of the above integrals, using the fact that 0(0) = 1 and Re 0 on U. To establish the bounds (8.3.4) we shall need some technical lemmas, as follows. Lemma 8.3.7. Let g G H(U) satisfy the assumptions of Definition 8.3.1 and let h € M.g. Then
(8.3.5)
exp Jo /
||z||2min{0(||z||),<7(-|NI)} < Re (h(z),z) < \\z\f m*t{g(\\z\\),g(-\\z\\)},
z€ B.
Proof. Let z E B \ {0}, and let p : U ->• C be given by
p(0 = i,
C = o.
Then p e H(U), p(0) = g(0) = 1, and since h € M.g we deduce that p(U) Q 9(U). Therefore p -< g, and from the subordination principle it follows that p(Ur) C p(J7r), r € (0, 1). On the other hand, combining the maximum and minimum principles for harmonic functions with (8.3.1), we deduce that min{<7(|C|), ,
C € U.
Setting (, = \\z\\ in the above relation, we obtain (8.3.5), as desired. Lemma 8.3.8. Suppose thatg satisfies the assumptions of Definition 8.3.1 and that h satisfies conditions (i)-(ii) of Definition 8.3.2. For z £ B and s > 0, let v = v(z, s, t) be the solution of the initial value problem
dv — = — h(v,t)
a.e. t>s, v(z,s,s) = z.
336
Loewner chains in several complex variables
Then INI
f \
1
1 dr 1 g±
/ I
1
I
mm
||t>CM,t)|| L
i9(x),g(-x)}
I
fl'T*
1 I
^
J x
for z £ B and t > s > 0. Proof. Fix s > 0 and z € B \ {0}, and let v(t) - v(z, s,t). Then
d||v||
1_
/dv
\
-^r- = ^Re ( — ,v } ,
a.e.
i > s,
and using the differential equation satisfied by u, one deduces that (8.3.7)
From (8.3.5) and the fact that Re g(() > 0 on [/, one obtains -—• < 0 dt a.e. t > s, which means that ||f(i)|| is strictly decreasing from \\z\\ = ||v(s)|| to 0 as t increases from s to oo. Integrating both sides of (8.3.7) with respect to t, making a change of variable and using (8.3.5), we obtain _ rlMI J\\z\\ =
dx xmm{g(x),g(-x)}
_ f* 1 Js mm{g(\\v(r)\\),g(-\\v(r)\\)}\\v(r)\\
d\\v(r)\\ dr
ftdT ~ Js
=t_tl
and
_ /J \\z\\
dx xmax{g(x),g(-x}}
i_
i;(r)||),p(-||t;(r)||)}||t;(r)||
«r)iu< rdr=t-a dr
' Js
Adding log ||v|| — log \\z\\ to both sides of the last two equations and exponentiating gives (8.3.6). We now return to the proof of Theorem 8.3.6. Since / 6 Sg(B) we have (8.3.8)
lim e*v(z,t) = f ( z )
8.3. Parametric representation
337
locally uniformly on B, where v = v(z,i) is the solution of the initial value problem
dv
— = —h(v,t), C/it
a.e. t > 0,
v(z,Q) = z,
for all z 6 B. Setting s = 0 in (8.3.6), we conclude that
/•Ml
r
i
i ,jT
/ - / -1 - < ./IK*,t)|| LmaxMz),^-*)} J x
dx IK*,t)|| [™™{9(x),9(-x)}
M z
for z & B and £ > 0. Since lim et||v(z,i)|| = ||/(z)|| < oo,
t—»oo
we must have lim |K*,t)|| = t->00 lim e~*||e*t;(*,*)|| = 0. If we now let t —>• oo in the above inequalities and use (8.3.8), we obtain the bounds (8.3.4), as claimed. This completes the proof of Theorem 8.3.6. Of particular interest in Theorem 8.3.6 is the case g(£) = (1 + C)/(l ~ C) for £ € U. This gives the following growth theorem for mappings in S°(B): (Cf. [Porl], [Koh9], [Gra-Ham-Koh]. Compare with Corollary 8.1.12.) Corollary 8.3.9. If f e S°(B) then
(i + IW
~ (i-W
7%ese estimates are sharp. Consequently, f ( B ) D #1/4 • As we have already seen, all normalized starlike mappings of B satisfy this growth result. Similarly, spirallike mappings of type a, a 6 (—7r/2,7r/2), belong to S°(B) and hence satisfy the same growth result. However, in general this is not true for spirallike mappings with respect to linear operators. To see this, we give an example of a family of spirallike mappings with respect to a diagonal matrix whose growth cannot be estimated from above [Ham-Koh4]. Example 8.3.10. Let n = 2 and / : B C C2 ->• C2 be given by f ( z ) = (z\ + a,Z2,Z2), z = (zi,Z2) G B. Define A e L(C 2 ,C 2 ) by A(z) = (2*1,32) for * = (*i,*2) G C2. Then it is easy to deduce that
338
Loewner chains in several complex variables
and hence / is a spirallike mapping relative to A for any a € C. However, if ZQ = (0, 1/2) it is easy to see that ||/(^o)|| -» oo as |a| —>• oo. Another consequence of Corollary 8.3.9 is that S°(B) is a normal family. This implies that S(B) is a larger class than S°(B), except in the onedimensional case, since only in dimension one is S(B) a normal family. Actually we can prove more, namely the compactness of S°(B) [Gra-Koh-Koh2] . Corollary 8.3.11. S°(B) is a compact set in the topology of H(B). Proof. It suffices to show that S°(B) is closed. For this purpose, let {/fc}fceN C S°(B) be such that fk —>• / locally uniformly on B as k —>• oo. Then for each k € N, there is a Loewner chain fk(z,t) such that {e~tfk(z,t)}t>o is a normal family and fk(z) = fk (z, 0), z € B. From Theorem 8.1.14 we deduce that there is a subsequence {/fcp(2,t)}PeN such that fkp(z,t) -* f ( z , t ) locally uniformly on B for each t > 0, f ( z , t ) is a Loewner chain and {e~tf(z,t)}t>o is a normal family. It is obvious that f(z) = /(z,0), z € B, and in view of Theorem 8.1.9, one concludes that / € S°(B). This completes the proof. The following example shows that S°(B) is also a proper subset of S1(B) in higher dimensions. Example 8.3.12. (i) As noted in Remark 8.1.7 (ii), if f ( z , t ) is a (normalized) Loewner chain and
whose initial element f(z) = f ( z , 0) satisfies
T
Choose r such that 7--2^ = p, where p is as above. It is evident that (1-r) is the first element of the Loewner chain $ o /(z, t), and hence $ o / G S1(B),
8.3. Parametric representation
339
but
In view of Corollary 8.3.9 we conclude that $ o / ^ S°(B). (ii) For example, let n = 2 and let 3>(z) = (zi,Z2 + z2), z = (21,22) € C2. Then $ is an entire normalized biholomorphic mapping of C2. Also if
then
is a Loewner chain. Let f ( z ) = /(z,0), z 6 B. Then for each r 6 (0, 1),
and hence $ o / ^ S0^). We remark that {e~"*$ o /(z, £)}*>o is not a normal family. We have seen that all normalized convex and quasi-convex mappings of B belong to S%(B), with g(Q = I + C, C € C/. The growth and covering theorems for convex and quasi-convex mappings (see Theorem 7.2.2 and [RopSu2, Theorem 4.1]) may therefore be deduced from Theorem 8.3.6: Corollary 8.3.13. Let g(C) = l + tandf€ Sj(B). Then
These estimates are sharp. Consequently, f ( B ) D -61/2We now pass to coefficient estimates for the class Sg(B). There are some generalizations of one- variable results, but also some unexpected phenomena. Poreda [Porl] obtained the second order coefficient bounds for mappings which have parametric representation on the unit polydisc of C71. The estimate (8.3.9) was obtained by Kohr [Koh9, Theorem 2.4]. (The reader may compare with [Gra-Ham-Koh] for the case of an arbitrary norm in C1.) The estimate (8.3.10) was obtained by Graham, Hamada and Kohr [Gra-Ham-Koh].
Loewner chains in several complex variables
340
Theorem 8.3.14. Let g : U —>• C satisfy the assumptions of Definition 8.3.1 andeSB. Then (8.3.9)
-(D*f(0)(w,w),w)
<\g'(0)\,
Consequently, (8.3.10) Proof. First we prove (8.3.9), using similar arguments as in the proof of [Porl, Theorem 3]. Since / € Sg(B) there is a g-Loewner chain /(z, t) such that f ( z ) = f ( z , 0), z G B. Also there is a mapping h = h(z,t) e Mg for each t > 0, measurable in t for each z e B, such that for almost all t > 0,
Moreover, if s > 0 and v = v(z,s,t) is the unique solution of the initial value problem
dv — = -h(v,t),
a.e. t>s,
v(z,s,s) = z,
then f ( z , s ) = lim e*v(z, s, t—>oo
locally uniformly on B, by Theorems 8.1.6 and 8.1.9. Fix z e B \ {0} and t0 > 0. Let
1,
C = 0.
Then pto <E H(U], pto(U) C g(U], and p to (0) = 5(0). Hence pto -< g, which implies |p£0(0)| < J5'(0)| by the subordination principle. A Taylor series expansion in £ gives
8.3. Parametric representation
341
and identifying the coefficients yields
We therefore obtain the estimate (8.3.12) Fix T > 0 and integrate both sides of the equality (8.3.11), to obtain
f ( z , T) - /(z, 0) = / Df(z, t)h(z, t)dt, Jo
z e B.
For z € B \ {0} define GZ,HZ:U-^^ respectively by
and
r
HZ(C,}= I Jo
Then Gz = Hz on U and Gz is clearly holomorphic on U. Differentiating twice the mapping Hz and noting that £)/(0,t) = e*I and /i(0, t) = 0, we deduce that d2Hz
(0)= f Jo
or equivalently D 2 /(0,T)(z,z)-D 2 /(0,i
= /" [2£>2/(0, Jo
, t ) ( z , z)]dt.
Next let ,z)- f Jo
for fixed z E B and £ > 0. Then it is obvious that qz(Q) = 0, and we also obtain for almost all t > 0 that
342
Loewner chains in several complex variables = e~2t [ - 2D2f(0, t)(z, z) + ^£>2/(0, *)(*, z) - etD2h(0, t ) ( z , z}] = 0,
making use of the fact that £>2/(0, t)( Z| Z) - £>2/(0, 0)(z, z) = f\2D2f(0, r}(z, z) + erD2h(0, r)(z, z)]dr. Jo Consequently g z (t) =
rT
I Jo
t
e-
D2h(0,t)(z,z)dt.
Prom this we conclude that e-2T(D2f(Q,T)(z,z),z)
(8.3.13)
rT
= I Jo
- (D 2 /(0,0)(z,z),z)
e-t(D2h(0,t)(z,z),z)dt.
Next, we note that Lemma 8.3.8 gives (8.3,4)
ll/t
Using Cauchy's formula ,
0 < r < 1,
for w € C1, ||u|| = 1, and taking into account (8.3.14), we easily obtain that lim e-2TD2f(0,T)(z,z)
T->oo
= 0.
Next, letting T —> oo in (8.3.13) and using the above equality and (8.3.12), we conclude that
It is now clear that this inequality is equivalent to (8.3.9), if we take into account the fact that f ( z ) — f ( z , 0) for z € B.
8.3. Parametric representation
343
Finally we prove (8.3.10). For this purpose, let P2(z) = -D2f(Q)(z,z). Then P2(z) is a continuous homogeneous polynomial of degree 2, and hence from Lemma 6.1.37 we deduce that \\P2\\ <*\V(P2)\. The estimate (8.3.9) yields that \V(P2)\ < |
< 2,
HI = 1.
This estimate is sharp. Consequently, 112
It would be interesting to determine whether the following conjecture is true for the class S°(B). This is a version of the Bieberbach conjecture in several complex variables.
Conjecture 8.3.16. If / € S°(B) then fc!x
for all w e C1, H| = 1, and k 6 N, k > 2. A second possible version of the Bieberbach conjecture for 5°(J3), namely
(8.3.15)
— Dkf(0)(wk)
< k,
\\w\\ = 1,
k e N,
is false in dimension 2 or greater, in fact it fails for k = 2 as Example 8.3.18 shows. However, in the case of the unit polydisc P of C™, the above inequality is
344
Loewner chains in several complex variables
true for k = 2 (and open for k > 3) and was proved by Poreda [Porl, Theorem 3]. Gong [Gon5, Theorem 5.3.1] has recently proved that if / is normalized starlike on the unit polydisc of C™, then the bound (8.3.15) holds for k = 2, 3, and is open for k > 4 (see also Section 7.1.2). We therefore formulate Open Problem 8.3.17. Find the sharp upper bounds for \\~Dkf(0)(wk} II K .
,
fc>2,
|H = 1,
when / e S°(B). We now give the example which disproves (8.3.15): Example 8.3.18. Let n = 2, a € C, |a| = 3\/3/2 and / : B C C2 ->• C2 be given by f ( z ) = (zi + az$, z 2 ),
z = (zi,z2) <E B.
Then / € S°(B) because / is normalized starlike on B (see Problem 6.2.1). However, a simple computation yields that
where a
ij
Therefore D2f(0)(w,w) (0, 1) we conclude that
=
( 2aw2, i = l, j = 2 I 0, otherwise.
S
= (2aty2,0) for w = (w\,wz) e C2, and for w =
'II
2
Another case of special interest in Theorem 8.3.14 is the case #(C) = 1 + £. This yields a coefficient estimate which is satisfied by the normalized convex and quasi-convex mappings (cf. Theorem 7.2.16 with p = 2): Corollary 8.3.19. If f e S°(B) with g ( ( ) = 1 + C then 2X
This estimate is sharp. Moreover,
8.3. Parametric representation
345
and for A; € N, k > 3, the estimates (8.3.16) hold.
Proof. It suffices to prove the bounds (8.3.16). To this end, fix k € N, k > 3, and w € C1, \\w\\ = 1. Using the Cauchy formula
lCI=r
and taking into account Corollary 8.3.13, we easily obtain
< Setting r = 1 — l/k in this inequality gives
Finally, in view of Theorem 7.2.19 the bounds (8.3.16) follow. This completes the proof. Remark 8.3.20. Note that if / : B —> C" is a normalized convex mapping, then in view of (7.2.12) we have the sharp bounds (8.3.17)
||^D fc /(0)||
fceN.
II
However, if / e S%(B) \ K(B], with g((,} = 1 + C, then (8.3.17) need not be satisfied in dimension n > 2. To see this, consider the case n = 2 and let f ( z ) = (zi + az$,z2) for z = (zi,z 2 ) G B and |a| = 3V5/4. Then / € Q by Problem 6.3.7. However, since |a| > 1/2, it follows from Example 6.3.13 that / is not convex on B. Moreover, if w = (0, 1), then
Hence
346
Loewner chains in several complex variables
We now give an example of a polynomial map in S(B) which does not satisfy the conclusion of Corollary 8.3.15. This gives another way of seeing that S°(B) is a proper subclass of S(B) in several variables. We also have seen that S°(B) is a proper subclass of Sl(B] in higher dimensions. Example 8.3.21. Let n > 2 and / : B -> Cn be given by f(z]
= (Zi + az\, Z2,..., Zn], Z = (zi,...,Zn)eB,
where a € C with |o| > 4\/2. Clearly / 6 S(B}, and a short computation yields the relation ±(D2f(Q)(w,w),w) Let w = (r, r, 0 , . . . , 0), where r = l/\/2. Since |a| > 4i/2, we obtain = |a|r3 = -^L > 2. ' 2v/2
Another case of special interest in the study of the set Sg(B) is the case = (1 + cO/(l — cO, C € U, where 0 < c < 1. Obviously, g is a univalent function on U that satisfies the assumptions of Definition 8.3.1. The image of the unit disc under g is the disc centered at (1 + c 2 )/(l — c2) and of radius 2c/(l - c2). Let S°(J3) denote the set S°(£) when 0(0 = (1 + <)/(! - <), c € (0,1). In the case of one complex variable such 0-Loewner chains, with 0(0 = (1 + cO/(l — cO> were intensively studied by Becker [Becl] and called c-chains. The interest of a c-chain f ( z , t ) arises from the fact that its first element f(z] = /(2,0) can be extended to a quasiconformal homeomorphism of the whole complex plane (see [Becl,2]). We next consider some examples of mappings in S®(B], c £ (0,1). Chuaqui [Chu2] introduced a subset of S*(B] which he called the set of strongly starlike mappings. He proved that these mappings can be extended quasiconformally toC 1 . Definition 8.3.22. Let z e C n , ||z|| = 1 and / € S*(B). We say that / is strongly starlike if the values of
8.3. Parametric representation
347
lie in a fixed compact subset of the right half-plane, independent of z. Next, let c € (0,1) and / be a normalized locally biholomorphic mapping on B such that 1_1 ,/
x
v
1 +
2c 1-c 2 '
C
1-C2
Clearly if / satisfies the above assumption, then / is strongly starlike. Moreover, since /(z, t) = etf(z) is a p-Loewner chain, with #(£) = (l+c£)/(l — c£), one deduces that / € «$
These estimates are sharp. Moreover, < 2c,
and <8c.
Remark 8.3.24. In this section we have studied the class Sg(B) for certain choices of the univalent function g, and we have seen that in higher dimensions S°(B) g S(B) and also S°(B) g Sl(B). However, the class S°(B) contains some important classes of mappings, including all normalized starlike and spirallike mappings of type a, \a\ < Tr/2. The normalized convex and quasiconvex mappings belong to Sg(B) with
348
Loewner chains in several complex variables
Problems 8.3.1. Show that if / : B -» Cn is a normalized locally biholomorphic mapping on the Euclidean unit ball B of C™ such that (l + \ \ z \ \ ) \ \ ( D f ( z ) ] - 1 D * f ( z ) ( z , . ) \ \ < 2 ,
z€B,
then / € 5J(B) with g(Q = l + ^^U. (Cf. Graham, Hamada and Kohr, 2002 [Gra-Ham-Koh]. See also [Ham-KohLic2].) Hint. Show the inequality
» f ( ( y 11) \ }
8.3.2. Let / € S and w G C1 with \\w\\ = 1. Also let F(z) = zj ^ '
/;
,
G 5. Show that F <E S°(£). In particular, if / € 5* then F € £*(£). 8.3.3. Let c <E (0,1) and fj <E # (U) be such that /j(0) = /J(0) -1 = 0 and :, C € U, j = 1, 2 , . . . , n. Also let f ( z ) = ( f i ( z i ) , . . . , / n («n)), C/j(0 1 z = (zi,... ,zn} e B, where B is the Euclidean unit ball of Cn. Show that
/ e SQC(B). Hint. Show that
Ikl
8.4
Applications of the method of Loewner chains to univalence criteria on the unit ball of Cn
There are some further applications of the method of Loewner chains to univalence criteria in several complex variables. The main result of this section is an extension of Becker's criterion (Theorem 3.3.1) to higher dimensions. This extension was obtained by Pfaltzgraff in 1974 in his original paper on Loewner chains in several variables [Pfal].
8.4. Univalence criteria
349
Theorem 8.4.1. Let f : B —> C71 be a normalized locally biholomorphic mapping which satisfies (8.4.1)
(1 - \\zf}\\[Df(z}]-Wf(z}(z,
0|| < 1, z G B.
Then f is univalent on B. Proof. We shall prove that the condition (8.4.1) enables us to embed f ( z ) as the initial element of a Loewner chain. Consider the mapping f ( z , t) = f(ze~i} + (e* - e-t)Df(ze-t)(z),
z<-B,
t>Q,
which obviously satisfies /(z,0) = f ( z } . We shall prove that f ( z , t ) satisfies the hypotheses of Theorem 8.1.6. Clearly /(-,*) 6 H(B), /(O,*) = 0 and D/(0,t) = e*I for t > 0. We note that f ( z , t) is a C°° mapping on B x [0, oo). It is elementary to verify that lim e~ij(z, t) = z J v '
locally uniformly on B, and hence the condition (8.1.23) holds for F(z) = z. It is also elementary to verify that Df(z,t) = eiDf(ze-i] + e\l - e'^D2 f (ze^ze'1 , 0 =
etDf(ze-t)(I-E(z,t}},
where, for fixed (z,£) 6 B x [0, co), E(z,i) is the linear operator (8.4.2)
E(z, t) = -(l- e
Since 1 - e~2t < 1 — ||^e~*||2 for z € B and t > 0, the assumption (8.4.1) immediately implies that ||.Z?(;z,£)|| < 1. Therefore / — E(z,t) is an invertible operator. On the other hand, using (8.4.2) we obtain -*)(z) - (et - e-^^ze-^ze^z) = etDf(ze-t}[I
+ E(z, t}](z) = Df(z, t)(I - E(z, t}]-l[I + E(z,
350
Loewner chains in several complex variables We conclude that f ( z , t ) satisfies the differential equation ^(z,t) = Df(z,t)h(z,t),
zeB,
t>0,
where h(z, t) = [I - E(z, t)]~l[I + E(z, t)](z). It remains to verify that h(z, t) satisfies the assumptions of Theorem 8.1.3. Clearly /i(z, t) is holomorphic in z and measurable (in fact smooth) in t, and satisfies /i(0, t) = 0, Dh(0, t} = I. Moreover, from the inequality
<\\E(z,t}\\-\\h(z,t) + z\\<\\h(z,t) + z\\, we obtain that Re (h(z, t), z) > 0, for z 6 B \ {0} and t > 0. Consequently, the conditions (i) and (ii) of Theorem 8.1.3 are satisfied. Prom Theorem 8.1.6 we therefore deduce that f ( z , t ) is a Loewner chain, and hence f ( z ) = f(z,Q) is univalent on B. This completes the proof. In the literature there are various generalizations of the preceding result of Pfaltzgraff. One of these is an extension to higher dimensions of Ahlfors' and Becker's result (Theorem 3.3.2), due Chen and Ren [Che-Ren] and Curt [Cul]. The proof proceeds along similar lines to the above, and we leave it as an exercise for the reader. Theorem 8.4.2. Let f : B —> C1 be a normalized locally biholomorphic mapping. Let c G C with \c\ < 1 and c ^ — 1. Suppose that ||(1 - \ \ z f ) [ D f ( z ) ] - 1 D 2 f ( z ) ( z ,
•) + c||z||2/|| < 1, z e B.
Then f is univalent on B. We finish this section with an n-dimensional version of Theorem 3.3.9, which generalizes both Theorems 8.4.1 and 8.4.2. This result was obtained in [Cu-Pas]. We only sketch the proof, since it suffices to use similar arguments as in the proofs of Theorems 3.3.9 and 8.4.1. Theorem 8.4.3. Let F : B x Cn -» Cn satisfy the following assumptions: (i) The function L(z,t) = F(e~tz,etz) is holomorphic on B for all t £ [0, oo) and is locally absolutely continuous in t £ [0, oo) locally uniformly with respect to z 6 B.
8.4. Univalence criteria
351
(ii) DvF(u,v) is invertible for all (u,v) € B x C71, and there exists a function a : [0,oo) ->• C such that a(t) ^ 0, t > 0, a(-) € C^QO.oo)), |a(-)| is increasing on [0,oo), lim \a(t)\ = oo and t—>oo
e~* DUF(Q, 0) + 6*^^(0,0) = a(t)I, where DuF(u,v) (respectively DvF(u,v)) is the n x n matrix for which the ( i , j ) entry is TT—-(u,v) (respectively -^-(u,v)). OUj
OVj
{
F(e~ z 6 z\ i —-—7-^ \ forms a normal family on a(t) ) t>o
B.
Let V >
'
II«HS
V
and
(2, 2) is univalent on B. Proof. Since L(z, t) = F(e~*z, etz) it follows that L(z, t) = a(t)z + ... for z € B and t > 0, and it is obvious that DL(z, t) = e'DvF^z, e*z)(7 - £(*, *)), where for each (z, t) e 5 x [0, CXD), E(z,t) = -e-2*[DvF(e-tz,et2)]-1DuF(e-*z,etz). Using the maximum modulus theorem for holomorphic mappings (see Theorem 6.1.27), it is easy to show that ||£?(z,i)|| < 1 for z € B and t > 0, and hence / — E(z, t) is an invertible operator. Moreover, if h(z,t) = [I-E(z,t)]-l[I + E(z,t)](z),
zEB,
t>Q,
then h(z, t) is holomorphic in z and differentiate in t, and a short computation gives O T-
— (2, t) = DL(z, t)h(z, t), ot
ztB,
t> 0.
352
Loewner chains in several complex variables Since < \ \ E ( z , t ) \ \ - \ \ h ( z , f ) + z\\<\\h(z,t) + z\\,
we obtain Re (h(z, t), z) > 0 for z e B \ {0} and t > 0. Hence from Problem 8.1.2 we deduce that L(z,t] is a Loewner chain, and thus L(z, 0) = F(z, z} is univalent on B. Remark 8.4.4. Let / : B —> Cn be a normalized locally biholomorphic mapping. Also let Fk : B x C1 -» C n , k = 1, 2, be given by
Fi(u, v] = f(u) + Df(u}(v - u],
Setting successively F = Fk, k = 1,2, in Theorem 8.4.3 yields the results of Theorems 8.4.1 and 8.4.2.
Problems 8.4.1. Prove Theorem 8.4.2. Hint. Prove that f ( z , t ) = f ( e ~ t z ] + (1 + c)"1^ - e~t)Df(ze-t)(z) is a (non-normalized) Loewner chain. 8.4.2. Complete the details in the proof of Theorem 8.4.3. 8.4.3. Let / : B —> C71 be a normalized holomorphic mapping such that \\Df(z)-I\\
\z\\2[[Dg(z}}-lDf(z)
-/]+(!-
< 1.
8.5. Quasiconformal extensions
353
Prove that / is univalent on B. (Cf. Curt and Pascu, 1995 [Cu-Pas].) Hint. Use Theorem 8.4.3 for F(u, v) = f(u)+Dg(u)(v-u), u € B, v e C". 8.4.5. Let / : B —> C1 be a normalized holomorphic mapping and let G(z) be a nonsingular n x n matrix which is holomorphic with respect to z 6 B. Assume G(0) = / and the following conditions hold for all z e B: \\[G(z)]-lDf(z)-I\\
-/]+(!- \\z\f)[G(z)]-1DG(z)(zt-)\\
< 1.
Prove that / is univalent on B. (Ren and Ma, 1995 [Ren-Ma].) Hint. Prove that /(z, t) = f(ze~t) + (et — e~t)G(ze~t)z is a Loewner chain.
8.5
8.5.1
Loewner chains and quasiconformal extensions of holomorphic mappings in several complex variables Construction of quasiconformal extensions by means of Loewner chains
In this section we obtain sufficient conditions for a holomorphic mapping on the unit ball of C71, which can be embedded as the first element of a Loewner chain, to have a quasiconformal extension to a mapping of R2n onto M2n. For this purpose, we briefly recall some definitions and results concerning quasiregular and quasiconformal mappings. For further details, the reader may consult [Boj-Iw], [Car], [Vai], [Vuo], [Ric], [Res], [Mart-Ric-Vail,2]. In this section we shall consider quasiconformal mappings in higher dimensions. Definition 8.5.1. Let G be a domain in E m , m > 2, and W^loc(G) be the Sobolev space of maps in L^C(G), whose first order weak partial derivatives exist and belong to L^G). Also let / : G —> Rm be a mapping. We say that / is quasiregular (qr) on G if / G W^/oc^) and there is a constant K > I such that (8.5.1) \\D(f]x)\\m < A"detD(/;z), a.e. x 6 G.
354
Loewner chains in several complex variables
Here \\D(f-x}\\ = sup{\\D(f'tx)(v)\\ : \\v\\ = 1} and D ( f - x ) is the (real) Jacobian matrix of /. The smallest constant K > 1 for which (8.5.1) holds is called the dilatation (outer dilatation) of / in G and is denoted by K 0 ( f } . If / is quasiregular on G, then the smallest K > 1 for which the inequality det£>(/;z)
holds a.e. in G is called the inner dilatation of / and is denoted by K i ( f ) . The maximal dilatation of / is the number K(f) = max{Ki(f),K0(f)}. If K e [1, oo) is such that -K"(/) < K, we say that / is K- quasiregular. We note that if f2 is a domain in C1, then a holomorphic mapping / : fi —> Cn is quasiregular as a mapping from E2n to R2n if and only if there is a constant K > 1 such that \\Df(z)\\n
zen.
This follows from (6.1.8). The geometric interpretation of quasiregularity is that a nonconstant quasiregular mapping maps infinitesimal spheres to infinitesimal ellipsoids whose major axes are less than or equal to a fixed constant times their minor axes. It is known that if / : G —>• Rm is a nonconstant quasiregular mapping, then detD(/;z) > 0 a.e. on G ( [Mart- Ric- Vail], [Boj-Iw]). Also if / is a nonconstant quasiregular holomorphic mapping, then / is locally biholomorphic (see [Mard-Ric]). In this section we study quasiregular holomorphic mappings. A homeomorphism / : G -> Mm which is quasiregular (A"-quasiregular) on G is called quasiconformal (qc) (-K"-quasiconformal (K-qc)). It is well known that there are many equivalent ways of defining quasiconformal mappings (see [Vai], [Vuo], [Res], [Ric], [Car], [Mart-Ric-Vail,2]). We recall that if E = {x € Em : a, < a;, < 6»,i = 1, . . . ,ra} is a closed interval, then a map / : £ ? — > • IRm is said to be absolutely continuous on lines (ACL) if / is continuous and is absolutely continuous on almost every line segment in E which is parallel to the coordinate axes. Moreover, if V is an open set in
8.5. Quasiconformal extensions
355
R m , then a map / : V -> IRm is ACL if f\E is ACL for each closed interval E of V. For further details, see [Vai]. The necessary and sufficient condition for .K"-quasiconformality in the following theorem is known as the analytical definition of .fiT-quasiconformality (see e.g. [Vai]). Theorem 8.5.2. Let G\ and GZ be domains in R m . A homeomorphism f of GI onto GI is a K-quasiconformal mapping if and only if the following conditions hold: (i) f is ACL; (ii) f is differentiate a.e. on G\; (Hi) \\D(f;x)\\m/K < |det£>(/;x)| < Kl(D(f;x})m, a.e. on GI. We next prove a beautiful quasiconformal extension result due to Pfaltzgraff [Pfa2], which states that if / is holomorphic and quasiregular on B and satisfies a slightly stronger condition on [Df(z)]~1D2f(z)(z, •) than (8.4.1), 1 then / extends to a quasiconformal homeomorphism of C onto Cn. For related results in one variable, the reader may consult [Becl-4]. The quasiconformal extension can be given explicitly in terms of an appropriate Loewner chain. Also we shall discuss a quasiconformal extension result of Hamada and Kohr [Ham-Kohl2] for quasiregular holomorphic mappings which can be embedded in Loewner chains f ( z , t ) such that {e~tf(z^t)}t>o is a normal family on B. Finally we shall apply this result to the study of quasiconformal extensions of strongly starlike and strongly spirallike mappings of type alpha on the unit ball of C". Before giving the main result of this section, we need the following lemmas. The first result is an n-dimensional version of a well-known theorem of Hardy and Littlewood (see [Gol4, p.411-413]), and will be useful in obtaining a continuous extension to B of a locally biholomorphic quasiconformal map on B. A proof may be found in [Pfa2]. Lemma 8.5.3. Let c € [0,1), M > 0, and h : B —>• C be a holomorphic function such that (8.5.2)
dh dzj (*)
M
Then h has a continuous extension to B, again denoted by h, and there is
356
Loewner chains in several complex variables
a constant K > 0 such that (8.5.3)
\h(z) - h(z'}\ < K\\z - z'\\l~c,
z, z1 € B.
An immediate consequence is the following [Pfa2]: Lemma 8.5.4. Let f : B —> Cn be a holomorphic mapping. Also let M > 0 and c € [0, 1) be such that ,
(8.5.4)
.
T/ien / /ias a continuous extension to B, again denoted by f , and there is a constant K > 0 such that \\f(z)-f(z')\\
z,z'£B.
We next consider a refinement of Theorem 8.4.1. The growth result (8.5.9) below is contained in [Pfal, Theorem 2.4] and the other statements can be obtained directly from Theorem 8.1.5 and Corollary 8.1.11. (See also the proof of [Pfal, Theorem 2.4].) We have Lemma 8.5.5. Let f : B —>• C71 be a normalized locally biholomorphic mapping and let c 6 [0, 1). Assume (8.5.5)
(1 - \\z\f )\\[Df(z}]~1D2f(z}(z,
-)\\ < c,
z e B.
Then f is univalent on B (in fact, f has parametric representation on B) and can be embedded as the first element of the Loewner chain (8.5.6)
f ( z , t) = f(ze-*) + (et - e-t)Df(ze-t}(z},
z € B, t > 0.
Further, let v = v(z,s,t) be the transition mapping associated to f ( z , t ) . Then pe t\\v(? uz sA t\\\ E
PeS\\Z\\ z (l-c\\z\
and
for all z € B and t>s>0. Consequently, v(B, s, t) C B for t > s > 0 and
(8 5 9)
--
^ l / ( z ' s ) l l 2 ' ^B,.>O.
8.5. Quasiconformal extensions
357
Proof. The fact that f ( z , t) is a Loewner chain has been proved in Theorem 8.4.1 since c < 1. On the other hand, since lim e ~ t f ( z , t) — z locally uniformly t— ¥00 on J3, we deduce from Theorem 8.1.5 and Corollary 8.1.11 (see also Remark 8.3.3) that / has parametric representation on B. Next, let E(z,t) be the linear operator given by (8.4.2), i.e. E(z,t) = -(1 - e-2t)[Df(ze-t)]-1D2f(ze-t)(ze-t,
•), z e B, t > 0.
In view of (8.5.5), we deduce that ||.E(2,£)|| < c, z G B, t > 0, and using Schwarz's lemma for mappings, we obtain that ||.E(2,£)|| < c\\z\\ for z e B and t > 0. Further, let h : B x [0, oo) -» C1 be denned by (8.5.10)
h(z, t) = [I- E(z, t)]~l[I + E(z, t)}(z),
zeB, t>0.
Since ||.E(,z,£)|| < c\\z\\, it follows that h ( - , t ) 6 H(B), t>0, and it is easy to see that \\h(z,t) - z\\ = \\E(z,t)(h(z,t) + z)|| < \\E(z,t)\\ • \\h(z,t) + z\\ < c\\z\\- 11/1(2, t) + z\\,
zeB,t>0.
Consequently, we obtain the relations 11\ (8.5.11) /
O K
II II ~ II H ^ IIL/ *MI / I I z IkIL , c\\z\\ !! ~< "IIM^OII v ' / n -< "\\ "l-c\\z\\ 1
C
2
and
(8.5.12)
IWI2^HS < R* <MM),z> < W2^!;!, '
II
II
—
II
II
for all z e B and t > 0. Indeed, the relation (8.5.11) is a simple application of the inequality
|||fc(*,*)l| - Pill < c||z||pCM)ll + Iklll,
z € B, t > 0.
The right inequality in (8.5.12) is obvious and in order to deduce the left hand inequality, we use ||/i(2,t) - z\\2 < c2\\z\\2 • \\h(z,t) + z\\2,
z£B,t>0,
358
Loewner chains in several complex variables
to obtain 2Re <MM),*>(1 + c 2 N| 2 ) > (1 - c2||z||2)(||MM)||2 + INI 2 )
The claimed relation now follows. Further, since v(z,s,i) is the transition mapping associated to f ( z , t ) , it follows from (8.1.30) that v(z, s,t) is the solution of the initial value problem
dv — = -h(v, t), t> s, v(z, s, s) = z, for all z e B and s > 0. Using similar reasoning as in the proof of Lemma 8.1.4 and taking into account the relations (8.5.11) and (8.5.12), we obtain (8.5.7) and (8.5.8). Finally letting t —> oo in (8.5.7) and (8.5.8), and using the fact that f ( z , s ) — lim etv(z,s,t] locally uniformly on B, we conclude (8.5.9). t—»oo It remains to prove that v(B,s,t)cB,
t>s>Q.
For this purpose, we use the equality r\
\\v(z,s,t)\\-l — \\v(z,s,t)\\ = -Re (h(v(z,s,t),t),v(z,s,t)),
t > s,
and in view of (8.5.12) we obtain for s
...
., .
. ,, 1 — c||t;(z, s.t}\\
sn.(*,.,on < -ii"(«.».«)ii 1+c j B |,; t : t jj
l —c < -— e-
Integrating this inequality with respect to t and using the fact that v(z, s, s) = z, we obtain *,M)|| < N|exp{ - [ Thus v(B,s,t) C B, as desired. This completes the proof. The following result of Pfaltzgraff [Pfa2] shows that an estimate on [Df(z}]~lD'2f(z, •) combined with a quasiregularity assumption allows one to extend a holomorphic map continuously from B to B.
8.5. Quasiconformal extensions
359
Lemma 8.5.6. Let f : B —> C™ be a holomorphic mapping which is quasiregular on B, and suppose there is a constant c < 1 such that (1 - \ \ z \ \ 2 ) \ \ [ D f ( z ) ] - l D 2 f ( z ) ( z ,
(8.5.13)
-)|| < 2c, zE B.
Then there is a constant M > 0 such that (8.5.14)
p>/(z)|| < _ _
Z€B,
and f has a Holder continuous extension to B (again denoted by /) which satisfies the relation \\f(z}-f(z'}\\
(8.5.15)
z,z'eB,
for some constant K > 0. Proof. Taking into account Lemma 8.5.4, it suffices to prove the relation (8.5.14). For this purpose, fix u 6 ~B and let V(0 = detjD/«) = -M^C), |£| < 1. Then ^ is a nonvanishing holomorphic function on the unit disc U. Considering the trace formula for differentiating the determinant of an n x n matrix- valued holomorphic function (see [Golb] and also Section 10.2.1), one deduces that C^'(C) = V'(Otrace{p/K)]-1Z>2/K)K, •)>. ICI < 1-
(8.5.16)
Further, if A = (ay)i <*,.,•<„ 6 L(Cl,Cri) and 0 < ai < a2 < . . . < an are the eigenvalues of A* A, then ||-A||2 = an- Moreover, since trace{A*A} = \ \Q>jk\ = ai + ... + an < nan = n\\A\\ , j,k=i we obtain (8.5.17)
|trace{A}| = I Va^- < fnV |a^fl1/2 < n||A||.
Hence, from (8.5.13), (8.5.16) and (8.5.17) we obtain
2nc
2nc
.
'
ICI
360
Loewner chains in several complex variables Elementary computations using the above inequality yield that
, ICKi. Next, setting u = 2/||z|| and (, = \\z\\, z € B \ {0}, in the above relation, we deduce that there is a constant MI > 0 such that (8.5.18)
|j /W |<_^_,
,efl.
Finally, using the quasiregularity of/ and the relation (8.5.18), we conclude that there is a constant M^ > 0 such that \\Df(z)\\n < M,\Jf(z)\
<
nc ,
z € B.
Letting M = \M\M-2\ 'n in the above relation, we obtain (8.5.14), as desired. This completes the proof. We are now able to prove the main result of this section, due to Pfaltzgraff [Pfa2j. (For the case of one variable see [Becl].) Theorem 8.5.7. Let f : B —>• Cn be a normalized holomorphic mapping which is quasiregular on B, and let c 6 [0,1). Assume f satisfies the relation (8.5.5). Then f can be extended to a quasiconformal homeomorphism o/IR2n onto IR2n. The extension F is given by (8.5.19)
F(z) = \ (' z' A ~ \/(R,log|N|J, W > 1 ,
where f ( z , t ) is the Loewner chain given by (8.5.6). Proof. We divide the proof into the following steps: Step 1. First we show that if /(z, t) is the Loewner chain given by (8.5.6), then ft(z) = f ( z , t } can be extended to a continuous and injective mapping on B (again denoted by ft(z)) for each t > 0, such that (8.5.20)
f(B, s) C f(B, t),
0<s
Indeed, it is obvious that /(•,£) is holomorphic on B for t > 0. Moreover, in view of the relation (8.5.5) and Lemma 8.5.6, we deduce that f ( z ) = f ( z , 0)
8.5. Quasiconformal extensions
361
has a continuous extension to B (again denoted by f ( z ) } which satisfies a Holder condition \\f(z}-f(z'}\\
z,JeB,
for some absolute constant A > 0. Moreover, if u(z, s, t) is the transition mapping associated to f(z,t), then using the inclusion v(B,s,t) C B for t > s > 0, which follows from Lemma 8.5.5, the equality f ( z , s ) = f(v(z,s,t),t) and the continuity of /(•,£) on B, t > 0, we deduce that f(v(B, 5, t), t) = f(v(B, s, t), t) C /(B, t),
t>s>0,
which yields (8.5.20). Further, the relation (8.5.20) and the continuity of /,(•) on B enable us to extend v(z, s,t) continuously in the first variable to B by the formula (8.5.21)
v(z,a,t) = f r l ( f s ( z ) ) ,
z£B,t>s>0.
For z € B and 0 < s < t, we obtain in view of (8.5.11) that
Using the continuity of v(-, s, t) on B, we obtain from the above inequality that 1 +c (8.5.22) \\z-v(z,Stt)\\<—-(t-8), \\z\\ < 1, t>s>0. Now we show the injectivity of /«(•) on B. For this purpose, let z,z' 6 B be such that f , ( z ) = fa(z'). Then ft(v(z,s,t)) = ft(v(z',s,t)) for t > s > 0, and since v(B,s,t) C B, t > s, and ft(-) is univalent on B, we deduce that v(z, s, t) = v(z', s, t). Letting t \ 5, we conclude that z = z' , as claimed. Moreover, using again the fact that fs(B) C f t ( B ) for t > s > 0 and the injectivity of /(•, t) on 5, it is easy to see that the mapping F given by (8.5.19) is univalent on C".
362
Loewner chains in several complex variables
Step 2. We next prove the continuity of F in C". For this purpose, we prove the following Holder conditions (8.5.23)
e-t\\f(z,t)-f(z',t)\\
(8.5.24)
\\f(z,t)-f(z,s)\\
z . z ' e B , t > 0, z£B,Q<s
where A\ and A^ are constants which are independent of s and t. In order to show (8.5.23), we use the equality (see the proof of Theorem 8.4.1) e ~ t D f ( z , t) = Df(ze~t)[I - E(z, £)], z € £, t > 0, the fact that ||J5(z,t)|| < c\\z\\, z 6 B, t > 0, and Lemma 8.5.6, to deduce that there exists a constant MI > 0 (independent of t) such that (8.5.25)
||e-'0/OM)|| = \\Df(ze-*)[I - E(z,t)]\\
In view of Lemma 8.5.6, we conclude that e~*/(-,£) has a continuous extension to B which satisfies (8.5.23) for some absolute constant A\ > 0. Furthermore, using the relations (8.5.21), (8.5.22) and (8.5.23), we obtain for z e 13 and 0 < s < t that , t) - f ( z , s}\\ = \\f(z, t) - f ( v ( z , s, t), t)\\ - vz s
C
-c
Setting A2 = A! ( \^ } gives (8.5.24). \l-cj Note that the relations (8.5.23) and (8.5.24) obviously imply the continuity of /(•, t) on B, and therefore in view of the definition of F, we deduce that F is continuous on C" . Step 3. We next prove that F is a homeomorphism of R2n onto R2n. Taking into account the left-hand side of (8.5.9), we deduce that \\F(*)\\ =
-> oo as \\z\\ —)• oo.
8.5. Quasiconformal extensions
363
Since F is continuous on M2n, injective on R2n and ||F(z)|| -> oo as \\z\\ ->• oo, we obtain that F is also a surjective map on R 2n . Let 52n = R2n U {00} be a one point compactification of E2n. We extend F to S2n by F(oo) = oo. Then F is a continuous bijective map from S2n onto S2n and since S2n is compact, we deduce that F is a homeomorphism from S2n onto itself. Therefore F is also a homeomorphism of E2n onto itself, as claimed. Step 4. We next prove that F is quasiconformal in R2n. For this purpose, we use a standard dilation argument (cf. [Becl, p.33-34], [Pfa2, p.21-24].)
For r > 1, let fr(z,t) = rf(z/r,t), and
hr(z,t] = rh(z/r,t) (\\z\\ < r, t > 0)
( fr(*,0),
\\z\\ <1
It is obvious that fr(z,t) satisfies the differential equation (8.5.26)
TJJ^O 8 '*) = D f r ( z , t } h r ( z , t ] ,
t > 0, ||2|| < r.
In particular the above equation is satisfied for all t > 0 and \\z\\ < 1. On the other hand, in view of (8.5.23) and the right-hand side inequality in (8.5.9), we obtain for r > 1, 0 < t < T, T > 0, and z € ~B that \\fr(z,t)-f(z,t)\\
=
\\rf(z/r,t)-f(z,t)\\
< \\rf(z/r,t) - f ( z / r , t ) \ \ + \\f(z/r,t) - f ( z , t ) \ \
1 r-
(l-c||2/r||) 2 ' — ""'•" 1
rr, T
eT
1
.
rr.fr
^f -
1\1~C
(I — (c/r))
Consequently, lim / r (^,t) = /(z,t) uniformly on B x [0,T], and therefore r\l
Fr —^ F uniformly on compact subsets of R2n as r decreases to 1. We next prove that Fr is absolutely continuous on lines, differentiable a.e. and has outer dilatation bounded a.e. by a bound which is independent of r. Since Fr ->• F uniformly on compact subsets of E2n as r \ 1 and F\B = f
364
Loewner chains in several complex variables
is nonconstant, we will conclude in view of [Vai, Theorem 21.7 and Corollary 37.4] that F is a quasiconformal homeomorphism of E2n onto R2n. Step 4.1. We show that Fr is ACL. For this purpose, we prove that Fr satisfies a Lipschitz condition on Cn. Since \\e-'Df -' r(z,t)\\
= \\e-'Df(z/r,t)\\ -'
= \\Df(e-'z/r)(I -'
- E(e~~ttz/r,t)]\\
for z € £, t > 0, by (8.5.25), we obtain
(8.5.27)e-'||/r(;M) - fr(w,t)\\ < ^J$e\\* ~ HI = M2(r)\\z - HI, for z, w e B and £ > 0, where M2(r) is independent of t. Moreover, using the fact that f ( z , s ) = /(v(z,s, £),£), z e B, t > s > 0, and the relations (8.5.22) and (8.5.27), we obtain ||/r(z, t) - f r ( z , s)\\ = r\\f(z/r, t) - f(z/r, s)\\ <e*rM 2 (r) --v(-,s,t}\\ r \r / II < eVM2(r)^-^(i - a) = e*M3(r)(t - s), z <E S, 0 < s < t. I —c Therefore, we have proved that (8.5.28)
\ \ f r ( z , t ) - f r ( z , s)|| < e*M3(r)(* - 5), ^ e B, 0 < 5 < t.
Combining the relations (8.5.27) and (8.5.28), and using the definition of Fr, it is not difficult to prove that Fr satisfies a local Lipschitz condition in Cn. Hence Fr is ACL in IR2n and in view of a theorem of Rademacher and Stepanov (see e.g. [Vai, Theorem 29.1]), we deduce that Fr is (real) differentiate a.e. in R2n. Step 4.2. It remains to show that Fr has outer dilatation bounded a.e. by a bound independent of r. For simplicity, we shall omit the subscript r and set G(z) = Fr(z) for fixed r > 1. Since Fr(z) = fr(z,0) for z € B, and / is quasiregular on B, it suffices to assume \\z\\ > 1.
8.5. Quasiconformal extensions
365
Let z = (x, y) = (xi, y i , . . . , xn, y n ), ||z|| > 1, be a point where the mapping G=(V,W) given by (8.5.29)
G:(x1,y1,...,xn,yn)^(V1,W1,...,Vn,Wn), Vk = Vk(xi,yi,...,xn,yn) = Re G fc (x,y),
" k ~~ VYA:\XI, yij • • • j Xnj yn) ^ im LT^^X, yj, fc = l,...,n, is differentiable. Also let £ = z/(r||z||) = ( C i > - - - > C n ) 5 Cfc = ^fc(^ 5 y) = rvfc(£,T/,£) with i = log ||z||. After some straightforward computations, based on the chain rule, we obtain D(V,W;x,y)=r
D(u,v,t,
77;
^ d ( >y}+dl(
Re /i(C, t]
grad
grad log||z||
Here we have used the relation (8.5.26) and the Cauchy-Riemann equations. (These relations hold since ||£|| < 1.) Other elementary computations using the above relation give
D(V,W;x,y)
(8.5.30)
\T+*(* \I + r* where / is the identity transformation. Next, let
It is not difficult to show that C has proportional columns, and thus C has rank 1. Hence det(7 + C) = l + trace{C}
3=1
3=1
366
Loewner chains in several complex variables = I + r 2 Re
Note that for the above inequalities we have used the left side of (8.5.12). Consequently, in view of (8.5.30) and the above relation, we obtain (8.5.31)
\6etD(V,W]x,y)\ =
For the last equality we have used the fact that /(•,£) is holomorphic on B, and thus |J/(C, t)\2 — detD(u,v;£,rj). On the other hand, from (8.5.30) we obtain (8.5.32)
\\D(V,W;x,y)\\ <
Dfav&nW • \\I + C\\
\\*\\ where we have used the equality ||£)(w,t;;^,77)|| = ||.D/(£,£)||. (This equality follows if we take into account the Cauchy-Riemann equations and the holomorphy of /(•, t) on JB.) Next, applying the Schwarz Lemma to the map h, we obtain ||J + C\\ < 1 + ||C|| < 1 + r 2 ||MC, *) - CIIIICII = 1 + r||MC, «) - CIIMoreover, since
and
we deduce that
III/ -
8.5. Quasiconformal extensions
367
and thus
Therefore,
Finally taking into account the above inequality, the relations (8.5.31) and (8.5.32), the quasiregularity of /, and the inequality
s (111)"-', we obtain \\D(V ?/»H2n —< II VV ||- L / V K » W-r > X ' y)\\
ln \\ntlt -\\J-\C\\2n • / V S > ' t\\\' '/ll U2 > °ll
||2n H
= ij^e*-|W(CO
4-
Hence we have proved that Fr has outer dilatation bounded a.e. by a bound independent of r. This completes the proof. Hamada and Kohr [Ham-Kohl2] have recently obtained the following sufficient condition for quasiconformal extension to C1 of the first element of a y-Loewner chain with g(() = (I + £)/(! — C), C € U. To this end, we recall that a mapping f ( z , t) is a p-Loewner chain with #(£) = (1 + C)/(l ~ 0 ^ and only if f ( z , t ) is a Loewner chain such that {e~tf(z,t)}t>o is a normal family on B. In this case there is a mapping (8.5.33)
h : B x [0, oo) -> C"
368
Loewner chains in several complex variables
such that /i(-, t) € M for t > 0, h(z, •) is measurable on [0, oo) for z € B, and -j£(z,t) = Df(z,t)h(z,t),
a.e. t > 0,
Vz e B.
We leave the proof of Theorem 8.5.8 for the reader and mention that the idea of the proof is to use similar arguments as in the proof of Theorem 8.5.7. (See Problem 8.5.1.) Theorem 8.5.8. Let f ( z , t} be a g-Loewner chain with g(() = C), C € U. Assume the following conditions hold: (i) There exist constants M > 1 and c G [0, 1) such that
(ii) There exists a constant c\ > 0 such that ci\\z\\2 < R e (h(z,t),z),
zeB,t>0,
where h(z,t) is given by (8.5.33). (Hi) There exists a constant c% > 0 such that
(iv) f ( - , t ) is quasiregular on B for each t > 0, and the outer dilatation is bounded independently o f t . Then for each t > Q, f ( - , t ) has a continuous and injective extension to B, again denoted by f ( - , t ) , and f ( z ) = f ( z , Q ) extends to a quasiconformal homeomorphism of lR2n onto IR2n . The extension F is given by
Remark 8.5.9. (i) Let f ( z , t ] be the chain given by (8.5.6) and f ( z ) = /(z,0) be a normalized quasiregular holomorphic mapping on B, which satisfies the assumption (8.5.5). Also let h(z,t) be the mapping given by (8.5.10). Then f ( z , t ) and h(z,t) satisfy the assumptions of Theorem 8.5.8, and hence
8.5. Quasiconformal extensions
369
/ can be extended to a quasiconformal homeomorphism of R2n onto R2n . We leave the proof for the reader. (ii) A large class of mappings h = h(z,t) which satisfy the conditions (ii) and (iii) in Theorem 8.5.8 is given by /i(z,t) = [/ - E(z,t)]-l[I + E ( z , t ) ] ( z ) , z e B, t > 0, where E(z,t) € £(€",0*) for z 6 B and t > 0, E(-,t) is holomorphic on J3, E(Q,t) = 0, t > 0, and ||.E(z,t)ll < c < 1, z e J5, t > 0. (iii) Another class of mappings which satisfy the conditions in Theorem 8.5.8 was considered in [Cu3]. Brodskii [Brod] proved the following simple sufficient condition for quasiconformal extension of normalized holomorphic mappings on the unit ball of C". We now give the proof of this result by applying Theorem 8.5.8 (see also [Cu3], [Ham-Kohl2]). Another application of Theorem 8.5.8, which generalizes both Theorems 8.5.7 and 8.5.10, is given in Problem 8.5.4. Theorem 8.5.10. If f : B —} C™ is a normalized holomorphic mapping such that
then f is univalent and quasiregular on B and extends to a quasiconformal homeomorphism of E2n onto E2n . Proof. We imbed / as the first element of the chain /(z, t) = f(ze~t) + (e* - e-*)z = etz + ...,
zeB,t>0,
and we shall show that f ( z , t ) satisfies the conditions in Theorem 8.5.8. First, we show that f ( z , t ] is a Loewner chain. (See also Problem 8.4.3.) It is not difficult to see that
and
where E(z, t) = e~2t[I - D f ( z e ~ t ) } ,
z£B,t>0.
Using the hypothesis, we deduce that there is a constant c € [0, 1) such that \\Df(z)-I\\
370
Loewner chains in several complex variables
and hence ||E(z,t)|| < c||z||, z € B, t > 0. This implies that / - E(z,t) is an invertible operator, and if h(z, t} = [I — E(z, t)]~l[I + E(z, t)](z} then ^(z, t) = D f ( z , t)h(z, t),
z&B,t>0.
Clearly h ( - , t ) is well denned and is holomorphic on B for each t > 0. Also h ( z , - ) is measurable on [0,oo) for each z e B. Since ||.E(z,t)|| < c||z||, we deduce as in the proof of Theorem 8.4.1 that Re (h(z, t), z) > 0 for z € B \ {0} and t > 0, and hence h ( - , t ) e M, t > 0. Using Remark 8.5.9 (ii), we deduce that h(z, t) satisfies the conditions (ii) and (iii) in Theorem 8.5.8. Moreover, since lim e~tf(z^ t) = z locally uniformly on B, we deduce that t->00
f ( z , t ) is a Loewner chain such that {e~tf(z^t}}t>Q is a normal family on B. On the other hand, since D f ( z , t ) = e*[7 — E(z,t)], we obtain
so the condition (i) from Theorem 8.5.8 holds. It remains to show that f ( - , t ) satisfies the condition (iv) in Theorem 8.5.8 for t > 0. Indeed, we have
\\Df(z,t)\\n = ent\\I - E(z,t)\\n < ent(l + cr
using the fact that | det[7 — E(z, £)]| > (1 — c)n. This completes the proof.
8.5.2
Strongly starlike and strongly spirallike mappings of type a on the unit ball of Cn
Chuaqui [Chu2] proved the following quasiconformal extension result for a quasiregular strongly starlike mapping on B (see Definition 8.3.22). We shall give a proof of this result based on the application of Theorem 8.5.8 (see [HamKohl2]). Hamada [Ham4] extended this result to the case of strongly starlike mappings on bounded balanced pseudoconvex domains in C™ for which the Minkowski function is of class C1 in Cn \ {0}. Also Hamada and Kohr [HamKohl4] have recently extended Chuaqui's result to the case of strongly starlike mappings on the unit ball of Cn with an arbitrary norm.
8.5. Quasiconformal extensions
371
Theorem 8.5.11. Let f : B —>• C"1 be a quasiconformal strongly starlike mapping such that \\[Df(z)]~lf(z)\\ is uniformly bounded on B. Then f can be extended to a quasiconformal homeomorphism of R2n onto R2n . Proof. Let /(z,t) = e t f ( z ) , z e B, t > 0. Since / is strongly starlike, f ( z , i ) is a Loewner chain. Of course lim e ~ t f ( z : t ) = f ( z ) locally uniformly t—>oo on B and where h(z,t) = w(z) and w(z) = [Df(z)]~1f(z). Also in view of Definition 8.3.22 and the remarks following this definition, there exists c € (0, 1) such that / e S? (B) and 1
,
, s
t
;(W(Z),Z}-
2c
1 + C
1-c 2
B\{0}.
O '
-—1
1 — C2
Then
M*^*«'>.^M'l±a. -*' and thus the condition (ii) from Theorem 8.5.8 holds. Also from the above relation, we obtain (8.5.34)
l ^ l l f l M I - IM*)"'
zeB
-
Moreover, from Theorem 8.3.23 and the fact that / £ 8^(3), we deduce that
(si t( ^o.o.oo;
tt\
.
and since Df(z)w(z)
I!'_2'!!i i _ i i \ 9 ^— iill/v f/-,\n <-• 2 ^!! -^
/-•
II-_2'!!i i _ u \ 2 '
/_ -0r>'
z fc
= f ( z ) , we obtain from (8.5.34) and (8.5.35) that
(**Tfi\ (8.5.36) Since / is quasiregular, there exists an absolute constant K > I such that (8.5.37)
\\Df(z)\\n < K\Jf(z)\,
z € B.
Combining this inequality with (8.5.36), we conclude that ||D/(2:)|| is uniformly bounded on B and (8.5.38)
\\Df(z)\\=
sup \\Df(z)u\\
ztB.
372
Loewner chains in several complex variables
Indeed, since [D f (z)]* D f (z) is a positive semi-definite matrix, its eigenvalues are real and nonnegative. Let 0 < ai < . . . < an be these eigenvalues. There is a unitary matrix V such that W = V*[Df(z)]*Df(z)V is a diagonal n matrix with diagonal components ai,...,a n . Let u € C be a unit vector. Then
\\Df(z)u\\2 = (Df(z)u,Df(z)u)
= (WV*u,V*u],
and since v — V*u is also a unit vector, we easily deduce from the above relation that \\Df(z)u\\2 > <*i. Thus if u = w(z)/\\w(z)\\, z e B \ {0}, we r 1 + c i2 obtain from (8.5.36) that ai < --^ . Further, in view of (8.5.37) we i(i. — c) J
obtain
a" = \\Df(z)\\2n < K2\ detDf(z}\2 = K2ai • • -a n . Therefore an < K2a\ and the inequality (8.5.38) now follows. Moreover, the inequality (8.5.38) gives
and hence the condition (i) in Theorem 8.5.8 occurs. Also since / is quasiregular, we obtain r = ent\\Df(z)\\n < Kent\Js(z}\ = K\Jf(z,t)\, zeB,t>Q. Therefore /(•,£) satisfies condition (iv) in Theorem 8.5.8 for each t > 0. Finally, since ||w(z)|| is uniformly bounded on B, we deduce that the condition (iii) from Theorem 8.5.8 holds. Hence all conditions from Theorem 8.5.8 are satisfied, and thus f ( z ) = f ( z , 0) extends to a quasiconformal homeomorphism of R2n onto E2n. This completes the proof. Hamada and Kohr [Ham-Koh6] have recently defined the notion of strong spirallikeness of type a € (—7r/2,7r/2) on the unit ball of C" (and more generally on bounded balanced pseudoconvex domains for which the Minkowski function is of class C1 in Cn \ {0}). Definition 8.5.12. Let / : B —> C71 be a normalized locally biholomorphic mapping and a € (—7r/2,7r/2). Also let a = tana. We say that / is strongly spirallike of type a if there is a constant c < I such that (8.5.39)
1-c 2
8.5. Quasiconformal extensions where (8.5.40)
373
h(z) = iaz + (I - i a ) [ D f ( z ) ] --f1( z ) ,
z e B.
Such a strongly spirallike map of type o; is of course spirallike of type a, and in view of Theorem 8.2.1, f ( z , t) = e-af(eatz),
z€B,
t > 0,
is a Loewner chain such that |£(z,t) = Df(z,t)h(z,t),
z e B, t > 0,
where
h(z,t) = iaz + (1 -
ia)e-iat[Df(eiatz)]-1f(eiatz),
for all z € B and t > 0. Then we obtain the following result by an argument similar to the proof of Theorem 8.5.11 ([Ham-Koh6]; see also [Ham-Kohl2]). We leave the proof for the reader. Theorem 8.5.13. Let f : B —>• C71 be a strongly spirallike mapping of type a e (— 7T/2,7r/2). Assume f is quasiconformal on B and \\[Df(z)]~1f(z)\\ is uniformly bounded on B. Then f can be extended to a quasiconformal homeomorphism of R2n onto M2n . Notes. For more information about quasiconformal extension results for biholomorphic mappings in C1 which can be embedded in Loewner chains, see [Chu2], [Cu3], [Ham-Koh6], [Ham-Kohl2], [Ham-Kohl4], [Pfa2].
Problems 8.5.1. Prove Theorem 8.5.8. Hint. Use the condition (ii) in the hypothesis to prove that there is a constant d > 0 such that \\f(z, s)\\ < des\\z\\ forz^B and s 6 [0, oo). For this purpose, fix z 6 B \ {0} and s > 0, and let v(t) = v(z, s,t) be the solution of the initial value problem dv — = — h(v,t) c/c
a.e. t > 5,
v(s) = z.
374
Loewner chains in several complex variables
Then prove that \\ei~sv(z, s,t)\\ < d\\z\\, t > s, for some d > 0. Letting t -»• oo in this inequality and using the fact that lim etv(z, s,t) = f ( z , s ) locally t—KX> uniformly on B, we obtain the claimed conclusion. Next, use the conditions (i)-(iii) to deduce that ft(z) = f ( z , t) has a continuous and injective extension to B (again denoted by ft(z)) for each t > 0, which satisfies Holder conditions similar to those in (8.5.23) and (8.5.24). We now use the condition (iv) and proceed as in Steps 2-4 in the proof of Theorem 8.5.7 to obtain the desired conclusion. 8.5.2. Let f ( z , t ) be the chain given by (8.5.6) and /(z) = f ( z , 0 ) be a normalized quasiregular holomorphic mapping on B which satisfies the relation (8.5.5). Also let h(z, t) be the mapping given by (8.5.10). Prove that /(z, t} and h(z, t) satisfy the conditions in Theorem 8.5.8, and thus / can be extended to a quasiconformal homeomorphism of R2n onto R 2n . (See also [Ham-Kohl2].) 8.5.3. (i) Let / : B C C2 -» C2 be given by
where a € (— TT, TT) and k 6 [0, 1). Show that / is strongly spirallike of type a/2. (ii) Let /i, . . . , fn be strongly spirallike functions of type a € (— 7r/2, 7T/2) on the unit disc U and / : B —> Cn be given by f ( z ) = (/i(zi), . . . , fn(zn)}, z = (zi, . . . , zn) € B. Show that / is strongly spirallike of type a on the unit ball B of C71. (Hamada and Kohr, 2001 [Ham-Koh6].) 8.5.4. Let / : B -> C" be a normalized holomorphic mapping and let G(z] be a nonsingular n x n matrix which is holomorphic with respect to z € B. Assume G(0) = / and there exist c € [0, 1) and K > I such that the following conditions hold for all z G B: [G(z)}-lDf(z)-I\\
+(l-\\z\\*)[G(z)]-lDG(z)(z,-)
8.5. Quasiconformal extensions
375
Prove that / is univalent and quasiregular on B and extends to a quasiconformal homeomorphism of R2n onto R 2n . (Ren and Ma, 1995 [Ren-Ma]. See also [Ham-Kohl2].) Hint. Prove that f ( z , t ) = f(ze~t) + (e* - e-t}G(ze~t}(z}, z € B, t > 0, is a Loewner chain which satisfies the conditions of Theorem 8.5.8 (see also Problem 8.4.5). To this end, we mention that the first and second conditions in the hypothesis imply that (1 - \\z\\2)\\[G(z)]-1DG(z)(zt
OH < 2c, z e B.
Then a similar argument as in the proof of Lemma 8.5.6 yields that there is a constant M > 0 such that <
This relation implies the condition (i) in Theorem 8.5.8. On the other hand, the third condition in the hypothesis implies that /(•,£) is quasiregular on B for each t > 0, and the outer dilatation is bounded independently of t. Next, it suffices to use similar arguments as in the proof of Theorem 8.5.10. Note that if G(z) = D f ( z ) , z € B, in Problem 8.5.4, we obtain the result of Theorem 8.5.7. In this case, the third relation in the hypothesis is equivalent to the condition that / is quasiregular on B. Also if G(z] = /, we obtain the result of Theorem 8.5.10. 8.5.5. Prove Lemmas 8.5.3 and 8.5.4. 8.5.6. Prove Theorem 8.5.13. Hint. Use similar arguments as in the proof of Theorem 8.5.11. 8.5.7. Verify that the mappings in Remark 8.5.9 (ii) satisfy the conditions (ii) and (iii) in Theorem 8.5.8.
Chapter 9
Bloch constant problems in several complex variables In this chapter we shall briefly discuss Bloch constant problems in higher dimensions. Let B be the Euclidean unit ball in C™. We shall be interested in subclasses of H(B), consisting of mappings from B into C™ for which there is a positive lower bound for the supremum of the radii of the schlicht balls in the image of all mappings in the given subclass. In particular we consider the set of Bloch mappings, although there is not as close a relationship as in one variable between Bloch constant problems for this class of mappings and Bloch constant problems for more general mappings.
9.1
Preliminaries and a generalization of Bonk's distortion theorem
We begin with the following definition: Definition 9.1.1. Let / : B ->• C™ be a holomorphic mapping and a£ B. (i) A schlicht ball of / centered at /(a) is a ball with center /(a) such that / maps biholomorphically a subdomain of B containing a onto this ball. (ii) Let r(a, /) be the radius of the largest schlicht ball of / centered at /(a). Also let r(f) = sup{r(a, /) : a € B}. (iii) If T is a family of holomorphic mappings from B to C™, the Bloch 377
378
Bloch constant problems
constant for T is defined by B(^) = inf{r(/) : / <E F}. As we have seen in Section 6.1.7, Bloch's theorem fails in C™, n > 2, if we do not require additional assumptions on the mapping besides the usual condition Df(Q) = I. The restrictions on mappings which have been considered in order to obtain nontrivial Bloch theorems are of the following principal types: quasiconformality (and similar) assumptions, boundedness assumptions, the class of Bloch mappings, and global geometric assumptions such as starlikeness and convexity. Quasiconformality assumptions were first considered by Bochner [Boc], then by Takahashi [Tak], Sakaguchi [Sakl], Wu [Wu], Hahn [Hah2], Harris [Harr3], and most recently by Chen and Gauthier [Che-Gau2]. Boundedness assumptions have been considered by Hahn [Hah3], Harris [HarrS], Harris, Reich, and Shoikhet [Harr-Re-Sh], and Chen and Gauthier [Che-Gau2]. Here we shall focus mainly on a generalization of Bonk's distortion theorem (see Theorem 4.2.4) obtained by Liu [LiuX], on a theorem for bounded holomorphic mappings due to Chen and Gauthier [Che-Gau2], and on some results for convex and starlike mappings obtained by Graham and Varolin [Gra-Var2]. Bloch functions and Bloch mappings may be defined as follows on the Euclidean unit ball of C71 : Definition 9.1.2. (i) Let / : B —> C be a holomorphic function. Then / is called a Bloch function if II/H = snp{\\D(f o ^(0)|| :
9.1. A generalization of Bonk's distortion theorem
379
various equivalent characterizations of Bloch functions and Bloch mappings. One of them is given in the following theorem [Timl], [LiuX] (compare with the one-variable case). Others are given in [Timl] and in the exercises at the end of this section. Theorem 9.1.3. Let f : B —>• C™ be a holomorphic mapping. Then f is a Bloch mapping if and only if the family Ff =
\\D(f o (p)(a}\\ < \\D(f o ip o a where we have used the fact that ||[Z)0a(0)]~1|| = !/(! — ||a||2) (see the relation (10.2.7)). Therefore, the family {\\D(f o
(ii) Liu [LiuX] showed that the quantity m(/) = sup{(l-|H| 2 )||D/(z)||: z G B}, is actually equivalent to the Bloch seminorm of /, i.e. there is a positive constant M such that m(f) < H/ll < Mm(/), V / 6 H(B).
380
Bloch constant problems
Consequently, / is a Bloch mapping if and only if m(f) < oo. Bloch functions have been studied on bounded homogeneous domains using the covariant derivative in the Bergman metric (see [Fit-Gon2], [Gon3], [Gon-Yu-Zh], [Timl,2]). Krantz and Ma [Kran-Ma] studied Bloch functions on strongly pseudoconvex domains using the Kobayashi metric. A somewhat analogous class of mappings may be defined as follows [LiuX]: Definition 9.1.5. Let / € H(B) and let (9.1.1)
||/||o = sup {| J/o^O)!1/" : v G Aut(B)} = sup {(1 - ||z||2)(n+1)/(2n)| Jf(z)\l/n
: z € #}.
Note that the second equality is a consequence of the relation (6.1.13). Also let VB={f£H(B): ||/|io
VB0 = {feVB:
H/Ho = 1 and J,(0) = 1}.
Then || • ||o is a seminorm [LiuX] which is invariant under composition with automorphisms of B. There is a generalization of Bonk's distortion theorem to the class V#o, due to Liu [LiuX]. Theorem 9.1.6. Let f G VB0. Then (9.1.2)
\Jf(z}\ > Re J f ( z ) > (1 - V^+2||z||)/(l - \\z\\/V^+2)n+2
for \\z\\ < 2\/n -f 2/(n + 3). This estimate is sharp. Proof. Fix w € C", ||ti;|| = 1, and let g : U -»• C be given by
, ICI < where
1 —C
1
It is easy to see that ij) maps the unit disc U onto the disc E = {£ G C : |1 - Cn£|2 < 1 - |£|2} C 17, and a short computation yields that g(l) = 1. Moreover, since «7/(0) = 1 and j|/|jo = 1, we deduce that \ = l+o(\\Z\\) 88 \\Z\\->0,
9.1. A generalization of Bonk's distortion theorem
381
and thus g'(l) = 1. On the other hand, using again the fact that ||/||o = 1 and V>(f7) = E, one deduces that
|0(C)| < (i - hKC)!2)(n+1)/V/(
into itself. The closed horodisc A(l, r) is also mapped to itself. Now any point £ € (—1,1) is a boundary point of exactly one horodisc of the form A(l,r). All other points in the closure of this horodisc have real part larger than £. Hence we obtain Re0(C)>C,
C €(-1,1),
and it is clear that this inequality also holds for £ € [—1, 1]. Next, let 77 = cn(l- 0/(l-cJO> -1 < C < 1- Then 0 < 77 < 2v/nT2/(n+ 3) and the above inequality is equivalent to
Finally, letting z e C", 0 < \\z\\ < 1\/n + 2/(n + 3), w = z/\\z\\, and 77 = ||z|| in the above, we obtain (9.1.2), as desired. To prove the sharpness of this relation, let / € H(B) be such that /(O) = 0 and Df(z)
= | ( 1 - *i/Vn + 2)B+a 0
0
I,
z = (zlt...,Zn)
JB_i
where In-i is the (n — 1) x (n — l)-identity matrix. Then J/(0) = 1 and it is elementary to see that ||/||o = 1. Moreover, for z = (r, 0, ...,0), r €. [0,2v/nT2/(n + 3)], we have ||z|| = r and Re Jf(z) = (1 - >/n + 2||z||)/(l ||z||/Vn + 2)n+2. This completes the proof. Since the relation (9.1.2) is an estimate for the Jacobian determinant of /(z) and not for D f ( z ) , one needs an additional assumption in order to get an estimate for r(0,/). Following Liu [LiuX], we introduce
382
Bloch constant problems
Definition 9.1.7. Let 1 < k < oo and Bn(k) = {/ : B ->• C" : / e fT(B), H / l l < fc}. Also let £n,0(fc) = {/ e B n (fc) : J,(0) = 1}. In the remainder of this section we shall obtain a lower bound for the Bloch constant B(B n; o(fc)) due to Liu [LiuX], which is actually a generalization of the one-variable estimate of Ahlfors in Corollary 4.2.5. First we need the following definition and lemma. Definition 9.1.8. Let / : B —>• Cn be a holomorphic mapping and ZQ e B. We say that ZQ is a critical point of / if J/(^o) = 0- In this case, /(ZQ) is called a critical value of /. A point WQ e Cn is called a boundary point of /(B) if there is a sequence {zfcjfcgN in B such that {£fc}fceN has no limit point in B and the sequence {/(zfc)KeN converges to WQ. (This use of the term "boundary point" occurs only in this section.) The following lemma is due to Liu [LiuX]. Lemma 9.1.9. Let f : B —> Cn be a holomorphic mapping, let £1 be a domain contained in B and a € 0. / / / maps 0 biholomorphically onto the schlicht ball S(/(a),r(a, /)), then either $1 and B have a common boundary point or else there is a critical value f ( z ) on the boundary of B(f(a),r(a, /)) such that the critical point z lies on the boundary o/Q. Consequently, r ( a , f ) is given by the Euclidean distance from /(a) to a boundary point of f ( B ) or to a critical value of f. Proof. Suppose that Ci and B do not have a common boundary point. In this case Q C B and we can find a family {^fc}fceN of open subsets of oo
B such that £lk D ft, fifc D fifc+i, and Q Qfc = £7. Using the definition of fc=i B(/(a),r(a,/)), we deduce the existence of points Zk and Wk in 17^ such that zk ^ wk and f ( z k ) = f(wk), k e N. Since {z/JfceN, {u>k}k£N are bounded sequences, we may suppose that zk -> z and wk —>• w as k -» oo. Obviously, z,w e 50, /(z) = f(w] and /(^) belongs to the boundary of JB(/(a),r(a, /)). There are two possibilities: either z = w, or z ^ w. If z = w, then it is easy to see that J / ( z ) = 0, and thus f ( z ) is a critical value of / and z is a critical point of / which lies on the boundary of fi. If z / ty, then at least one of the points z and w must be a critical point of /. Otherwise / must be locally univalent near z and iu, and combining this fact
9.1. A generalization of Bonk's distortion theorem
383
with the equality f ( z ) = f ( w ) , yields easily that / cannot be univalent on fi. However, this contradicts the assumption that 1) is mapped biholomorphically onto B(/(a),r(a,/)). This completes the proof. Now we are able to prove the following result [LiuX]: Theorem 9.1.10. Let I < k < oo and let f € Bn(k] n VB0. Also let (9.1.3)
c(k,n) /•l/Vn+^ = fc1-" / [(1 ./O n+l
>k l-n
1+
en
-2
n +1
Thenr(Q,f) >c(k,n). Proof. In view of Lemma 9.1.9, r(0, /) is given by the Euclidean distance from /(O) to a point p which is either a boundary point of f ( B ) or a critical value of /. Let F be the straight line segment from /(O) to p. Also let 7 be the inverse image of F under /. Since there are no critical values of / on F except possibly for p, we deduce that 7 is a smooth curve from the origin to the boundary of B or a curve from the origin to a critical point ZQ of / in #, smooth except possibly at ZQ. In the latter case \\ZQ\\ > l/\/n+~2 by (9.1.2). Hence in all cases, \[ dw = f \\dw\\ =
\Jr
•/
Jr
Df(z)
dz
J-v
f\\Df(z}dz\\
\\dz\\ > f J*i
\Jf(z)\
VM\
IP/GOir-1 " "• Here we have used the fact (see Lemma 9.2.2) that ||^4w|| > for any nonzero n x n matrix A and any unit vector u € C". Combining the above arguments and using the fact that ||.D/(z)|| < k/(l — \\z\\2), z 6 B, since / G Bn(k), we deduce that r(0,/) > c(fc,n), where c(k,n) is given by (9.1.3). This completes the proof.
384
Bloch constant problems
Using this result together with a version of Landau's reduction (see Chapter 4) to reduce to the case j|/||0 = 1, Liu [LiuX] proved the following theorem. We leave this proof for the reader. Theorem 9.1.11. B(5n,0(fc)) > c(fc,n). Analogous results on classical matrix domains were obtained by FitzGerald and Gong [Fit-Gon2]; see also [Gon-Yu-Zh].
Problems
9.1.1. Let / : B —>• C be a holomorphic function. Show that / is a Bloch function if and only if the family {f oh\ h \ U -> B, h holomorphic} is a family of Bloch functions with uniformly bounded Bloch seminorm. (Timoney, 1980 [Timl].) 9.1.2. Let / : B —> C be a holomorphic function. Show that / is a Bloch function if and only if the directional derivative o f / is O(l/(l — ||.z||2)) in the radial direction and O(l/(l — ||z||2)1//2) in the directions orthogonal to the radial direction. (Timoney, 1980 [Timl].) 9.1.3. Prove Theorem 9.1.11. 9.1.4. Show that || • ||o given by (9.1.1) is a seminorm which is invariant under composition with automorphisms of B. (Liu, 1992 [LiuX].)
9.2
Bloch constants for bounded and quasiregular holomorphic mappings
In this section we shall discuss a Bloch constant theorem of Chen and Gauthier [Che-Gau2] for bounded holomorphic mappings on the Euclidean unit ball of Cn. It is a generalization of the result of Landau [Lan] given in Problem 4.2.3. The first step is to generalize Landau's theorem to the case of holomorphic mappings from the unit disc U into Cn. The following lemma is due to Chen
9.2. Bounded and quasiregular mappings
385
and Gauthier [Che-Gau2]. Lemma 9.2.1. Let g : U —>• C™ be a holomorphic mapping such that \\g(z)\\ < M, z e U, g(Q) = 0 and \\Dg(0)\\ = a > 0. Then (i) g is injective on UPQ and Dg(z) ^ 0 for z € UPQ, where
a M+ (ii) For any positive number p < PQ, g(dUp) lies outside the ball BR, where R = M—— M — ap
> MpQp.
Proof. Let g = (pi,..., gn) and let h : U —> C be given by
Then h is a holomorphic function on U, h(Q) = 0, /i'(0) = a and an application of the Schwarz inequality shows that |/i(C)| < 11^(011 < M for C € U. Hence Problem 4.2.3 applies to h and all of the desired conclusions about g follow from this. We leave the proof of the next lemma for the reader since it suffices to use elementary properties of matrices and eigenvalues. (A proof may be found in [Che-Gau2].) Lemma 9.2.2. If A = (aij)i
The next result is due to Chen and Gauthier [Che-Gau2]. Lemma 9.2.3. Let f : B —> C1 be a holomorphic mapping and let M > 0 be such that \\f(z)\\ <M,z€B. Then (9.2.1)
and (9.2.2)
(1- \\z\f )\\Df(z)\\<Mt (1 - \\z\\2)*P\Jf(z)\
zeB,
< Mn,
z € B.
Proof. Fix w € C" with ||iu|| = 1. Applying the Cauchy integral formula -dC,
r 6(0,1),
386
Bloch constant problems
and using the inequality ||/(2)|| < M for all z € B, we easily deduce that ||D/(0)(io)|| < M. Since w is arbitrary, we conclude that ||£)/(0)|| < M, and thus we obtain (9.2.1) for the case z = 0. Next, fix a e B\{0} and let (pa(z) =
||D/(a)|| = pMOMD^O)]-1!! < \\Dh(0)\\
making use of the fact that IKltyaCO)]"1!! = l/(l-||a||2) (see formula (10.2.7)). On the other hand, since ||/i(2)|| < M, z e 5, we deduce in view of the first step of the proof that ||.D/i(0)|| < M, and from (9.2.3) and this relation we obtain (9.2.1). Now we show (9.2.2). Clearly (9.2.2) for the case z = 0 is a consequence of (9.2.1). Thus, we have to prove (9.2.2) for z € B\{0}. Again fix a € B\{0} and let h be defined as above. Since \\h(z)\\ < M on B it follows that | Jh(0)| < Mn. But from (6.1.12) we have
and thus (l-\\a\\2)^\Jf(a)\<Mn. This completes the proof. The following lemma [Che-Gau2] is related to a result of Takahashi [Tak]. We have Lemma 9.2.4. Let f : B —> C" be a holomorphic mapping such that |J/(0)| = a > 0 and \\f(z)\\ < M, z e B, for some M > 0. Then f is univalent on the ball Bpo, where po = , and b « 4.2 is the minimum of the function (2 - r 2 )/(r(l - r 2 )) for 0 < r < I. Proof. Since ||/(z)|| < M, z e B, we can apply (9.2.1) to deduce that
< M - - + i = M -,
ZZB.
9.2. Bounded and quasiregular mappings
387
2-r2 Now, the function g(r) = —-=r-, 0 < r < 1, attains its minimum r(l — r*) b « 4.2 at ro w 0.66. Combining this with the Schwarz lemma for holomorphic mappings (see Lemma 6.1.28), we deduce that
\\Df(z)-Df(Q)\\
\\z\\
On the other hand, from Lemma 9.2.2 and the relation (9.2.1) we obtain (9.2.4) Moreover, in view of Lemma 9.2.2 we have | J/(0)| = a < Mn, and thus Po < !/&• We also note that TO > 1/6. Next, let 51,^2 G -Spo De two distinct points. Then (9.2.5)
\\Df(z) - I>/(0)|| < bM\\z\\ < bMPo = ^L_, z € [ft,®],
where [51,92] is the closed line segment between q\ and qi. Prom (9.2.4) and (9.2.5), we therefore deduce that
Thus /(<7i) 7^ /(?2)> as desired. This completes the proof. Now we are able to prove the main result of this section, due to Chen and Gauthier [Che-Gau2]. Theorem 9.2.5. Let f : B —t C1 be a holomorphic mapping such that /(O) = 0, J/(0) = a > 0, and ||/(>z)|| < M, z 6 B, for some positive constant M. Let PQ be defined as in Lemma 9.2-4 ana ^
Then f maps the ball BPQ biholomorphically onto a domain which contains the ball B^ . Proof. Taking into account Lemma 9.2.4, we deduce that / is univalent on BPQ. Next, fix z 6 dBpQ and let w — z/||.z|| = z/po = (^i> • • • > ^n) G dB.
388
Bloch constant problems
We have already noted that po < l/b < 1. Let g : U —>• C™ be given by = /(C«0, ICI < I- Then g is holomorphic on C7, 0(0) = 0, ||0(C)|| < M, < 1, and by Lemma 9.2.2 and the relation (9.2.1) we obtain \\Dg(Q)\\ =
'"^Mn-i-
Let c = ||^(0)|| and n = M + Jtft _ #' Then ri > ^W = Pi > Applying the result of Lemma 9.2.1, we deduce that
Since z was arbitrarily chosen on d-B^, we conclude that /(dB^) lies outside the ball B^ . This completes the proof. We conclude this section by discussing some Bloch constant results for quasiregular holomorphic mappings on the unit ball of C" (cf. Section 8.5). Such results have been obtained by Bochner [Boc], Takahashi [Tak], Sakaguchi [Sakl], Wu [Wu], Hahn [Hah2], Harris [HarrS], and recently by Chen and Gauthier [Che-Gau2]. The condition of quasiregularity or quasiconformality was the earliest condition considered in studying Bloch constant problems in several complex variables. It can be formulated in various ways. Using the terminology of [CheGau2], we introduce the following definitions. Definition 9.2.6. Let / : B —> C71 be a holomorphic mapping and K > 1. (i) We say that / is a Wu K-mapping if \\Df(z)\\ < K\Jf(z)\l/n,
zeB.
Obviously, a Wu K-mapping is also a quasiregular mapping. Let FjK,n be the set of Wu K-mappings and for K > I and n > 2, let ,n) = inf (r(/) : / € FK,n, J/(0) = l}
be the Bloch constant for the n-dimensional Wu K-mappings. (ii) We say that / is a Bochner K-mapping if
9.2. Bounded and quasiregular mappings
389
i In
n
Y^ l a y| 2 for any (nxn) complex matrix A = (ojj)i
where ||^4||2 =
max \\Df(z)\h < JT maxr \ J f ( z ) \ l f n , r e [0,1). IMI<»INI< If / is a Wu /^-mapping, then / is also a Takahashi X-mapping with K = ^/nK. (iv) Finally we say that / is a Hahn K-mapping if
max A.(z) < K nmax X(z), 0 < r < 1, ii ^ ' ~~" n ~"~
ii
\\*\\=r
\\z\\=r
where A2 (z) and A2 (z) are the smallest and the biggest eigenvalues of the Hermitian matrix [Df(z)]*[Df(z)], z 6 B. Then a Wu -Remapping is a Hahn .K"n-mapping. We mention that Takahashi [Tak] proved that the Bloch constant for Takahashi ^"-mappings /, for which J/(0) = 1, is bounded below by (n — l)n~2/[12.K"2n~1]. Sakaguchi [Sakl] improved this result by showing that the Bloch constant for Takahashi A"-mappings /, for which «7/(0) = 1, is at least (n - l^-VfSA"2"-1]. Hahn [Hah2] showed that the Bloch constant for Hahn ^-mappings /, for which J/(0) = 1, is at least K^n/[4K(2K + 1)]. Wu [Wu] proved that B(^rR:,n) > 0 for all K > 1 and n > 2. The above result of Sakaguchi [Sakl] applied to Wu K-mappings yields that
while Harris [HarrS] proved the estimate
Chen and Gauthier [Che-Gau2] improved the above results, as follows: and
390
Bloch constant problems
Finally we remark that an interesting Bloch constant problem for holomorphic mappings of the unit ball of a Banach space with restricted numerical range has recently been solved by Harris, Reich and Shoikhet [Harr-Re-Sh] ; see also [HarrS] for various norm restrictions. Notes. For more information about Bloch constant problems in several variables, the reader may consult [Che-Gau2], [LiuX], [Fit-Gon2], [Gam-Che], [Gon-Yu-Zh], [HarrS], [Harr-Re-Sh], [Wu].
9.3
Bloch constants for starlike and convex mappings in several complex variables
We conclude this chapter with some results concerning Bloch constants for normalized convex and starlike mappings. For normalized convex mappings, the Bloch constant is known precisely on the unit polydisc in C™, and on the Euclidean unit ball it is known if one adds the assumption of fc-fold symmetry, k > 2. For normalized starlike mappings of the Euclidean unit ball, the Bloch constant is known precisely in certain cases of fc-fold symmetry depending on the dimension, and it can be estimated in the general case. We begin by recalling the covering theorem of Szego [Sze] for convex functions in one variable proved in Chapter 4: if / e K then the image of / contains a disc of radius 7T/4, and this result is sharp. It is natural to consider generalizations to several variables. In the case of the unit polydisc P in C n , this is straightforward, as shown by Graham and Varolin [Gra-Var2]: Theorem 9.3.1. Let f : P -» Cn be a convex mapping with -D/(0) = /. Then f ( P ) contains a polydisc of radius Tr/4, and this result is sharp. Proof. Using Theorem 6.3.2 and the fact that Df(Q) = /, we easily deduce that
f(z) = (0i(2i),...,0 n (*n)),
z = (zi,...,zn) € P,
where gj is a convex function of one variable such that gfj(0) = 1, j = 1, . . . , n. The image of U under each function gj contains a disc of radius ?r/4 by Theorem 4.3.7, so the image of P contains a polydisc, each of whose radii is 7T/4.
9.3. Starlike and convex mappings
391
In order to prove the sharpness, it suffices to consider the convex mapping / : P ->• C1 given by
The corresponding problem for the Euclidean unit ball B in C71 has not yet been solved (cf. [Gra-Var2]): Open Problem 9.3.2. If f : B —¥ C71, n > 2, is a normalized convex mapping then does f ( B ) contain a ball of radius Tr/4? However, the answer is positive if / is odd, in view of Theorem 7.2.9 for the case k = 2. More generally, we have [Gra-Var2] Theorem 9.3.3. The Bloch and Koebe constants for normalized convex mappings of B with k-fold symmetry (k > 2) coincide and have the value dt where rk
-i:
Proof. It suffices to observe that if fi is a convex domain in C" with fc-fold symmetry and V C fi is a ball, then the convex hull of the union of the balls e27ry'/feV, j = 1,..., k, contains a ball centered at 0 which is at least as large asV. Another important special case in which the solution of Open Problem 9.3.2 is known is given in Theorem 11.2.9. We next give an upper bound for the Bloch constant for normalized starlike mappings on the Euclidean unit ball of C", n > 2, which decreases with the dimension and which tends to 1/4 (the value of the Koebe constant for starlike mappings of the ball B} as n —>• oo. This result was obtained by Graham and Varolin [Gra-Var2]. Theorem 9.3.4. The Bloch constant ~B(S*(B)) for normalized starlike mappings on the unit ball B o/C™, n > 2, satisfies B(5*(B)) < - ( —j=
(
z~\
z
\
(l-^1)2'-"'(1_^n)2 1
I .
4 VV™-1/
fOT
Z
= (Zl'---'*n)
e
B. This mapping is normalized starlike and omits the hyperplanes Wj = — 1/4, j = 1,... ,n. If there is a ball of radius r > 1/4 in the image of /, the center of this ball must be at a distance at least r from each of these hyperplanes. Let (ci,..., Cn) be the coordinates of the center of such a ball.
392
Bloch constant problems
Now the Koebe function fc(0 = 7= — — + —I
ici,
TTTT, C £ t/, may be written as
u-O 2
2
1 . This representation shows that for a given value is maximized when £ is positive real. Conversely, for a given
1 is minimized when £ is positive real. Thus for the 4 purposes of bounding r above we may assume that ci = ... = Cn = r — 1/4. If we solve the equation x/(l — a:)2 = r — 1/4 and require that x < n"1/2,
value of
we deduce that r < - [ ^-=
1 . This completes the proof.
4 \Vn-lJ
A similar argument gives an upper bound for the Bloch constant for normalized odd starlike mappings on the Euclidean unit ball of C" (see [GraVar2]). Theorem 9.3.5. The Bloch constant ~B(S^2JB)) for normalized odd starn +1 like mappings on the unit ball B ofCn, n>2, satisfies B(5/*2\(B)) < — -^. v /
£(71/ ~~~ Lj
Proof. Let / : B -KG" be given by
(
1
1
~1
z
n
\\
l2'"-'i 12 ) ' ~~ l -1 ~~ zn / z
i
\
75
Z — (Zi,. ..,Zn) t: &.
Then / is odd and starlike, since each of its components is odd and starlike on the unit disc. Now if we consider the function C i->
C -~ and fix \CI,
i-C 2
then the modulus of the image point and its distance from the omitted rays are maximized when £ is real. The image of the above mapping / omits the hyperplanes Wj = ±i/2, j — 1,... ,n. Suppose there exists a ball of radius r > 1/2 in the image of this mapping. For the purpose of estimating r we may assume that the center of this ball is (c, c, ...,c), where c > 0. Then r < \/c2 + 1/4. Next, setting c — x/(l — x2) and requiring 0 < x < n"1/2, we deduce that r < (n + l)/[2(n — 1)]. This completes the proof. We conclude with a result due to Graham and Varolin [Gra-Var2], concerning the Bloch constant for normalized starlike mappings with fc-fold symmetry. Theorem 9.3.6. The Bloch constant for normalized starlike mappings on the unit ball B ofU1, n > 2, with k-fold symmetry is given by B(S/*M(B)) = 4~1/*: when k = 3 and n > 4 and when k > 4 and n > 2.
9.3. Starlike and convex mappings
393
Proof. Let F : B ->• C1 be given by
This mapping is fc-fold symmetric starlike on B and the image of the unit ball under this mapping contains the ball B i . The one-variable function fk(C) = 7(1,. 0/ . covers a disc of the same radius centered at 0. However, — ^KY/K for k > 3, r(fk, w) has a local maximum at w = 0. We discuss the cases k — 3 and k > 3 separately. If k = 3 the endpoints of the three rays omitted by fa are located at the points 4-1/3e2Tt7/3j j _ i } 2,3, and we must move a distance 4"1/3 from 0 to find a nonzero w such that r(/3,iy) is as large as 4"1/3. To minimize |£| such that /jfe(C) is at distance 4"1/3 from 0 we take C3 to be positive, hence we may take £ > 0. Solving Requiring that £ < n"1/2 gives n < I —=.- ] w 3.24. Hence only n = 2, 3 y V2 —1 are possible. For all other values of n, the largest ball in the image of F is centered at 0. If k > 3 the entire omitted rays of fk come into play instead of just the endpoints. A point on the positive real axis which is at distance a > 4~1/fc from the nearest omitted ray must have distance ocsc(7r/fc) from 0. Solving fk(x) = 4"1/* csc(7r/fc) and requiring that 0 < x < n"1/2 gives
I + Y 1 + (csci I
/I
I
/
The right-hand side has the approximate value 1.6 when k = 4 and is a decreasing function of k. Hence there are no values of n > 2 which satisfy this relation. This completes the proof. Remark 9.3.7. For the cases k = 3, n — 2,3, not covered by Theorem 9.3.6, one can estimate the Bloch constant by using the reasoning in the proof of Theorem 9.3.4. This gives B(5(*3)(£)) « 0.8340 for n = 2 and B(5(*3)(S)) w 0.6486 for n = 3.
Chapter 10
Linear invariance in several complex variables In this chapter we are going to demonstrate that the one variable notion of linear-invariant families has a useful generalization to several complex variables. As we have seen in the fifth chapter, linear invariance, introduced by Pommerenke ([Poml], [Pom2]), has been a powerful tool in extending many ideas of univalent function theory to the study of locally univalent functions on the unit disc. Here we shall present some of the most important applications of this method in geometric function theory of several variables. In this chapter we shall study growth and distortion results for linearinvariant families on the Euclidean unit ball of C" as well as certain examples, open problems, and conjectures which involve the order of a given L.I.F. In fact we shall consider two definitions for the order of a L.I.F. F. The trace order leads to bounds for the growth of the Jacobian determinant of mappings in F, and in fact is determined if sharp bounds for the growth of |J/(^)|, / € T, are known. The norm order, recently introduced by Pfaltzgraff and Suffridge [Pfa-Su4], contain more information and gives bounds for ||.D/(z)|| and an upper bound for ||/(^)||, as well as geometric results such as the radii of convexity, starlikeness, and univalence of a L.I.F. of known order. In one variable, both of these notions of order reduce to the usual one. We briefly consider L.I.F.'s on the unit polydisc as well. 395
396
10.1
Linear invariance in several complex variables
Preliminaries concerning the notion of linear invariance in several complex variables
10.1.1
L.I.F.'s and trace order in several complex variables
Let C™ be the space of n complex variables equipped with the Euclidean structure. As usual, let B be the unit ball of Cn with respect to the Euclidean norm || • ||. The underlying set of mappings and transforms which are the subject of the study of linear invariance are the set CS(B) consisting of normalized locally biholomorphic mappings on the Euclidean unit ball B and the Koebe transform A^(/) given by (10.1.1)
A,(/)(z) = [IWOr^/WO))]-1^*)) - /(0(0)))
for / 6 £S(B) and
In this chapter we shall use an alternative notation for the automorphisms of B in order to conform with source papers. We shall write (10.1.2) and
TJ-a — —
p( (1 " Sa)aa * + Sa||a||2/}'
Sa =
V1 - Nl 2 -
Since a*z = (z,a), z € C71, the operator Ta is the same as in (6.1.10), and we have tpa(z) = (j>-a(z), z e B, where 4>a is given by (6.1.9). The Koebe transforms formed with the automorphisms
10.1. Preliminaries
397
We introduce the following definition (see [Pfa5]; [Bar-Fit-Gon2]): Definition 10.1.1. The family F is called a linear-invariant family (L.I.F.) if (i) F C CS(B) and (ii) A^(/) e T for all / € T and 0 € Aut(B). In the case of one variable, this definition coincides with Definition 5.1.1. (Rudin [Rud2] used the term M-invariant family to mean a family of holomorphic mappings on the Euclidean unit ball B of C11 that is invariant under the group Aut(-B).) The following result of Pfaltzgraff and Suffridge [Pfa-Su4] is useful in various situations. Recall that by U we denote the set of unitary transformations fromC" toC". Theorem 10.1.2. Let f C CS(B). Then T is a linear-invariant family if and only if for each f £ F, the mapping V~lf(Vz;a] e T for all a e B and V £U. Proof. Let / € F. Taking into account Theorem 6.1.23, we observe that the condition (ii) in Definition 10.1.1 is satisfied if for each a £ B and V £ U, Av,(/) € F for (f> —
= V-l[Dva(Q)Tl[Df(a)}-\f(va(Vz}}
- /(a)) =
V-lf(Vz;a).
Therefore A^(/) € T if and only if V~lf(Vz;a) € :F, as desired. This completes the proof. Pfaltzgraff [Pfa5] defined the order of a L.I.F. in the following way: Definition 10.1.3. For a linear-invariant family .F, let (10.1.4)
ord^ r =sup sup trace
= sup sup
A j = l 1=1
398
Linear invariance in several complex variables
be the order of J- '. We shall also refer to ord T as the trace order of 3- ' . We note that the trace order may be expressed in some alternative forms, using unitary transformations from C71 to C™ . If V £ U and / 6 T, then also g e T, where g(z) = A v (/)(z) = V~lf(Vz), ztB, and D*g(Q)(v, •) = V-l[D2f(Q)(Vv, .)]V,
t; 6 C".
Using the fact that the trace is a similarity invariant, one obtains that
Next, if v and A; are fixed, 1 < A; < n, we may choose V such that Vv = e^, where efc is the unit vector in C71 having the fct/l-coordinate equal to 1. Thus we obtain
In view of (10.1.4) and the above equality, we deduce that the order of the L.I.F. f can be written equivalently as follows (see [Pfa5]): (10.1.5)
= sup{i
(0)
If we set k = 1 and n = 2 in (10.1.5), we obtain the expression for which appears in [Bar-Fit-Gon2]: ord T — sup
+
when / = (/i, /2) G J7 is normalized by /,-(*) = ^i + 4^1 + 4^1^ + 4^2 + • • • , J = 1, 2, 2 = (zi, z2) G B. A similar expression to (10.1.5) was also used by Liu [LiuTl] (cf. [Gon4, p.lll]).
10.1. Preliminaries
399
Consequently, we have proved the following equalities: (10.1.6)
ord:F=supsup trace! -D2f(0) (w,-)\ \ /e:F|MI=i ^2 Jl fZF \\w\\--
= sup sup
j=l for each A; = 1,..., n, where fL(0) = J
filf.J Q
Q
OZjOZk
(0).
Remark 10.1.4. In order to define the notion of linear-invariance on a domain in C", one needs the domain to have a transitive automorphism group. In this case one can introduce the Koebe transform as in formula (10.1.1). The theory of linear-invariant families was extended to bounded symmetric domains by Gong and Zheng [Gon-Zhl-4] (cf. [Gon4]). They obtained a general distortion theorem involving the order of a L.l.F. Some interesting results for the case of the polydisc were obtained by Pfaltzgraff and Suffridge [Pfa-Su2]. There are some differences from the case of the ball; for instance, it is quite easy to compute the order of the L.l.F. K(P). We shall give some results for the polydisc in this chapter, and we refer the reader to [Gon-Zhl-4] and [Gon4] for the more general case of bounded symmetric domains.
10.1.2
Examples of L.l.F.'s on the Euclidean unit ball of C1
Next, we give some examples of linear-invariant families (L.I.F.'s) on the Euclidean unit ball of C" (see [Pfa5]). Example 10.1.5. The family S(B) (of normalized biholomorphic mappings of B) is a L.l.F. of infinite order in dimension n > 2. To see that the order is infinite, let n = 2, I e N, / > 2, and let f(z) = (zi, -—2-^rr) = (zi,z2 + Iz2zi + ...), \ (l-zi)lJ Clearly / e S(B) and since 2
the conclusion follows (see also [Bar-Fit-Gon2]).
z= (zi,z2) e B.
400
Linear invariance in several complex variables
Example 10.1.6. The family K(B] of normalized convex mappings of B is a L.I.F. Its behaviour as a L.I.F. is quite different in higher dimensions than in one. For example, the trace order of K(B) in dimension at least two does not give the minimum possible order of a L.I.F., and its exact value is unknown. Moreover, a L.I.F. of minimum order need not be a subset of K(B). Example 10.1.7. The set £S(B) is a L.I.F. of infinite order. The set Un(<*) Q £S(B), consisting of the union of all L.I.F.'s in £S(B) of order not greater than a, is also a L.I.F. on B. This is the n-dimensional generalization of the universal linear-invariant family U(oi) = Ui(a) (see Example 5.1.6). Example 10.1.8. Let Q be a nonempty subset of CS(B) and let denote the L.I.F. generated by Q, i.e.
To see that A[£?] is a L.I.F., it suffices to observe that the Koebe transform has the group property. Also it is obvious that A[£] = Q if and only if Q is a L.I.F. We shall see that many interesting questions about linear- invariant families arise through this process. For example, if Q = S* (B) (this set is not a L.I.F. since the Koebe transform translates the point /(O)), then A.[S*(B}} is a L.I.F. whose order is unknown in dimension n > 2. Any individual element / 6 £S(B] generates the L.I.F. A[{/}]. Its order is called the order of f , and is denoted by ord/ (see [God-Li-St]). Example 10.1.9. It is also possible to generate L.I.F.'s in higher dimensions beginning with sets of normalized locally univalent functions on the unit disc. For example, let / € CS and a > 0, and define Fa : B —> C™ by
Fa(z} = (/(*i), z'(f'(Zl))a) , z = (zi, z') e B, where z' = (z%, • • • > zn) and the branches of the power functions are chosen such that (/'(zi)) a U 1=0 = 1. It is obvious that Fa e £S(B). If we begin with a subset T C £5, we can construct the corresponding set (F}n,a C £S(B) given by (10.1.7)
(f)n,a
= (Fa(z) = (/(zi),z'(/'(2i)) a ) : / e F\
We then obtain a L.I.F. by taking A[(^")n>a]. (We note that even if T is a L.I.F., it is unlikely that (^n.a is a L.I.F.)
10.2. Distortion results
401
The set (F)n,i/2 will De denoted by (.F)n. This set was recently investigated by several authors (see [Pfa5], [Gra-Koh2], [Gra-Ham-Koh-Su], [Lic-St2]). In Section 11.5 we shall study the order of A[(.F)n(a], when T is a L.I.F. on the unit disc of a given order and a £ [0, 1/2]. An important special case occurs when T = K. In this case (K)n (and hence A[(.K")n]) is a subset of K(B), as shown recently by Roper and Suffridge [Rop-Sul]. (We shall prove this result in Chapter 11.) We remark that ord A[(.K")n] = (n+ l)/2, but as already noted, in several variables the exact value of ord K(B) is unknown. Example 10.1.10. If fj € £S, j = 1, . . . , n, then the mapping (10.1.8)
F(*) = (/i(zi), . . . , /n(*n)),
Z = (Zi, . . . , 2n) € B,
belongs to £S(B). Hence if we begin with subsets J-j of £5, j = 1, . . . , n, we may construct the subset of £S(B) given by : fk £ Fk, k = 1, . . . ,
Prom this we obtain the L.I.F. A[{^i, . . . , jFn}} generated by {.Fi, . . . , .
10.2
Distortion results for linear-invariant families in several complex variables
10.2.1
Distortion results for L.I.F. 's on the Euclidean unit ball ofC 1
In this section we are going to obtain a distortion theorem for linearinvariant families on the Euclidean unit ball B of C" . This theorem gives upper and lower bounds for the growth of the Jacobian determinant of mappings in a L.I.F. in terms of the trace order. Later, in Section 10.4, we shall see that there is a distortion theorem for ||.D/(;z)|| in terms of the norm order. Both of these results are generalizations of Theorem 5.1.8. We need to deduce some computational results concerning the automorphisms of B. We have seen in Theorem 6.1.23 that up to multiplication by a unitary transformation, the set Aut(B) consists of all mappings (pa given by
402
Linear invariance in several complex variables
(10.1.2). If Ta is the linear operator defined by (6.1.10), then it is not difficult to check the following relations (see [Pfa5], [Pfa-Su4]): (10.2.1)
T0(a) = a,
(10.2.2)
r-1 = ^(1 - aa*)Ta, s
o*T0(0 = <»*(•), Ta2 = aa* + all.
a
Also straightforward computations yield the following formulas: (/-aa*)- 1 = / + - a a * ,
(10.2.3) (10.2.4) and (10.2.5)
(l + a*z)2 •a)(a*u)a*(-)
Hence, combining (10.2.4), (10.2.5), (10.2.1) and (10.2.2), we obtain (10.2.6) (10.2.7)
s
a
and (10.2.8)
[D<pa(Q)]-lD2<pa(0)(v, •) = -(a*u)J - va*.
We give the details of the proof of (10.2.8) and we leave the proofs of the other relations for the reader:
=
s
a
(aa* + s^)(-^a* - (a*v)I + 2a(a*v)a*)
= ~2 | - (a*v)aa* - sfaa* - (a*v)aa* - sl(a*v) s
a •-
+2(a*a)(a*t;)aa* + 2s£(a*t;)a
10.2. Distortion results
403
= 4 [ ~ 2(a*v)oa* + 2||a||2(a*
a
L
J
+2(a*v)aa* - va* - (a*v)I 2
= -g (—1 + ||a|| )(a*u)aa* + 2(a*v)aa* — va* — (a*v)I = —(a*v)I — va*. The following lemma, due to Pfaltzgraff [Pfa5], is very useful and plays a key role in the proof of the distortion theorem for L.I.F.'s on B. Lemma 10.2.1. /// e CS(B], (p € Aut(B), v € C", andg(z) = A^( then (10.2.9)
Proof. By differentiating the formula g(z) =
we obtain
Dg(z) = and
+&f(
Taking the trace in both sides of this relation and using the fact that the trace is a similarity invariant gives (10.2.9). Remark 10.2.2. The formula (10.2.9) is a local result and hence a similar formula is valid on the poly disc.
404
Linear invariance in several complex variables
Another fact needed to prove the distortion theorem for L.I.F.'s on B is the trace formula for differentiating the determinant of an n x n matrix-valued holomorphic function of one complex variable (see [Golb]). If A : £ (->• A(£) is such a function, then
where -A*(C) is the adjoint matrix of A(£). Moreover, if A is nonsingular, then we can divide both sides of the above equality by det(A(£)) to obtain — log(det A(C)) = trace Further, if / is a normalized locally biholomorphic mapping on B and \\w\\ = 1, then applying the above equality to A(p) = Df(pw), p e 17, we obtain (cf. [Pfa2]): (10.2.10)
£- \og[Jf(pw)]
= trace{[D/(p^)]-1D2/(pu;)(u;, •)}•
In particular, this result is true when 0 < p < 1. We are now able to prove the distortion theorem for the Jacobian determinant on the unit ball in C". The first result in this direction was a theorem for L.I.F.'s on the unit ball in C2 due to Barnard, FitzGerald, and Gong [BarFit-Gon2]. The estimate (10.2.12) was obtained by Liu [LiuTl] and Pfaltzgraff [Pfa5]. Another interesting proof was recently given by Gong and Yu [GonYu]. The estimate (10.2.11) was given in a preprint of Liu [LiuTl] (cf. [Gon4]), and is also contained in a more general distortion theorem of Gong and Zheng [Gon-Zhl, Theorem 2] (cf. [Gon4, Theorem 5.4.1]). (The work of Gong and Zheng is carried out on bounded symmetric domains in C71 and measures the growth of | Jf(z)\ relative to the Bergman kernel function. Originally they considered only L.I.F.'s consisting of biholomorphic maps; locally biholomorphic maps are considered in [Gon4].) The proof below uses similar arguments as in [Pfa5, Theorem 5.1].
Theorem 10.2.3. Let T C £S(B) be a L.I.F. with ord.F = a < oo and f € T. Then (10.2.11)
10.2. Distortion results
405
where the branch of the logarithm on the left is chosen to have the value 0 when z = 0. Consequently,
a + wr+ (10.2.13)
-
- a - nr
|argJ/(z)|
\i
ll^ll/
2 e B.
Proof. Let
w € C1,
and hence in view of (10.2.9) we deduce that (t>, •)} = trace{-(a*v)I - va*}
Letting
in the above, we obtain trace a a trace < D2g(Q) I s-^a, • 1 > = —2 {( * )-^ + aa*} s I \ a / J a
+trace{[D/(a)]-1D2/(a)(a,.)}
Now, fix w £ B \ {0} and set a = pw, p e (0,1], in the preceding formula. Applying the relation (10.2.10) to the term tTace{[Df(pw)]~1D2f(pw)(pw, •)} we obtain 2 ,- , ^ P2HI -P 2 HI 2
d +p—\og(Jf(pw)),
406
Linear invariance in several complex variables
or equivalently,
1^22o(0) /^ = trace { -£> 2 yv ; Therefore, we deduce that
w d ( 7-^7, •\}> — i log S d
d ,
(10.2.14)
-PlMI Since ord.F = a, we have trace
< a,
and hence from (10.2.14) we obtain
If we integrate both sides of this inequality with respect to p over the interval [0, 1] we obtain (10.2.11), as desired. Finally, it suffices to observe that (10.2.11) implies both relations (10.2.12) and (10.2.13), by taking real and imaginary parts. This completes the proof. We remark that equality in (10.2.11) and (10.2.12) is achieved by the mapping (10.2.15)
FnM
for z = (zi,z') € B, where z' = ( ^ 2 , . . . , zn) and (10.2.16)
fft(zi)
= -^ { (\±^2 P [\l - zi
Note that the mapping Fn,a is convex on B for a = (n + l)/2 (in this case Fn^/2(z) = z/(l — zi), z — (zi,...,zn) G B), biholomorphic on B for ( n + l ) / 2 < a < n + l , and locally biholomorphic on B for a > n + 1.
10.2. Distortion results
407
Remark 10.2.4. In one variable we have seen that equality in the distortion result (5.1.8) holds if and only if / is a rotation of the generalized Koebe function, i.e. / is given by (5.1.9). In several variables, the situation is different and there are many mappings for which equality in (10.2.12) holds. To see this, we consider the following example from [God-Li-St]. Let n > 2 and let /!,...,/„ be holomorphic functions on U such that /fc(0) = 1, k = 1, . . . , n, and
For j fixed between 1 and n, let Gj : B —¥ C1 be the mapping whose jth z ri component is / /j(C)^C> and whose other components are given by Zkfk(zj), Jo k ^ j. It is easy to see that Gj € £S(B), and a short computation yields Z
= *! • • • *n € B.
fc=l
For each Gj, equality occurs in (10.2.12) when Zj is real and Zk = 0, fc ^ j. Among many important consequences of Theorem 10.2.3, we mention the following generalization to higher dimensions of Theorem 5.1.12. This result was obtained in [Pfa5]. Theorem 10.2.5. Let T C £S(B) be a L.I.F. with ord-F = a. Then a> (n + l)/2. Proof. Suppose a < (n + l)/2. From the lower estimate in (10.2.12) we conclude that |«//(^)| —>• oo as \\z\\ —t 1, for all / € F. On the other hand, since each mapping / € F is locally biholomorphic on B, Jf(z) ^ 0 for z e B. Therefore we obtain a contradiction with the minimum principle for holomorphic functions. Hence we must have ord^7 > (n+l)/2. This completes the proof. Theorem 10.2.3 also leads to a characterization of the order of a L.I.F. T in terms of upper and lower bounds for the growth of the Jacobian determinant of the mappings in F. This result was proved by Gong and Zheng in the context of bounded symmetric domains [Gon-Zh3, Lemma 2.1], [Gon4, Lemma 5.4.2]. Also Godula, Liczberski, and Starkov [God-Li-St] gave a proof for the case of the ball in which they showed that the order could be characterized using
Linear invariance in several complex variables
408
only the upper bound for the growth of the Jacobian determinant. We shall use similar arguments as in [God-Li-St, Theorem 1]. Theorem 10.2.6. Let F c £S(B] be a L.I.F. such that ordJF = a < oo. Then a is the smallest positive number such that the estimate (10.2.12) holds for all f € T and z^B. Proof. Let /3 > (n + l)/2 be such that
(i - H»| for all / E f and z e B. These inequalities are equivalent to the following:
We have to show that ft > a. For this purpose, let g € f. In view of the above distortion result, we have
- INI for z G B. Next, let z = pw with w fixed, ||ty|| = 1, and 0 < p < 1. Then we deduce that -P
that is, -/31og
< Re log
I—
Dividing these inequalities by p and then letting p \ 0, and using the fact that Js(0) = 1, we obtain
p=o Taking into account (10.2.10), this relation is equivalent to
10.2. Distortion results
409
Since g 6 J- and w, \\w\\ = 1, were arbitrarily chosen, we deduce that sup sup -Re < t r a c e * D 2 g ( Q ) ( w , O f f g€.J~ ||iu||=l 2
v
I
< ft-
JJ I
Finally it suffices to remark that
sup sup -
= sup sup -Re { tracej D2g(Q)(w, •) f r h g€F\\w\\=l 2 L L JJ ' since we are free to multiply w by a complex scalar of modulus 1. Consequently, a < (3, as desired. This completes the proof. Remark 10.2.7. Let T be a L.I.F. of finite order. From Theorem 10.2.6 one may deduce another equivalent definition of the order of T (cf. [God-Li-St] , [Gon-Zh3],[Gon4]):
(10.2.17)
ordJF = inf (a : C1 ~ ND" 0 '
I a + iuin *^)
< |j
(z)|
—- / € ; F , 2 € J 3 . This latter definition of the order of L.I.F.'s has an interesting consequence, due to Godula, Liczberski and Starkov [God-Li-St]. Corollary 10.2.8. Let /i, /2 € CS(B) be such that J f l ( z ) = Jf2(z), zeB. Then ord /i = ord /2. Proof. Let € Aut(B) and gt = A^(/j), i = 1,2. It is obvious that
and hence Jgi(z) = J92(z), z € B. Taking into account (10.2.17), the conclusion follows.
410
Linear invar iance in several complex variables
10.2.2
Distortion results for L.I.F.'s on the unit polydisc of C™
In this section we obtain a distortion theorem for L.I.F.'s on the unit polydisc P of C71 , and thus we consider Cn with the maximum norm || • ||oo- For the proof we use similar arguments as in the proof of Theorem 10.2.3, including the fact that Lemma 10.2.1 is valid on the polydisc. The distortion estimate (10.2.19) was obtained in [Pfa-Su2], and (10.2.18) as well as (10.2.19) can be obtained from the work of Gong and Zheng [Gon-Zhl] (cf. [Gon4, Theorem 5.4.1]), if we interpret their result for the polydisc. We let £S(P) denote the set of normalized locally biholomorphic mappings from P into C" . The Koebe transform and the notions of linear- invariance and order of a L.I.F. on P are completely analogous to the case of the unit ball. The automorphisms of the polydisc are given in Theorem 6.1.25. Theorem 10.2.9. Let T C £S(P) be a L.I.F. with ord.F = a < oo. // / 6 T, then (10.2.18)
log \Jf(z) f[(l - N 2 )l I < a log f l ^ l l j ' 0 0 ) , -_i
'
\ •*•
Halloo/
for all z = (zi,..., zn) 6 P, where the branch of the logarithm on the left is chosen to have the value 0 when 2 = 0. Consequently, < I Jf(z\\ <
/i _ u. < \jf(z)\ < 1 1 _ | | | |oo J 11U \Z3 A
oo
.=
for all z e P. Proof. Let a = (ai, . . . , a n ) e P \ {0} and let if> e Aut(P) be given by
in the notation of Theorem 6.1.25. Let g — A^,(/). Using the trace formula (10.2.9) and taking into account (6.1.14) and (6.1.15), we obtain
for all v eC1. Setting
10.2. Distortion results
411
in the above gives
Further, if we fix w € B \ {0} and set a = pw, 0 < p < 1, in the above, and use similar arguments as in the proof of Theorem 10.2.3, we deduce the relation
where A = (Ai, . . . , A n ) and
Since g e F, ||A||oo = 1 and ord^" = o;, it follows that
and hence
3=1
Integrating both sides of the above inequality in p between 0 and 1, we obtain (10.2.18). Also, taking the real part inside the left-hand side of (10.2.18) and using the fact that the minimum of |«//(2)| on the closed polydisc \\z\\oo < r, r < 1, occurs on the distinguished boundary, we easily deduce (10.2.19). This completes the proof. Note that equality in (10.2.18) and (10.2.19) is achieved by the mapping = (fa/n(zi),
• • • , /«/n(^n)),
Z = (zi, . . . , Zn) G P,
412
Linear invariance in several complex variables
where /^ is given by (10.2.16). We remark that the mapping Fa is in K(P] for Q: = n, is biholomorphic on P for n < a < 2n, and is locally biholomorphic on P for a > n. The lower bound for the distortion in (10.2.19) and the minimum principle for holomorphic functions give a lower bound for the order of a L.I.F. on the unit polydisc (see [Pfa-Su2]): Theorem 10.2.10. If F C £S(P) is a L.I.F. , then ordF > n. Moreover, similar reasoning as in the proof of Theorem 10.2.6 yields the following result (cf. [Gon4, Lemma 5.4.2]; [Ham-Kohll]): Theorem 10.2.11. Let T be a L.I.F. on P such that ordF = a < oo. Then a is the smallest positive number for which (10.2.19) holds for all f € T and z 6 P. Proof. Let (3 > n be such that
+
3 — -1
for all / € F and z = (z\, . . . , zn) 6 P. We have to show that (3 > a. For this purpose, let / e F. It is not difficult to see that the above inequalities imply the relation
for all z = (zi, ...,zn) £ P. Next, setting z = pw with \\W\\QQ = 1 and p G (0, 1), we obtain -/31og (i^) < Re log [jf(pw) f[(l -
2 P
K|2)] < /31og (fr^)-
Dividing these inequalities by p and letting p \ 0, and using the fact that Jy(0) = 1, we obtain " -2
With similar reasoning as in the proof of Theorem 10.2.6, we deduce that sup sup
-R
10.2. Distortion results
413
Since sup sup
itrace{£>2/(0)(«v)}| 2 2 e {trace{D /(0)(uv)}}|,
= sup sup
we conclude that ord J- = a < /3, as desired. This completes the proof. Therefore, as in the case of the Euclidean unit ball, we can give an alternative definition of the order of a L.I.F. on the unit polydisc in terms of the growth of the Jacobian determinant (see [Gon4, Lemma 5.4.2] and [HamKohl 1]. Compare with Remark 10.2.7): Remark 10.2.12. Let T be a L.I.F. of finite order on P. Then
{
(1 — \\z\\ }a~n a : 7, . , < (1 + N|oo)a+n ~
\Jf(z)\ f
._]_
Moreover, there is also an analog of Corollary 10.2.8 on the unit polydisc. We have [Ham-Kohl 1] Corollary 10.2.13. Let /i,/ 2 € £S(P) be such that Jfl(z) = J/2(z), z £ P. Then ord/i = ord/2, where ord/j = ord A[{/j}], j = 1,2.
Problems 10.2.1. Let F C £S be a L.I.F. on the unit disc with ord T = a < oo and let (^)n = (•^r)n,i/2 De tne set of locally univalent mappings of the Euclidean unit ball B generated as in Example 10.1.8. Show that a(n+J) .
(Pfaltzgraff, [Pfa5,6], Liczberski and Starkov, [Lic-St2].) 10.2.2. Prove Theorem 10.2.10. Also prove Corollary 10.2.13. 10.2.3. Let / € Un(a) and M(r, J/) = max \J/(z)\, 0 < r < 1, where is the Euclidean norm of C" . Show that
l\*\\=r
414
Linear invariance in several complex variables
is a non-increasing function of r G [0,1). Show that for each v € C n , ||u|| = 1, IJf(™)|
T
a_/n±i}
is also a non-increasing function of r 6 [0,1). (Liczberski and Starkov, 2000 [Lic-Stl].) 10.2.4. Verify the relations (10.2.1)-(10.2.7). 10.2.5. Show that the mapping Fn>a given by (10.2.15) is biholomorphic on the Euclidean unit ball B of Cn for (n + l)/2 < a < n + 1, and locally biholomorphic on B for a > n 4-1.
10.3
Examples of L.I.F.'s of minimum order on the Euclidean unit ball and the unit polydisc of Cn
10.3.1
Examples of L.I.F.'s of minimum order on the Euclidean unit ball of Cn
In this section we shall prove two important and unexpected results, due to Pfaltzgraff and Suffridge [Pfa-Su2]: • The Cayley transform does not give bounds for the growth of the Jacobian determinant of all normalized convex mappings of the Euclidean unit ball of Cn, n > 2 . • The n-dimensional analog of Corollary 5.2.5 (i.e. T has minimum order if and only ij T C K] does not hold on the Euclidean unit ball of Cn, n > 2. Indeed, K(B] does not have minimum order. At this time, the order of K(B] is unknown when n > 2. The first estimates for ordK(B) in several variables were obtained by Barnard, FitzGerald and Gong [Bar-Fit-Gon2] in dimension two. Theorem 10.3.1. Let n = 2 and let B C C2 denote the Euclidean unit ball of C 2 . Then (10.3.1)
^ < ordK(B) < 1.761.
Proof. It is obvious that oidK(B) > 3/2, in view of Theorem 10.2.5. Thus we have only to show the upper bound in (10.3.1). For this purpose, let
10.3. Examples of L.I.F.'s of minimum order
415
/ e K(B). From Theorem 6.3.4 we have 1 - Re ([Df(z)]-lD*f(z)(v,
(10.3.2)
u), z) > 0
for z € B and v € C2, \\v\\ = 1 and Re (z,v) = 0. Since / is normalized, / can be written as follows: B, j = l,2.
Short computations combined with (10.3.2) yield that
"i ^\ (10.3.3) Re { 1 - z* I ^2,0
**!,!
"1,1
2a0)2
Fix z € B \ {0}. Let ci and 02 be positive numbers and define v = (vi, •
by V-2 =
Then ||u|| = 1, Re (z,i'} = 0 and in view of (10.3.3), we obtain c\zl \ 1,1
We may write z in the form z = £w where £ € C, 0 < |£| < 1, and t/; = (wi,W2) € C2 with ||ty|| = 1. Then the above inequality becomes cX \ / ZCA A a; ; a; ; ZCA A \ 1
V
ciio2
/
Clearly the left-hand side of the above inequality can be written as Re /i(C)> where h is a holomorphic function on U such that h(0) = 1. Since Re h(£) > 0, C 6 U, the coefficient of £ in the power series expansion of h about the origin may be expressed as
:^\
416
Linear invariance in several complex variables /•27T
= 2 / e-« ./o where fj, is a non-decreasing function on [0, 2?r] such that ^(27r) — fj,(0) = I (see
the proof of Theorem 2.1.5). Hence (10.3.4)
/•27T
/ Jo l&1
%e
Let wi = \Wi\e and w2 = \W2\e *, where 0\, 62 € [0,2?r]. Multiplying both sides of equation (10.3.4) by w\ and then integrating with respect to ddi/2'jr, we deduce that
and hence 102
,(2)
(10.3.5)
ci
\Wl\
Let (c2/ci)|u;2/tyi|2 = ex and |iui| = x. Thus (10.3.5) is equivalent to
Since
^T
ori
^(0) ^
3—^
e K(B)
we only need to consider the case a = 1/2 in the estimate for In this situation we obtain (10.3.6)
—
x
^ 7 1
4 1 — or9 '
0 < x < 1.
+
10.3. Examples of L.I.F.'s of minimum order
417
It is easy to see that the function on the right-hand side of (10.3.6) has a minimum at x = A / (9 — \/33) /6 and that this minimum is strictly less than 1.761. Thus from (10.3.6) we obtain that o<x
41 - x
Since / is arbitrary, the conclusion now follows. This completes the proof. Remark 10.3.2. Liu [LiuTl] showed that if B is the Euclidean unit ball of C71 , then the upper estimate
holds (cf. [Gon4]). Therefore, we know that n ^ Jt-/m^ , v ^ ~Y~ < ordK(B) < -y- + —3— /fa - 1),
when B is the Euclidean unit ball of C" , n > 2. Barnard, FitzGerald and Gong [Bar-Fit-Gon2] conjectured that in the case of n complex variables, n > 2, However, this conjecture is false, as shown by Pfaltzgraff and Sufrridge [Pfa-Su2]: Theorem 10.3.3. ordK(B) > (n + l)/2 for n > 2.
Proof. First step. Let F : B ->• C1 be given by
(10.3.7)
F(z) = ((I + (V2/2)zn)Zl, . . . , (1 + (V2/2)zn)zn-i, zn).
We first show that F is convex on the unit ball B. It suffices to prove this assertion in the case n = 2, since only minor modifications are needed for n > 2 (see also Example 6.3.14 and [Gon4]). Since
F(z) = (zi + — we have
DF(z) =
A/2 -x/2 \ 2 *2 2 Zl 0
1
)
418
Linear invariance in several complex variables
V
o
i
/
Consequently F is a normalized locally biholomorphic mapping on B and D2F(z)(v,v) = (V2vivz,o) ,
v = (ui,u 2 ) € C2.
Moreover, for z e J3, v 6 C 2 , ||v|| = 1 and Re {2,v} = 0, we obtain
1*11 1
2
Re ([DF(z')]- D F(z)(v,v),z)
= Re
Z2 1 + V/2
<
1-
J
Now, let h : [0,1/V2] ->• [0, CXD) be given by
i-y
,
0 < y < 1/V2.
Then it is elementary to see that h assumes its maximum value 1 when y — 1/2, and hence we deduce that
N l
2
Re ({DF(z)}- D F(z)(v,v),z}
< 2\vi\\v2\
^
<
max h(y) = 1.
v/2 In view of Theorem 6.3.4, we conclude that F is a convex mapping of B. Second step. We now prove that ord K(B] > (n + l)/2 for n > 2. For this purpose, let x € (—1,1) and 0 e Aut(B) be given by
-xz
1
Also let F be given by (10.3.7) and c = \/2/2. Straightforward computation shows that the Koebe transform A^(F) is given by C
~r
X
I — ex
C— X ":
I — cx
z
n
10.3. Examples of L.I.F.'s of minimum order Let G =
419
. Then G 6 K(B] and it is easy to deduce that c —x
1 — ex'
dzjdzn
2x,
~~ ~~ j = n.
Hence
2-w
nx +
n— 1
c —x 1 — ex
3=1
Taking into account (10.1.6) as well as the above equality, one concludes that ordK(B)>
sup
nx +
n— 1
c —x 1 — ex
-Kz
Let
n—1 c — x -1 < x < 1. 2 1-cx Straightforward computation yields that g has a maximum value at xn = - l)/(2Vn). Then nx +
n +1
^n — 1
or:dK(B)>g(xn) = ^-7^
for
n > 2.
Consequently, ordK(B) > (n +1)/2 for n > 2, as claimed. This completes the proof. Note that for n = 2, g(x2) = 5\/2/2 - 2 w 1.535 > 3/2 and for n = 3, g(x3) = 4>/2 - 2v/3 w 2.19 > 2. Open Problem 10.3.4. Find ord K(B} when B is the Euclidean unit ball o/C", n > 2. The following example of Pfaltzgraff and Suffridge [Pfa-Su2] shows that *y the Cayley transform f ( z ) = , z e B, does not give bounds for the 1-21 growth of the Jacobian determinant of all normalized convex mappings of B when n > 2. In fact, sharp bounds for |J/(z)|, / € K(B}, are not known. Example 10.3.5. The inequality (10.3.8)
—
1
" \n+l -
l
(i-IN \n+l
is false for some values of z G B and some / G K(B), when B is the Euclidean unit ball of C1 with n > 2.
Linear invariance in several complex variables
420
Proof. Let F and 0 be as in the proof of Theorem 10.3.3. Also let G(z) = A.tj)(F)(z). We prove that for the mapping G, the distortion estimate (10.3.8) does not hold for certain z sufficiently near 0. We have /
4.-a l + ^r-
G(z) =
\
+
-xzn
(l-xzrf \
A simple computation yields that /
n-l
1
JG(Z) =
Let Zj = 0 for 1 < j < n — 1 and zn = t, — 1 < t < 1. For n > 3 and x = l/\/2, we obtain d
[/-.M
dt Y
l
n+i1
(i-*) L- 0 ~
1
dt
d
1 2n
(!-*)n+1
(l__L\
J t=0
for t suffi-
Consequently, when n > 3, we have |JG(Z) (1
ciently near 0 and positive, and |JG(^)| < 7":
\\\n+i for t sufficiently near
0 and negative. When n = 2, choose x = \/2 - 1/2. Then d I" dt ["U^~J
1 1 d 'l + ^ ( i _ ^ 3 j fa (1 - xt)4
! (l-t)3
J <=0
= 4x - 3 +
10.3. Examples of L.I.F.'s of minimum order
421
Therefore, again we deduce that |Jcz(z)| > —^
M OHN
positive, while |«/G(Z)| < T:
3
for t near-0 and
...3 for t near 0 and negative.
Open Problem 10.3.6. Find the sharp bounds for \J/(z)\ when f is a normalized convex mapping of the Euclidean unit ball in C", n > 2. We now recall that if T is a L.I.F. on the unit disc, then in view of Corollary 5.2.5, o r d ^ = l & FCK. In several complex variables the analogous property (i.e. T has minimal order if and only if T C K(B}} is not true, as shown recently by Pfaltzgraff and Suffridge [Pfa-Su2,3]. They constructed some examples of L.I.F.'s T such that ord^" = (n + l)/2, but f <JL K(B) for n > 2. Also recently Graham and Kohr [Gra-Koh2] gave another such example. Their construction involves the operators (10.3.9)
*n,a (/)(*) = Fa(z) =
where z' = (z%, ...,2 n ), a € [0,1] and / is a normalized locally univalent function on the unit disc U such that /(£) ^ 0 for £ e U \ {0}. We choose the branch of the power function such that
\
\
^
Z\
= 1.
I
I
'
zi=0
We remark that these operators preserve some interesting geometric properties. For example, in [Gra-Koh2] it is shown that ^n,a(S*} C S*(B] and also ^n,a(S} C 51(B), where Sl(B) is the set of normalized univalent mappings of B which can be embedded in Loewner chains (see Section 8.3). However, the operator ^fn,a does not preserve convexity when n > 2, and the example of Graham and Kohr is based on this fact. To see that convexity is not preserved, we note the following Example 10.3.7. Let a € [0,1], n = 2, and / : U -»• C be given by
Linear invariance in several complex variables
422
Then / is convex, but *2,a(/) is not convex on the Euclidean unit ball in C2.
Proof. Let
and i 1 . '
V = Z2 \ ~— log 6
If ^2,a(f}(B) is a convex domain, then so is its intersection with the plane Im u = 0, Im v = 0. This intersection contains the entire real w-axis and precisely the interval (—1,1) of the real v-axis. In order to show that convexity is not satisfied, it suffices to show that if z\ —> 1 along the real axis then we are constrained to have \v\ —>• 0. If z\ —> 1 along the real axis then Re u —» oo and Im u — 0, and
1-zi Let Z2 = £ > 0, with e small and let z\ — \/l — e2. Then it is elementary to show that for a. G (0,1], we have 1
2
M =
log
/i+,/rr72M2Q
== (log ( log
l +
Now it is easy to check that |i>|2 —> 0 as e —>• 0. On the other hand, if a = 0 then =e
0
as e -» 0.
Hence \v2\ -> 0 as e —>• 0, for a € [0, 1]. This completes the proof. Using the idea in Example 10.1.8, we are now able to prove the following result [Gra-Koh2]. Another example of a L.I.F. with minimum order will be
10.3. Examples of L.I.F.'s of minimum order
423
given in Chapter 11. We shall use similar arguments as in the proof of [Lic-St2, Theorem 5]. Theorem 10.3.8. The L.I.F. A[^n>Q(K)] has minimum order, that is ordA[#n,oC?0] = (n + l)/2. However, A[#n)0(#)] is not a subset of K(B) for n>2. Proof. The fact that A[\l>n)o(.fiT)] £ K(B] for n > 2 is a direct consequence of Example 10.3.7. Thus we have only to show that ord A[#n)0(/0] = (n+l)/2. For this purpose, let £ = A[*n>0(^)] and let (3 = ord£. From Theorem 10.2.6 we know that /3 is the smallest positive number for which the inequality
a-
+
holds for all G € <7 and z € B. Let G 6 Q. Then G(z) = A^(F}(z) for some F e Vn,o(K) and e Aut(£). Taking into account Theorem 6.1.23, we may write 4>(z) = Vha(z) for some V € U and a = (ai, . . . , on) € 5, where U is the set of unitary transformations inC n , ha(z) = -(/>a(z) = Ta
L —a z
, z e B,
and 0a, Ta and sa are given by (6.1.9), (6.1.10) and (6.1.11), respectively. Straightforward computation yields the relation
< i. Here we have used the fact that n+l
by (6.1.12). Now since F e tynio
, there exists a function f E K such that
424
Linear invariance in several complex variables
It follows that
\M*)\ = If and thus (10.3.11) |J G (zi,0,...,0)| = Further, let b = (b\, . . . , 6n) = Va and let v = (vi, . . . , vn) denote the first column of the unitary matrix V. Then ||6|| = ||a|| and a short computation shows that — zib*v = hb(ziv),
\zi\
where s = sa = s&. If we let
and
then we have
61 — z\c
bi — z\c
There are two possibilities: either 61 ai ^ c or b\a\ — c. In the first case, ip is a univalent function on the unit disc U and il>(U) is a disc contained in U. Taking into account (10.3.11), we obtain (10.3.12)
|J G (zi,0,...,0)| =
1 11-
7T, l"+l
n-l
Setting
we deduce that g £ K.To show this, it suffices to observe that if / is a convex univalent function on U and D is any disc such that D C [7, then /(£)) is a
10.3. Examples of L.I.F.'s of minimum order
425
convex domain. (This follows from the fact that f ( U r ) is convex for 0 < r < 1.) In particular, f(ip(U)) is a convex domain. We also note that g is normalized. Consequently, the distortion result
holds on U (see (2.2.7)). Thus, if we let 7 = (n + l)/2 and use the above relation and the equalities (10.3.12), we deduce that
for aU zi € U. In the second case, if b\ai = c, then il)(z\) = &i> \*i\ < 1>
Now, it is obvious that if 7 = (n + l)/2, we have
7 a
--** 1
- ^ , ,^ 1
(1 + |2l|r ^ - 11 - siSil** - (1 - Izij)
'
'
l! <
Therefore, again the relation (10.3.13) holds. On the other hand, since the equality I./AHKC?)^)! = l^c(W^)l holds for all W 6 U and z e B, we conclude from (10.3.13) that
Since G was arbitrarily chosen, it follows from (10.3.10) and the above estimate that (3 = ord£ < 7. However, Theorem 10.2.5 gives ord£ > 7, and so we must have ord (/ = 7. Another interesting example of a L.I.F. of minimum order on the unit ball of C1 is the following due to Pfaltzgraff and Suffridge [Pfa-Su3]. Using Corollary 10.2.8, we can give a simpler proof than the original of Pfaltzgraff and Suffridge. This proof was obtained by Godula, Liczberski and Starkov [God-Li-St]. We have
426
Linear invariance in several complex variables
Theorem 10.3.9. Let T = {F G CS(B) : JF(z] = 1} and let A[^j be the L.I.F. generated by T. Then ord A[J] = (n + l)/2. Proof. Let fo(z) = z, z € B. Then ord/o = (n + l)/2, and in view of Corollary 10.2.8, one deduces that ord/ = ord/o, for all / € F . This completes the proof. Note that Theorems 10.3.8 and 10.3.9 are particular cases of the following result ( [Ham-Kohl 1]; compare with [Lic-St2]). We leave the proof for the reader, since it suffices to use arguments similar to those in the proof of Theorem 10.3.8. Theorem 10.3.10. LetF be a nonempty subset of £S(B) such that F e F if and only if there is f e K for which JF(Z) = f'(zi), z = (z\, z'} e B. Then ordA[:F] = ( n + l ) / 2 . We mention that other examples of L.I.F. 's of minimum order on the Euclidean unit ball of C™, n > 2, have recently been obtained in [Pfa-Su2,3] and [Gra-Ham-Koh-Su] .
10.3.2
Examples of L.I.F. 's of minimum order on the unit polydisc of C1
We next consider some examples of L.I.F.'s on the unit polydisc P of C™, n > 2. Again we find that there are L.I.F.'s that have minimum order and which are not subsets of K(P). However, as shown by Pfaltzgraff and Suffridge [Pfa-Su2] , in contrast to the situation for the Euclidean unit ball it is very easy to obtain the exact value of the trace order of K(P), and ord K(P) is minimal. Theorem 10.3.11. ordK(P) = n. Proof. Let / e K(P}. In view of Theorem 6.3.2, there exist /_,- 6 K, j = 1, . . . , n, such that , . . . , /„(*„)), Z = (*i, ...,Zn)
It follows that D2f(Q)(w, •) has the diagonal matrix /{'(O)t^ ...
0
......... 0
...
f»(Q)Wn
10.3. Examples of L.I.F.'s of minimum order
427
and hence trace Now, since fj 6 K, we have |/j'(0)/2| < 1, and therefore /j'(0)
trace < -
< n\\w oo.
j=i Consequently, ordK(P) < n. On the other hand, in view of Theorem 10.2.10, ord.K"(P) > n, and hence the conclusion follows. The next result shows that, as in the case of the ball, there exist L.I.F.'s of minimum order on the unit poly disc P of C1, n > 2, which are not subsets of K(P). (See [Pfa-Su2] and compare with Theorem 10.3.8. and Corollary 5.2.5.) Theorem 10.3.12. For n > 2 there exist L.I.F. 's F C £S(P) such that ord^-" = n, but T is not contained in K(P). Proof. Consider any mapping F : P —*• C" of the form
(10.3.14)
F(z) = (z, zn + g(z)),
z = (z, zn) e P,
where z = (zi,... ,z n _i) and g is a nonconstant holomorphic function from Halloo < 1 into C such that 0(0) = 0 and Dg(Q) = 0. These mappings are not convex on P since they do not satisfy the criterion given in Theorem 6.3.2. However, we show that ordA[{F}] = n. For this purpose, we consider the set of mappings {F^z) = F(4>(z)) : (j> e Aut(P)}. From (10.3.14) we obtain
Short computations yield that : -
7
, , . . Wj.
2^^(0) Now, suppose that / € A[{F}] and V is a unitary transformation which preserves the polydisc, i.e. V is the composition of a diagonal unitary transformation and a permutation matrix. Then if h = Ay(/), we have trace{£>2/i(0)(i;, •)} = trace{D2/(0)(Vv, •)}•
428
Linear invariance in several complex variables
It therefore suffices to consider automorphisms 0 of P which have the form (f>j(zj) = el9i(zj - a,j}/(l- ajZj) for \Zj\ < I, where |o,-| < 1 and Oj € R, j = 1,..., n (see Theorem 6.1.25). In this case we deduce that •
2
"
'
•
•
>
•
'
"•
—
jr
II
< n to
I
Therefore trace ^ r
< n,
||w||oo = 1?
and hence ordA[{F}] < n. Finally, applying Theorem 10.2.10 we deduce that ord A[{F}] = n, as claimed. This completes the proof. We note that the example constructed in the proof of Theorem 10.3.12 is a particular case of the result below [Ham-Kohl 1]. (Compare with Theorem 10.3.9.) Theorem 10.3.13. Let T = {/ € £S(P) : J/(z) = 1}. Then ordAf^j = n, but A[f] <£ K(P) when n>2. Proof. Let fo(z) = z, z € P. It is not difficult to check that ord/o = n. Since each / e F has the property that J/(^) = 1, it follows from Corollary 10.2.13 that ord/ = ord/o, and thus ordA[^"] = n, as desired. The fact that A[.F] ^ K(P] when n > 2 is obvious, since the map F given by (10.3.14) belongs to f but is not convex on P.
Problems 10.3.1. Let a € [0,1] and let *n>a denote the operator defined by (10.3.9). Show that (ii) ord A[*B,0(5)] = ord A[*B,0(S*)] = (n + 3)/2. (Graham and Kohr, 2002 [Gra-Koh2].) 10.3.2. Complete the details in the proof of Theorem 10.3.13. 10.3.3. Let P be the unit polydisc of C", a = (01,... ,a n ) € P and let ij>a € Aut(P) be given by (6.1.14)-(6.1.15). Show that a subset T C £S(P)
10.4. Norm order of linear-invariant families
429
is a linear-invariant family if and only if for each / € .F, the mapping ^~1-^V'a(/)(^(')) e F f°r all a € P and all unitary transformations V of C™ which preserve P.
10.4
Norm order of linear-invariant families in several complex variables
In this section we are going to study a new notion of order, recently introduced by Pfaltzgraff and Suffridge [Pfa-Su4], for a linear-invariant family of locally biholomorphic mappings of the Euclidean unit ball in C" . The reader will see throughout this section that the norm order has a much broader range of applicability to the study of geometric properties of locally biholomorphic mappings than does the trace order. For example, using the norm order we may deduce results about the radius of convexity, starlikeness and univalence of mappings in a L.I.F. In the case of one variable both the norm order and the trace order reduce to the usual order of a L.I.F. The basic source for this section is [Pfa-Su4]. We shall keep the notation from the previous sections of this chapter. In particular we recall the notation f ( z ; a) (see (10.1.3)) for the Koebe transform of a mapping / with (f>a e Aut(B) given by (10.1.2). First we give a lemma of PfaltzgrafT and Suffridge [Pfa-Su4] which is useful in the computation of the norm order. Lemma 10.4.1. Let T be a L.I.F. and f € T. Let a € B, V € U, and let g(z) = f ( z ; d ) and G(z) = f(Vz;a), z£B. Then (10.4.1)
sup{||D2G(0)(i;,w)|| : ||v|| = 1, V € U} = sup{|p2<7(0)(7,7): 11711 = 1}-
Proof. Since G(z] = g(Vz) for z € B, it follows that DG(z) = Dg(Vz}V and £>2G(0)(-, •) = D2g(0)(V(-),V(-)). Hence D2G(Q)(v, v) = £>2s(0)(7,7),
7 = Vv,
430
Linear invariance in several complex variables
and (10.4.1) follows, because it is obvious that ||7J| = ||v|| = 1. This completes the proof. ~ fc Recall that if Ak : JJ Cn —>• Cn is a symmetric fc-linear mapping, then by j=i Theorem 7.2.19, we have (10.4.2)
||Ifc||=
sup \\Ak(zM,...,zW)\\=
sup ||Ifc
We are now able to define the norm order of a L.I.F. on the Euclidean unit ball of C™ . This notion has recently been introduced by Pfaltzgraff and Suffridge [Pfa-Su4]. Definition 10.4.2. If T is a L.I.F., we define the norm order of T by
In view of Lemma 10.4.1 and (10.4.2) we have
Next we compute the second order Frechet derivative of /(z; a) at z = 0. For this purpose, it suffices to use formulas (10.2.6), (10.2.7), (10.2.8), and the equalities Df(z-a) =
[DvaMr^DfWri
and
Evaluating at z = 0, we deduce for v e Cn that £>2/(0; a)(u, u) = -(a*v)v+
(10.4.3)
~Ta
10.4. Norm order of linear-invariant families
431
where Ta and sa are defined by (6.1.10) and (6.1.11). The next result of Pfaltzgraff and Suffridge [Pfa-Su4] gives the minimum possible value of the norm order of a L.I.F. on B. The corresponding result for the trace order is given in Theorem 10.2.5. In Theorem 10.5.1 we shall see that the lower bound is assumed by the L.I.F. of normalized convex mappings on B. Theorem 10.4.3. ||ord|| J" > 1 for every L.I.F. F'. Proof. Let / € T and o 6 B \ {0}, and let v = a/\\a\\ in (10.4.3). Since D<pa(Q)a = s2aa by (10.2.6), we obtain
= -o
2||a|
;Ta[Df(a)}-lD2f(a}(a,a).
Further, let
Then g is a holomorphic function on the disc |£| < l/||a|| and since g(0) = 0, for every r with r < l/||a|| there is a point £, |£| = r, such that Re
,a
We now replace a by £a where £ is chosen so that |£| = 1 and Re #(£) < 0. Then
: |||a||2 - Re p(C)| = ||a||2 + |Re where we have used the fact that T^a = Ta and Re 0(0 < 0. Hence ||ord|| T > \\a\\2 for all a 6 B \ {0}, which implies ||ord|| T > 1, as desired.
432
Linear invariance in several complex variables
The following distortion theorem, due to Pfaltzgraff and Suffridge [PfaSu4], is a generalization to higher dimensions of the distortion estimate (5.1.8). It involves estimating ||D/(z)||, which is considerably more difficult than estimating |J/(^)|. Since the proof is rather long, we leave it for the reader. However, it is a very interesting proof. Theorem 10.4.4. If F is a L.I.F. with ||ord||.F = a < oo, then
Dg(z) \a-l
< IP/0011 < for all f e F and z 6 B (with z ^ 0 where z/\\z\\ occurs). An upper estimate for the growth of mappings in a L.I.F. follows from Theorem 10.4.4., as shown by Pfaltzgraff and Suffridge [Pfa-Su4j. Only an upper bound is obtained, since a general L.I.F. can include maps with zeros in B \ {0}. Theorem 10.4.5. If f is a L.I.F. with ||ord|| T = a < oo, then
\\M\\ < 2a I V l - l l z l for all f € T and z 6 B. Equality is achieved by the mapping (10.4.4)
F(z) = (/a(*i), z'h(Zl)),
z = (z^z1) E B,
-1 and h(zi) = where z1 = (z2,...,Zn), fa(zi) = — 2a - z\ - z\ Proof. Let z e B \ {0} and / e F. Assume that /(z) ^ 0 and let G : [0, 1] -)• R be given by
Using the upper bound for ||D/(-)|| from Theorem 10.4.4, we obtain the estimate = G(l) = f 1 G'(t)dt = f 1 Re ( D f ( t z ) z ,
10.4. Norm order of linear-invariant families
2a
433
\l-\\z\\
This completes the proof. It is of course possible to obtain upper and lower bounds for the Jacobian determinant of maps in a L.I.F. of given norm order. To do so, it suffices to note that ord^7 < n||ord||.F (since |trace{A}| < n\\A\\ for any n x n matrix A), and to apply Theorem 10.2.3. Corollary 10.4.6. Let F be a L.I.F. with ||ord||^ = a < oo. If F e T then
a+ Problems 10.4.1. Let T be a L.I.F. and let cl(T} be the closure of T with respect to the usual topology of local uniform convergence. Prove that d(F) is also a L.I.F. on B and ||ord|| d(F) = ||ord|| T. (Pfaltzgraff and Suffridge, 2000 [Pfa-Su4].) 10.4.2. Let a > 1 and let B denote the Euclidean unit ball of C". Also let F : B -» C" be given by (10.4.4). Let A[{F}] denote the L.I.F. generated by {F}. Prove that ||ord|| d(A[{F}]) = a. (Pfaltzgraff and Suffridge, 2000 [Pfa-Su4].) 10.4.3. Prove that if T is a L.I.F. with ||ord|| T = a < oo and if / € T then
(\ (Pfaltzgraff and Suffridge, 2000 [Pfa-Su4].)
C".
434
Linear invariance in several complex variables
10.5
Norm order and univalence on the Euclidean unit ball of Cn
In this section we determine the norm order of the linear-invariant family K(B), and we investigate some radius problems which are analogous to results in one variable obtained in Chapter 5 (radius of convexity, starlikeness, and univalence). These results indicate that the norm order is much more closely related to geometric properties of the mappings in a given L.I.F. on the Euclidean unit ball in C" than the trace order. We begin by showing that, in contrast to the situation for the trace order of the L.I.F. K(B) (see Open Problem 10.3.4), it is possible to give a complete characterization of ||ord|| K(B}. This result was obtained recently by Pfaltzgraff and Suffridge [Pfa-Su4]. Theorem 10.5.1. ||ord|| K(B] = I. Proof. It suffices to apply Theorem 7.2.16. The next result gives a lower bound for the radius of convexity of a L.I.F. in terms of its norm order [Pfa-Su4]: Theorem 10.5.2. Let F be a L.I.F. with ||ord|| T = a < oo and let f € T. Then f maps the ball Bi/^a) biholomorphically onto a convex domain. Proof. Let / e F and g(z) = /(z;a), z € B, a £ B \ {0}. In view of (10.4.3), we have D2g(0)(v,v) = -2(v,a)v +
^Ta[Df(a)}-1D2f(a)(D^a(0)v,D(pa(Q)v).
s
a
We will show that the sufficient condition for convexity in Theorem 6.3.4 holds if ||a|| < l/(2a). For this purpose, let w 6 Cn, ||w|| = 1, be such that Re (w,a) = 0. Also let v = [D(pa(Q)]~lw — (l/s%)Ta(w) in the above formula. Then v) = -^(Ta(w},a}Ta(w) + S
a
= ~(w,a)Ta(w) + s
a
-^Ta[Df(a)]-LD2f(a)(w,w)
S
a
\Ta[Df(a)]-lD2f(a)(w,w).
s
a
On the other hand, a straightforward computation yields that (D2g(Q)(v,v),a} = — j ( w , a ) ( T a ( w ) , a } + -= (C
10.5. Norm order and univalence
=^ {((Df(a)}-1D2f(a)(w,w),a) s a (.
435
+ s^\(w,a)\2} , a
)
where we have used the fact that Re (w, a) = 0, and thus (w,a)2 = — \(w,a)\2. Since ||ord|| T = a, it follows that ||£>20(0)(u,v)|| < 2a||v||2. Also since Ml2 =
we obtain in view of the above arguments that |{l - ([D/Ca)]-1^2/^,™),"}} - {l + j^Ka,' = s2a\(D2g(0)(v,v),a)\ < s2a\\D2g(0)(v,v)\\ • \\c s
a
Therefore, we deduce that l-Re([Df(a)]-lD2f(a)(w,w),a)
for ||a|| < l/(2a). Prom Theorem 6.3.4 we conclude that / maps the ball Z?i/(2a) biholomorphically onto a convex domain. This completes the proof. Pfaltzgraff and Suffridge [Pfa-Su4] formulated the following conjecture for the radius of convexity of a L.I.F. of given norm order. If this conjecture is true, it would be a generalization of Theorem 5.2.3. We note that in dimension n > 2, there is in general no such connection between the trace order of a L.I.F. on B and its radius of convexity. (Theorem 10.5.6 below gives a weaker result in the direction of the conjecture.) Conjecture 10.5.3. If F is a L.I.F. on the Euclidean unit ball B of C*, n > 2, with ||ord|| F = a < oo, then the radius of convexity of f is a—Vet2 — 1.
436
Linear invariance in several complex variables
A covering theorem for L.I.F.'s of given norm order may be deduced from Theorem 10.5.2 [Pfa-Su4]: Theorem 10.5.4. If F is a L.I.F. with ||ord|| JF = a < oo, then for each f G J- the image f ( B } contains the ball BI^^. Proof. Since / 6 JF, g(z) = 2af[ — ) is a normalized convex mapping of \2a/ the unit ball JB, by Theorem 10.5.2. In view of Theorem 7.2.2, g(B) contains the ball BI/%, and hence f ( B ) contains the ball .£?i/(4a). This completes the proof. A lower bound for the radius of starlikeness of a L.I.F. in terms of its norm order was also obtained by Pfaltzgraff and Suffridge [Pfa-Su4] (compare with Corollary 5.2.6): Theorem 10.5.5. If F is a L.I.F. with ||ord|| F = a < oo, then each f e F maps the ball -Bp(a)> where p(a) = 4a/(l + 4a2), biholomorphically onto a starlike domain. Proof. Let ZQ e B with ||zo|| < 4a/(l + 4a2) and choose a € B such that ZQ = 2a/(l + ||o||2). Then ||a|| < l/(2a). If / G 7 then the Koebe transform /(z;a) belongs to F too. Taking into account Theorem 10.5.2, one deduces that /(z; a) maps the ball -Si/(2a) biholomorphically onto a convex domain. Because ||a|| < l/(2a), one concludes that the mapping f ( z ; a ] — /(— a; a) is a starlike mapping on the ball ||^|| < l/(2o:). Therefore, using Theorem 6.2.2, one obtains that Re (lDf(z;a)}-(f(z;a)-f(-a;a)),z)
> 0,
||z||
Setting z = a in the above, one deduces that 0 < Re ( [ D f ( a ; a ) ] - l ( f ( a ; a )
- /(-a; a)), a)
Since ZQ is arbitrary, we conclude that / is starlike on the ball -Bp(a) p(a) = 4a/(l + 4a2). As a further application of the norm order, we show that it is possible to solve a radius problem involving quasi-convexity of type B. We recall (De-
10.5. Norm order and univalence
437
finition 6.3.20) that if / € £S(B), then / is quasi-convex of type B (or bquasiconvex) if Re (z + ( D f ( z ) ] - l D 2 f ( z ) ( z ,
z), z) > 0, z € B \ {0}.
The following result was obtained in [Pfa-Su4] (compare with Theorem 5.2.3): Theorem 10.5.6. // T is a L.I.F. with ||ord|| T = a < oo and if f € F, then f is b-quasiconvex on the ball Bp(a), where p(a) = a — \lo? — 1. Consequently, if \\ord\\ F = 1 then T is a subset of the b-quasiconvex maps of B. Proof. Let g(z) = f ( z ; a) be the Koebe transform of / with the automorphism ipa, a e B \ {0}, given by (10.1.2). Then D2g(Q)(v,v) = -2(v,a)v + s-^Ta a
by (10.4.3). Since ±D2g(0)(v,v}
we have (10.5.1)
\\-(v,a)v+^Ta
using the fact that Dy?a(0) = s^T"1, by (10.2.7). Let v = a/||o||. Since T~ 1(a) = Ta(a) = a, we deduce from (10.5.1) that
i-NI 2 , and hence Re ( T a ( D f ( a ) ] - l D 2 f ( a ) ( a , a ) , a )
- -*t > II
11
-f^} I I I
Using the fact that T£ (a) = a, the above relation is equivalent to '
'
—
]_ _
438
Linear invariance in several complex variables
and the last quantity is positive for ||a|| < a — ^ot2 — 1. This completes the proof. We close this section with another result due to Pfaltzgraff and SufTridge [Pfa-Su4] which relates the radius of univalence r\ of a L.I.F. of finite norm order with its radius of nonvanishing TQ (compare with Theorem 5.2.7 in the case n = 1). More precisely, let TQ denote the largest number such that no mapping in the L.I.F. F has zeros on Bro \ {0}. Also let r\ denote the largest number such that every mapping in the L.I.F. T is univalent on Bri. Then we have the following connection between TQ and r\: Theorem 10.5.7. If F is a L.I.F. of finite norm order, then r\ =
Proof. Let / e T, r < r0/(l + y^l - rg) and z', z" & Br be such that z' ^ z". Also let /(z; a) be the Koebe transform of / with (pa given by (10.1.3), i.e. f(z;a) = where
"~z'
in the above, we obtain that (10.5.2)
/
On the other hand, it is easy to verify the relation
(Also see Lemma 6.1.24.)
10.5. Norm order and univalence
439
If we choose a = — z' and z = z" in this equality, and use the fact that \z'\\ < r, ||;z"|| < r, we easily obtain
- \\TZ
(1-r 2 ) 2
and therefore 2r
^ - ro-
Combining this inequality with (10.5.2), we conclude that f ( z " ) ^ f ( z ' ) . Consequently / is univalent on Br, and hence ri > ro/(l + \/l — TQ). Conversely, let ZQ € B be such that 0 < poll < 2ri/(l + r%) and choose a € B such that z0 = 2a/(l + ||a||2). Then it is obvious that 0 < ||a|| < n, and short computations yield the relations
/(a; a) =
~ /(a))
and
/(-a; a) = Hence
a;a) - /(-a; a)). Since 0 < ||a|| < ri, we have /(a; a) 7^ /(—a;a), which implies that /(20) 7^ 0. Hence ro > 2ri/(l + r 2 ), or equivalently ri < ro/(l + y'l — r(j). This completes the proof.
Problems 10.5.1. Show that if f is a L.I.F. with ||ord|| T = a < oo, then T is a normal family. (PfaltzgrafT and Suffridge, 2000 [Pfa-Su4].) 10.5.2. Does the set of quasi-convex maps of type B of the Euclidean unit ball in C", n > 2, form a linear-invariant family?
440
10.6
Linear invariance in several complex variables
Linear-invariant families in complex Hilbert spaces
We end this chapter with a few ideas concerning the notion of linear invariance in complex Hilbert spaces. Recently Hamada and Kohr [Ham-Koh8] generalized the notion of norm order to the case of a L.I.F. T on the unit ball B of a complex Hilbert space X. As in finite dimensions, we have the following Definition 10.6.1. The family T is called a linear-invariant family if (i) F C CS(B] — {f : B —> X\ f is normalized and locally biholomorphic}, (ii) A e Aut(B). (The Koebe transform A^(/) of / is defined in a similar manner as in the finite dimensional case.) Definition 10.6.2. The norm order of the L.I.F. T is
An essential fact is that Hormander's result in Theorem 7.2.19 can be applied in the case of complex Hilbert spaces, and thus 2 = sup{i||D /(0)(ti;,ti;)||: / € T, Nl = l) . Iz J
Using this observation and the fact that the set Aut(B) of biholomorphic automorphisms of B has similar behaviour as in the finite dimensional case (see Section 6.1.5), Hamada and Kohr [Ham-Koh8] obtained the following generalizations of some of the results contained in the last two sections of this chapter. We leave the proofs for the reader. Theorem 10.6.3. Let T be a L.I.F. on a complex Hilbert space X. Then ||ord|| T>\. Also ||ord|| K(B) = 1. For the following result we need the definition of quasi-convexity of type B (or 6-quasiconvexity) in the case of complex Hilbert spaces. This definition is the same as in the case X = C71. Then we have Theorem 10.6.4. Let T be a L.I.F. on a complex Hilbert space X with ||ord|| F — a < oo. // / 6 F then f is b-quasiconvex on the ball -Br(a), where r(a) = a — \/a2 — 1.
10.6. L.I.F.'s in complex Hilbert spaces
441
Theorem 10.6.5. Let T C S(B) be a L.I.F. on a complex Hilbert space X with ||ord|| F = a < oo, and f e F. Assume that for each r € (0,1) there exists M = M(r) > 0 such that \\(Df(z}]-\f(z}
- /(u))|| < M(r), z,u 6 BP.
T/ien / maps the ball -Bi/(2a) biholomorphically onto a convex domain in X. Moreover, f maps the ball -Bp(a) with p(a) = 4o:/(l + 4a2) biholomorphically onto a starlike domain in X. In addition, certain results involving the norm order of a L.I.F. on the unit polydisc of C71 have recently been obtained by Hamada and Kohr [HamKohl3].
Problems 10.6.1. Prove Theorem 10.6.3. 10.6.2. Prove Theorem 10.6.4. 10.6.3. Prove Theorem 10.6.5.
Chapter 11
Univalent mappings and the Roper-Suffridge extension operator The aim of this chapter is to study an operator, introduced by Roper and Suffridge, that provides a way of extending a (locally) univalent function / G H(U) to a (locally) biholomorphic mapping F on the Euclidean unit ball B of C1. This operator has the remarkable property that if / is a convex function on U then F is a convex mapping on B. We shall investigate its behaviour on other classes of functions as well. In particular, if / is starlike on U then F is starlike on B, and if / is a univalent Bloch function on U then F is a Bloch mapping on B. Growth and covering theorems for the extended mappings may also be obtained, including covering theorems of Bloch type. There are some interesting connections with the theory of Loewner chains, which we shall study in the context of a generalization of the Roper-Suffridge extension operator, denoted by $n,a> a € [0,1/2]. We shall prove that every mapping Fa e $n,a(S) may be embedded in a Loewner chain, and in fact has parametric representation. We shall also consider radius problems associated with the operator $„,<*• Related conjectures and open problems will be discussed. Finally we study the concept of linear-invariant families as it relates to families generated by the operator $n,a» and we obtain generalizations of 443
444
The Roper-Suffridge extension operator
recent results of several mathematicians concerning the order of the families so generated.
11.1
Convex, starlike and Bloch mappings and the Roper-Suffridge extension operator
Let C1 be the Euclidean space of n-complex variables with the usual inner product (•, •} and the induced norm || • ||. As usual, let B be the Euclidean unit ball of C". Let z' — (z%, . . . , zn] so that z = (zi, z'). The Roper-Suffridge extension operator is defined for normalized locally univalent functions on the unit disc U by (11.1.1) We choose the branch of the square root such that \//'(0) = 1. Note that if / € S then $ n (/) ££(£). This operator was introduced in [Rop-Sul] as a means of constructing a convex mapping on the Euclidean unit ball in C™ given an arbitrary convex function on the unit disc. It is surprisingly difficult to do this. If /i , . . . , / „ are normalized starlike functions on the unit disc U then it is very easy to show that F(z] = (/i (21), . . . , /„(*„)),
z = (zi, . . . , zn) € B,
is a normalized starlike mapping on B (see also Problem 6.2.5). However, if /i > • • • > fn are convex functions then F need not be convex on the Euclidean unit ball B of C1, n > 2. The following well known example illustrates this phenomenon (see also Problem 6.3.3): Example 11.1.1. The mapping -ZI
is not convex on the Euclidean unit ball B of C™, n > 2, even though /(C) = — £) is a convex function on U.
11.1. Convex, starlike and Bloch mappings
445
Proof. The following proof is due to Gong, Wang and Yu [Gon-Wa-Yu3]: Suppose F is convex on B. Then obviously
for all r e (0, 1), where ek is the fc
k—l
However, a short computation yields that
r 1 and if r is sufficiently close to 1, then ---r > —7=, which implies that
r + n(l — r)
yn
This is a contradiction. As observed by Roper and Suffridge [Rop-Sul], similar arguments show that there is no convex mapping of the form (11.1.2)
F(s)
In fact, prior to the work of Roper and Suffridge, there was no general way of constructing convex mappings of B out of convex functions on U. Subsequently their extension operator was used in [Gra-Var2] to show that certain constants in covering theorems for convex mappings are sharp (see Theorem 7.2.9), and by Pfaltzgraff [Pfa5] in studying linear-invariant families (see Sections 10.1 and 10.2). We shall present a simplified proof of the theorem of Roper and Suffridge which is due to Graham and Kohr [Gra-Kohl]. Another interesting and quite different proof has recently been given by Gong and Liu [Gon-Liu3].
446
The Roper-Suffridge extension operator
Theorem 11.1.2. Let f € K and let F : B -» C" be defined by (11.1.1). Then F is convex. Proof. We shall give the proof when n = 2; this case contains all the essential features of the general case. It is clear that F is locally biholomorphic on B, since JF(Z) = (/'(zi))3//2 ^ 0 for z = (21,22) € B. Therefore, using the criterion for convexity given in Theorem 6.3.4, we have to prove that 1 - Re ([DF(z)]-lD2F(z)(u,u),z)
> 0,
for all z — (21, 22) € B and u — (ui,uz) € C2 with \\u\\ = 1 and Re (z,u) = 0. We may assume that z = (zi, 22) & B, z% ^ 0, because the case z = (21, 0) is easily handled. Indeed, if z = (21,0), (21 1 < 1 and u = (111,112) € C2 are such that \\u\\ = 1 and Re (2, u) = 0, a short computation yields 1 - Re ([DF(z)]-iD2F(z)(u,u),z)
= 1 - Re
Here we have used the fact that ziu\ + ~z\u\ = 0 and / € K. It is obvious that F is holomorphic at all points z = (21, 22) 6 B such that 22^0.
Now we can write z = (21,22) € B \ {0}, 22 7^ 0, and u as 2 = aV and u = (a/|a|)W respectively, where a G C, 0 < |a| < 1, ||F|| = ||W|| = 1, V = (Vi, V2), F2 ^ 0, and Re (V, W} = 0. Hence we obtain 1 - Re
([DF(z)}'lD2F(z)(u,u),z)
= l-Re The expression 1 - Re {a([DF(aV)]-lD2F(aV)(W, W}, V)} is the real part of an analytic function of a and thus is harmonic. Applying the minimum principle for harmonic functions, this function can only attain its minimum when | a | = 1. Therefore it suffices to prove that (11.1.3)
1 - Re ([DF(z}}-lD2F(z)(u, u),z}> 0,
11.1. Convex, starlike and Bloch mappings
447
for all z = (zi,z2) € C2, |zi|2 + |z2|2 = I , z ? (zi,0), u = («i,u 2 ) € C2, |wi|2 + |u2|2 = 1 and Re (z,u) = 0. After short computations we deduce that the relation (11.1.3) is equivalent to (11.1.4) 1 > Re for all zi,z2 € C, \zi\2 + \z2\2 = 1, z2 ^ 0, wi,w 2 € C, |wi|2 + |w2|2 = 1 and Re {uiz\ + u2z2} = 0, where {/;zi} denotes the Schwarzian derivative of / at z\. We consider the following three cases: (i) ui = 0, \u2\ = 1. (ii)t«2 = 0, |ui| = l. (iii) 0 < |ui| < 1, 0 < |u2| < 1, K|2 + |w2|2 = 1. In fact the first case is obvious, so we need to consider only the second and third possibilities. (ii) Assume u2 = 0, |ui| = 1. In this case the relation (11.1.4) becomes
for zi, z2 € C, z2 ^ 0, |zi|2 + \z2\2 = 1, wi 6 C, |wi| = 1 and Re {wi^i} = 0, i.e. uizi + u\z\ = 0. Therefore we have to show that (11.1.0)
1 > -Be
+
Re {«?{/;,,}},
for z\ e C, \z\\ < 1 and wi € C, |ui| = 1, Re {wi^i} = 0. We need to make use of an estimate for the Schwarzian derivative of a convex function which we considered in Chapter 2 (see (2.2.16) and the preceding remarks): if / € K then (11.1.6) (l-|*i| 2 ) 2 K/;2i}l<2U-
f'M
Then we obtain r *, f"(v+\}
i _ i*, i2
r _
>.
-l
The Roper-Suffridge extension operator
448
i-N 2
Re
f'W
Thus the assertion (11.1.5) is proved. (iii) Assume that 0 < |wi| < 1, 0 < \U2\ < 1 and \ui\2 + \U2\2 = 1. In fact we can suppose that 0 < u\ < 1, 0 < u^ < 1. For if the statement (11.1.4) fails for some choice of /, z, and u, then choose 0\ and #2 such that u\el&l > 0, W2C102 > 0, and consider the rotation of / given by /-^(zi) = etfflf(e~l&lzi). Then since =e and
we see that (11.1.4) also fails for /_ fll , Hence we have to prove that
, and
for all wi,u 2 and
(0,1), u\ + w2, = 1 and zi,z2 € C, z2 ^ 0, \zif
Let 2j = 1, we have
for j — 1,2. Then nixi+u 2 x 2 = 0, and since ,2^2 J.
and hence (11.1.7)
o
-
x
o
n
l ~ Vl = ul
449
11.1. Convex, starlike and Bloch mappings
Also we can suppose that u\y\ + 1123/2 > 0, since the other case can be treated in a similar manner. Using these remarks, we need to show that
.f'M\
-----
2
Taking into account the relations (11.1.6) and (11.1.7), we obtain
f'M + w2y2)Im
2 l-\zi\2\f"(zi) f"W
Zl-\ _
~7^)- S
Zl
1-
f'M 2^
1-IZTI2
i-M*f»M 2 f'M
,
< 1.
Here we have used the fact that the last inequality is equivalent to + ti2lfe)
I2
>0.
This completes the proof. It seems to be difficult to perturb either the Roper-Suffridge operator or the domain without losing the convexity-preserving property. We shall give two illustrations of this. One is an example of Graham, Hamada, Kohr, and Suffridge [Gra-Ham-Koh-Su], and the other is a theorem of Roper and Suffridge [Rop-Sul]. Example 11.1.3. Let n = 2 and let F : B C C2 -> C2 be given by
where g is a non- vanishing analytic function on U such that 0(0) = 1. We show that the mapping F is only convex for g(zi) = 1.
450
The Roper-Suffridge extension operator
Proof. Observe that the line L = {(it,Q) : t 6 R} C F(B). Since F(B) is convex, an elementary argument shows that for every W e F(B), the line W + LcF(B}. Let zi , ^25(21) u—-- and v = ———-. I- zi 1-^1 Then Re u > -1/2 and -
2
l + 2Reu
so the mapping F is of the form . (u, pelVl + 2Rew/i(n)),
p < 1, h(u) =
u
It can be seen from the nature of the mapping F and the remark at the beginning of the proof that
(11.1.8)
(u, v) e F(B) & (u, M) 6 F(B) &(u + it, |u|) e F(B),
for all t e R For each u, let
M(u) = sup{|v| : (u,u) e F(B)} = >/l + 2Re u\h(u)\. Then (11.1.8) implies that M(u) is independent of Im u, so that |/i(u)| is constant on the lines Re u = constant > —1/2. By the Schwarz reflection principle, we may reflect the function h with the domain restricted to the right half-plane across the unit circle (it is the unit circle since /i(0) = 1) by h(—u) = l/h(u). This extended function is entire because h(u) ^ 0. Write u = a + ir and observe that h(u) = Re1^ where R is independent of r. Using the Cauchy-Riemann equations, it is easy to see that 0 is independent of a and in fact h(u) = eau for some real a. Using the convexity of the mapping, it follows that the set (u, v) such that u > —1/2 and 0 < v < \/l + 1ueau — k(u) is convex. By elementary calculus, since k"(u) > 0 for large u when a 7^ 0, the above set cannot be convex unless a = 0. This completes the proof.
11.1. Convex, starlike and Bloch mappings
451
Theorem 11.1.4. Let f e K, let B(p) be the unit ball in C2 with respect to the p-norm, 1 < p < oo, and let F : B(p) —>• C2 be given by 6 B(p).
/
Then F is convex if and only ifp = 2. Proof. The case p = oo follows from Theorem 6.3.2. For the case 1 < p < oo we will use a similar argument as in [Rop-Sul]. For this purpose, let /(C) = ~ log I ], C € U. Then / is convex on U. 2 \1 ~ C/ Also let 1 /1 + z\ \ and z* u = o loS 1 v= If F(B(p)) is convex then so is its intersection with the plane Im u = 0, Im v = 0. This intersection contains the entire real u-axis and precisely the interval (—1,1) of the real v-axis. In order to show that convexity is not satisfied, it suffices to show that if z\ —>• 1 along the real axis then we are constrained to have \v\ —> 0 if p < 2, while it is possible to have \v\ —t oo if p > 2. It suffices to consider the images of the points ((1 — ep)1/p, ±e) for small e > 0. We have
M2 = /
2 £
V
p
2
P
£
*
1 - (1 - £P)2/P
2 /2
\
\
P\P
/
/
--1+M-\ V P
2 (1 + (1 - -} ep] + 0(e*P}
\
\
P/
J
Now it is easy to see that if p = 2 then \v\2 = 1, if p < 2 then |v|2 -> 0 as e —> 0, and if p > 2 then |i;| 2 —^ooase—)-0. This completes the proof. Theorem 11.1.2 can be extended to the case of complex Hilbert spaces without any essential new difficulties (see [Rop-Sul]). Let X be a complex
452
The Roper-Suffridge extension operator
Hilbert space and let B denote the unit ball of X. In view of Theorem 6.3.8 (ii), we obtain Theorem 11.1.5. Let f € K and u <E X with \\u\\ = 1. Let Fu : B ->• X be given by (11.1.9)
Fu(z] = f((z,u))u + ^f'((z,u})(z
- (z,«)«),
z e B.
Then Fu is convex. Proof. It is not hard to show that Fu is biholomorphic. Moreover, using the convexity of / and Theorem 2.2.8, we deduce that for each r e (0,1) there exists M = M(r) > 0 such that \\[DFu(z)]-l(Fu(z)
- F»)|| < M(r),
||z|| < r,
||v|| < r.
Finally, similar reasoning as in Theorem 11.1.2 yields that \\v\\2 - Re ([DFu(z)]-lD2Fu(z)(v,«),
z) > 0,
for z e B and v e X \ {0}, Re (z,v} = 0. Hence Theorem 6.3.8 (ii) implies that Fu is convex. This completes the proof. We now turn to other properties of the Roper-Suffridge extension operator. First we show that the operator
11.1. Convex, starlike and Bloch mappings
453
After short computations, we obtain
. Hence we need to show that
f(z ^ for \zi\ < 1. Now, the function p defined by p(z\) = —jj-—-, z\ 6 £/, satisfies p € H(U), p(0) = 1 and Re p(z\) > 0, i.e. p e P. Therefore from the Herglotz formula (2.1.3), we have
P(ZI)= r\ Jo
i
z\e
where /z is a non-decreasing function on [0,2ir] with /i(2?r) — /u(0) = 1. Also, since
>"<^- = i-vwit follows from (11.1.10) that we have to prove the relation (11.1.11) (1 + |zi|2)Re p(zi) + 1 - \zi\2 + (1 - \z} for z\ € U. Using the fact that rlit
ZIP'(ZI) = 2 / Jo
z e-i6
TQ (1 - z\e~™
it suffices to observe that the relation (11.1.11) is equivalent to (11.1.12)
\zi\ zie d^fQ\ > 0? (I — z\e ™y
Re / Jo
Zie
u.
Since /z is non-decreasing on [0,2?r] and Re
1-HV ' ' 2 > 0,
, , |w| < 1,
the inequality (11.1.12) follows. This completes the proof.
454
The Roper-Suffridge extension operator
In Theorem 11.1.4 we have seen that the operator <&n, regarded as an extension operator from U to B(p), does not preserve convexity if p ^ 2. It is natural to see what happens in the case of starlikeness. In this situation, we can prove that Theorem 11.1.6 does not hold in the unit poly disc P of C2 [Gra-Kohl]. Remark 11.1.7. Let / e S* and let F : P C C2 ->• C2 be given by F(zi,Z2) = i f ( z i ) , Z 2 \J j'(z\ ) J . Then F need not be starlike on P. Proof. Since
F will be starlike if and only if the following relations are satisfied on the unit disc (see Theorem 6.2.3):
The first inequality is obvious since / € S*. However, the second is not always true. To see this, it suffices to consider f ( z \ ) — --r~, \z\\ < 1. (1 - z\Y Then / G 5*, but (ii) is equivalent to Re --r~ > 0, \z\\ < 1. Since this (1 + ziY relation is not satisfied everywhere in the unit disc, F is not starlike on P. As in the case of convex maps, Theorem 11.1.6 may be extended to complex Hilbert spaces (see [Gra-Kohl]). To do so, it suffices to apply Theorem 6.2.6 and similar reasoning as in the proof of Theorem 11.1.6. Let X be a complex Hilbert space and let B denote the unit ball of X. We have Theorem 11.1.8. Let f € S* and u € X, \\u\\ - 1. Define Fu : B ->• X by (11.1.9). Then Fu is starlike. Since we have proved that $n(S*) C S*(B) and $n(K) C K(B), when B is the Euclidean unit ball of Cn , it would be interesting to give an answer to the following: Conjecture 11.1.9. If / : U —> C is a normalized close-to-convex function with respect to a normalized starlike function g on U (i.e. Re 1—T-T- J > 0, 9(Q C 6 U), then $ n ( f ) is close-to-starlike with respect to $n(g) on the Euclidean unit ball of C1, n > 2.
455
11.1. Convex, starlike and Bloch mappings
Finally we consider the extension of Bloch functions using the RoperSuffridge operator. Let B denote the set of Bloch functions on C7, and let BI denote the subset of B consisting of functions which are normalized and satisfy (11.1.13) (See Chapter 4.) We have the following result obtained in [Gra-Kohl]: Theorem 11.1.10. /// €. S C\ BI then F = &n(f) is a Bloch mapping. Proof. It suffices to prove that the quantity m(F) given by
is finite (see Remark 9.1.4). For this purpose, we shall again restrict our attention to the case n = 2. Short computations yield that DF(,)u =
«,/•(„),
for all z = (zi, z?) € B and u = (ui, 112) G C2. Since / € 5, it follows that 2
-kl
, < 2, zi e u.
f'M f'M
Hence we obtain + U-2
f'M for all z = (zi,22) G B and u G C2, ||u|| = 1. From (11.1.13) we have 1
\fM\
+ 1 + f'M f'M
456
The Roper-Suffridge extension operator
and hence < (1 - N2)2!/'(*i)|2 + (1 - l*i| 2 )|/'(*i)l(6M + 9) < 16, for all z € B. Therefore m(F] < 4 and we conclude that F is a Bloch mapping, as claimed. This completes the proof. Problems
11.1.1. Show that there is no convex mapping of the form (11.1.2) on the Euclidean unit ball of C2 .
11.2
Growth and covering theorems associated with the Roper-Suffridge extension operator
In this section we shall prove that if / belongs to a class of univalent functions in the unit disc which satisfy a growth theorem and a distortion theorem, then the extension F = 3>n(/) of / satisfies a growth theorem and consequently a covering theorem on the Euclidean unit ball of Cn. We shall also prove a Bloch-type covering theorem for convex mappings obtained using the operator $n. For this purpose, we need to extend the definition of the Roper-Suffridge operator slightly. If / is not normalized, so that there is not a canonical choice of the branch of the square root \X/'(zi), there are two possible ways to define $ n (/). Both of them have the same range, however, and their growth properties are identical. Without specifying a rule for choosing between them, we shall sometimes write 3> n (/) m discussing results which are valid for both choices of the branch of the square root. Other properties of 3>n which are easily verified include the following: • If f ( z \ ) — z\/(l—zi) thenF = $ n (/) is the normalized Cayley transform, i.e. F(z] = z/(l - zi) for z = (z\, . . . , zn) € B. • If f is an automorphism of U then F = 3>n(/) (i-e. either choice of F) is an automorphism of B.
11.2. Growth and covering theorems
457
• If f is odd or more generally k-fold symmetric, then so is F = $„(/). Another property of the Roper-Suffridge operator which shows that this operator is to some extent canonical among possible extension operators is the following [Gra6]: Lemma 11.2.1. If f : U —>• C is a locally univalent function and T is an automorphism ofU then $ n (/) and $n(/ oT) have the same range. Remark 11.2.2. For specific / and T, we may choose the branches of the square roots such that $n(/ o T) = $ n (/) ° $n(T). However it is not possible to choose the branch of the square root in a systematic way for all / so that this formula is always valid. Proof of Lemma 11.2.1. Using the definition of $n, we have (11.2.1)
*„(/ o T)(z) = ((/ o T)(zi), >/(/oT)'(ziy)
Whether the correct choice of sign (for a particular / and T) is plus or minus is of no consequence, for in either case f T(ZI), ±^/T'(zi)z' j gives an automorphism of B. Hence the right-hand side of (11.2.1) has the form (/(Ci)> \//'(Ci)C') f°r some C € B, and £ = (Ci,CO ranges over B as z does. We next show that, under conditions which are quite commonly encountered in practice, if f is a subset of S whose members satisfy a growth theorem and a distortion theorem, then the mappings in $n(-^) satisfy a similar growth theorem. As we have seen in Section 6.1.7, this is in contrast to the behaviour of the full class S(B), and additional assumptions are needed to obtain positive results (see Chapter 7). The following result was obtained by Graham [Gra6]: Theorem 11.2.3. Suppose T is a subset of S such that all f e F satisfy
(n.2.2) (11.2.3) where (11.2.4) (11.2.5)
(p^ are twice differentiate (f>(r) < r,
0,
on [0,1),
[0, 1)
458
The Roper-Suffridge extension operator
(11.2.6)
^(r)>r,
il>'(r}>Q,
Then all mappings F e &n(F) (11.2.7)
ip"(r)>Q
on [0,1).
satisfy
ztB.
Furthermore if for some f G T the lower (respectively upper) bound in (11.2.2) is sharp at z\ £ U', then the lower (respectively upper) bound in (11.2.7) is sharp for $„(/) at (21,0, ... ,0). For the proof of this result we need the following lemma [Gra6] : Lemma 11.2.4. Suppose (p and if) are functions which satisfy the conditions (11.2.4)-(11.2.6). Then for fixed r e [0,1), the minimum of (y?(t)) 2 + (r2 — £ 2 )y/(£) for t € [0, r] occurs when t — r; the maximum of (-0(i))2 + (r2 — £ 2 )t//(t) for t € [0, r] occurs when t = r. Proof. It is easy to check the sign of the first derivative in [0,r]. Proof of Theorem 11.2.3. Let ||z|| = r < 1. Taking into account the result of Lemma 11.2.4, it is not difficult to obtain the sharp upper and lower bounds for
As particular cases of Theorem 11.2.3, we obtain the following growth results [Gra6]: Corollary 11.2.5. /// e 5 and r € [0, 1) then
(1 + r)2 - " // / € S is k-fold symmetric then
r (1 _|_ r fc)2/fc ~ II
// / e K then
IffeK
and f"(Q) = 0 , . . . , /W(0) = 0 t/ien =r.
11.2. Growth and covering theorems
459
SHBi then (11.2.9)
1 l-exp^
< ||*B(/)(*)|| < 5
for \\z\\ = r. Remark 11.2.6. The bounds in (11.2.8) are clear because we have already proved that &n(K) C K(B). All of the above estimates are sharp except for the lower estimate in (11.2.9). (We mention that the lower estimate in (11.2.9) is not sharp in one variable for functions in SC\Bi. The corresponding distortion estimate |/'(C)| >
fi -/ Jo 'o
dt i
(vi) IffeSnBi then $„(/)(£) 5 Bl/2. All results are sharp except for the last. Finally we obtain covering theorems of Bloch type for the families $n(S) and $n(K). These results were obtained by Graham [Gra6]. Theorem 11.2.9. (i) The image of every F e $n(K) contains a ball of radius n/4. This result is sharp.
460
The Roper-Suffridge extension operator
(ii) The image of every F € $n(S) contains a ball of radius 1/2. Proof. These results may be obtained from Corollary 11.2.8 (iv) and (vi) respectively, if we show that, in addition to the stated assumptions, it is possible to assume that / € Sr\Bi (which implies /"(O) = 0 (see Remark 4.2.3)). To do so we apply Landau's reduction in the z\ variable and use Lemma 11.2.1. For this purpose, suppose / € 5 and / is not in B\. By dilating we may assume that / is holomorphic on U, and hence (1 — |^i|2)|/'(^i)| has an interior maximum on U, say at a. Since / £ B\, it follows that a ^ 0. Let T be a disc automorphism such that T(0) = a and let (/ o T)'(O) = a. Since a ^ 0, we must have a > 1. Let
a and consider the mapping $ n (pr(/)) given by
/ ° T)(*) - 6), where A = diag(a~1, or1/2, . . . , a"1/2), the branch of aT1/2 is determined by the choice of branch of >/(/ o T)'(ZI) so that we have
6 = ((/oT)(0),0, . . . ,0), and the action of .4 on <J> n (/oT)(z) — 6 is determined by writing the latter as a column vector. From this we see that if the image of contains the ball (11.2.10)
\wi - ci|2 + \\w' - c'||2 < p2,
then the image of 4> n (/ ° T) or of $ n (/) contains the ellipsoid M~ 2 N - ci - (/ o T)(0)|2 + lal^Hti;' - c'||2 < p2. This ellipsoid contains the ball \wi - ci - (/ o T)(0)|2 + |K - c'||2 < p2|a|, which is larger than the ball (11.2.10). Further, note that gr(f) is convex whenever / is, so the same is true for $ n (<7T(/))- Thus given any / € S (resp.
11.3. Loewner chains and the operator $n,a
461
.fiT), we can produce a function g € 5 n B\ (resp. g € K C\ #1), such that the largest ball in $n(g)(B) has smaller radius than that of the largest ball in 3>n (/)(£). Consequently, we may indeed reduce to the case / 6 S D B\ and apply the statements (iv) and (vi) from Corollary 11.2.8. The sharpness of part (i) follows by considering the mapping $ n (/) 5 where Izil < 1Remark 11.2.10. In Chapter 9 (and originally in [Gra-Var2]), the question of whether the image of any convex mapping of B contains a ball of radius 7T/4 was considered, and an affirmative answer was obtained for the case of odd mappings. We have just proved that the same result holds for convex mappings in the family $n(K). However, the general case of the problem remains open.
11.3
Loewner chains and the operator 3>n,a
In this section we study the embeddability in Loewner chains of mappings constructed using the Roper-Suffridge extension operator. In fact we shall consider simultaneously the family of extension operators given by (11.3.1) *„,«(/)(*) = Fa(z) = (fM, (/'(2i))V),
* = (*i, z') e B,
where a € [0,1/2] and / is a normalized locally univalent function on U. We choose the branch of the power function such that (f(zi))a\zi=o = 1. Of course when a = 1/2 we obtain the Roper-Suffridge extension operator 3>n. We remark that all of the 3>n,a fall into the general class of operators of the form $g(f)(z) = (f(zi), z'g(zi)), where / is a normalized locally univalent function on [7, and g is a non-vanishing holomorphic function on U such that (jf(0) = 1. If / is univalent on U and g is analytic and non-zero on U, then the mapping $ g ( f ) is univalent on B. Among such operators, the Roper-Suffridge operator has some canonical properties as we have seen, but it is of interest to determine to what extent operators of the more general type have useful properties. At a minimum, some connection between the functions / and g
462
The Roper-Suffridge extension operator
appears to be necessary if we wish the extended mapping to satisfy some geometric condition. We shall obtain a number of extension results for the operators $„,«, a € [0, 1/2], on the unit ball of Cn with the Euclidean structure. Our main result is that if / € S then 3>n,a(/) can be embedded in a Loewner chain, and moreover $n,a(/) 6 SQ(B), where S°(B) consists of those normalized univalent mappings on B which have parametric representation. In particular, our proof shows that if / <E S* then $n,a(/) € S*(B). On the other hand, Example 11.1.3 implies that convexity is preserved only when a = 1/2. Thus the dependence of extension operators from S to S(B) on parameters appears to be an interesting subject. We begin with the main result of this section, obtained recently by Graham, Kohr and Kohr [Gra-Koh-Kohl]. We give a different proof from that originally obtained in [Gra-Koh-Kohl] . This proof is based on the application of Theorem 8.1.5 instead of Theorem 8.1.6, and gives the conclusion that Fa has parametric representation on B. Theorem 11.3.1. Suppose that f e S and a € [0,1/2]. Then Fa = Proof. It suffices to give the proof in the case n = 2. Since / € 5, there exists a Loewner chain /(zi,£) = e*zi+ 02(^)2^ + . . . such that f ( z i ) = /(zi,0), zi € U (see Theorem 3.1.8). Let v = v(zi,s,t) be the transition function associated to the Loewner chain f(z\,i). Then v'(0,s,t) = es~*, 0 < s < t < oo, by the normalization of /(zi,t), and f ( z \ , s ) = f(v(zi,s,t),t) for z\ € U and 0 < s < t < oo. In view of Theorem 3.1.12, there exists a function p(z\, t) such that p(-, t) 6 P for each t > 0, p(z\,t) is measurable in t G [0,oo) for each z\ £ t/, and for almost all t > 0, ;
Vzi € U.
Moreover, v(t) = v(zi, s, i) satisfies the initial value problem (11.3.2)
dv — = -vp(v,t],
a.e. t > s,
v(s) = z\,
11.3. Loewner chains and the operator $n,a
463
for all s > 0 and z\ € Z7, and
(11.3.3)
lim e*v(zi,s,*) = /(zi,s)
locally uniformly on U. Now, let h = h(z, t) : B x [0, oo) ->• C" be given by
h(z, t) = (zip(zi,t), Z2 \1 - a + ap(zi, t) + azip'(zi, i)J J for z = (zi, z2) € B and t > 0. Clearly h(-,t) € H(B), h(Q,t) = 0, Dh(Q,t) = /, and we shall show that h(-,t) & M. Indeed, we have
Re (h(z,t),z) = |zi|2Rep(*i,*) + (1 -a)|z2|2
(11.3.4)
+a|z2|2Re p(zi,t) + a|z2|2Re [zip'(zi,t)],
z £ B, t > 0.
We may assume that z = (zi, z2), z2 ^ 0, because it is obvious that (11.3.4) is positive when z2 = 0 and z\ 6 U \ {0}. Applying the minimum principle for harmonic functions, it suffices to show that Re (h(z,t),z) > 0, z = (zi,z 2 ) € C2, |zi|2 + |z2|2 = 1, z ^ (zi,0), t > 0. Since p(Q,t) = 1 and Re p(zi,t) > 0, zi € C/, t > 0, the estimate (2.1.6) gives , •I-
,
i
|2 Rep(zi,t),
zi€Z7,
t>0.
Consequently, Re
and using (11.3.4) and the fact that a € [0, 1/2], we deduce that Re (/i(z,t),z) > (1 - a)(l - |zi|2) + Re p(zlt t)[(l - a)|zi|2 - 2a\Zl\ + a] > 0 for z = (zi, z2) € C2, |zi|2 + |z2|2 = 1, z2 ^ 0, and t > 0. Thus /i(-, t) G >1 for t > 0, and hence h(-, t) satisfies the assumption (i) of Theorem 8.1.3. Obviously the measurability condition (ii) in Theorem 8.1.3 is also satisfied.
464
The Roper-Suffridge extension operator Next, for z — (z\, z^} G B and 5 > 0, let V(t) = V(z, s, £), t > s, be given
by (11.3.5)
V(t) = V(z,s,t) = u(zi,a,t),« 2 e (1 - a)( - t) (t; l («i,a,t)
The branch of the power function is chosen such that (v'(zi,s,i))a\zl=Q = ea(a-t) for each t e [S,OQ). Then F(-,s,t) e #(£), F(0,s,£) = 0 and Prom (11.3.3), (11.3.5) and Weierstrass' theorem we deduce that the limit Fa(z,8) = lira e*V(z,s,t) t—>oo
exists locally uniformly on B for each s > 0, and (11.3.6)
Fa(z,8) =
for z = (21,2:2) G 5 and s > 0. The branch of the power function
(f'(zi,s))a
is chosen such that (/'(2i,s)) a | 2l= o —easMoreover, from (11.3.2) we obtain for almost all t > s,
dtdzi = -v'(zi, 5, t)p(v(«i, s, t), t) - v(zi, s, f)p' (v(zi, s, t), t)v'(zi,s, t). Here we have made use of the fact that v(zi,s,t) is a Lipschitz continuous function of t locally uniformly with respect to z\ £ U, and Vitali's theorem, to conclude that the order of differentiation can be changed. Then after straightforward computations, we obtain for almost all t > s,
1 - a + apK z i> s, t), t) + av(zi, s, t)p'(v(zi, s, t), t) = - (vi(z,s,t)p(Vi(z, s,t),t),V 2 (2,s,t)x
11.3. Loewner chains and the operator $n,a
465
Therefore V(i) = V(z, s, t) is the solution of the initial value problem dV — (z, a, t) = -h(V(z, s, t), t),
a.e. t>s, V(s) - z.
Since h satisfies the assumptions (i) and (ii) of Theorem 8.1.3, we conclude from Theorem 8.1.5 that the mapping Fa(z, t) given by (11.3.6) is a Loewner chain. The conditions of Definition 8.3.2 (for g(Q = (1 + £)/(! - C), ICI < 1) are therefore satisfied, and we have Fa(z) = Fa(z,Q) € S°(B), as claimed. This completes the proof. In the previous section we have seen that the Roper-Suffridge extension operator $n has the following properties: $n(K) C K(B) and $n(S*) C S*(B). We have now added another one: $n(S) ^ S°(B). As consequences of Theorem 11.3.1, we can also prove that the operator $n,a> & G [0,1/2], preserves starlikeness and spirallikeness of type 7 with J7| < 7T/2. In particular, for a = 1/2 we obtain another proof that the RoperSuffridge operator $n preserves starlikeness. The following results were obtained in [Gra-Koh-Kohl]: Corollary 11.3.2. Let f € 5* and a € [0,1/2]. Then Fa = $„,«,(/) 6 S*(B). Proof. Since / € 5*, f(z^t) — etf(z\] is a Loewner chain. Let Fa(z,t) be given by (11.3.6). Theorem 11.3.1 implies that Fa(z, t) is a Loewner chain, and since Fa(z,t) = etFa(z) we deduce from Corollary 8.2.3 that Fa is starlike. This completes the proof. Corollary 11.3.3. Assume f is a normalized spirallike function of type 7, where 7 € K, |7J < Tr/2, and let Fa = $„,«(/), with a e [0,1/2]. Then Fa is a spirallike mapping of type 7. Proof. Since / is spirallike of type 7, f ( z i , t } = e^1~ta^tf(eiatzi) is a Loewner chain, where a = tan 7 (see Corollary 3.2.9). A short computation using (11.3.6) yields that Fa(z,t) = e^-ia^Fa(eiatz). From Theorem 11.3.1 one concludes that Fa(z,t) is a Loewner chain, and Theorem 8.2.1 therefore yields that Fa is spirallike of type 7. This completes the proof. As we have already noted, Example 11.1.3 shows that the only case in which $n,a maps K into K(B) is the case a = 1/2. Another example which leads to the same conclusion is the following. We leave the proof for the reader.
466
The Roper-Suffridge extension operator Example 11.3.4. Let n — 2 and
Then / is convex, but $n,a(/) is not convex, for a £ [0,1/2). Remark 11.3.5. It seems the value a = 1/2 plays a unique role in preserving convexity under the operator n;Q on the domain
fc=2
when a G (0,1], and ftn>i/a is the unit polydisc of Cn when a = 0, then $n,a preserves both starlikeness and convexity on ftU)i/a for a G [0,1]. Their method of proof is completely different from that which we have used in the case of the unit ball B and is of interest for the reader.
11.4
Radius problems and the operator &n^a
In this section we continue the study of the operator 3>n,a with a G [0,1/2], and we obtain the radius of starlikeness of $n,a(5r) and the radius of convexity of $n(S) (i.e. of
11.4. Radius problems
467
univalent on jBr, then / must also be univalent on Ur. Also if 3>n,a(/) is starlike on BT (resp. convex on Br) then / will likewise be starlike on Ur (resp. convex on Ur). The following result is due to Graham, Kohr and Kohr [Gra-Koh-Kohl]. Theorem 11.4.2. r*($n>0(5)) = tanh ^, for all a € [0, 1/2]. Proof. Using Corollary 3.2.3, we have r*(S) = tanh . Let / € S. Then
and this quantity can be negative if \z\\ > tanh — . Now let Fa = $n,a(/). Taking into account Corollary 11.3.2 and Remark 7T 11.4.1, we deduce that Fa is starlike on Br with r = tanh — , and further that 7T
Fa may fail to be starlike in any ball Bri with r\> r. Therefore r = tanh — is the largest radius for which each Fa 6 3>n,a(S) is starlike on Br. This completes the proof. Since *„,«(£) C S°(B) for a <E [0,1/2], we must have r*(S°(B)) < r* ($n,a(S)) = tanh — for n > 2. Hence Theorem 11.4.2 leads to the following [Gra-Koh-Kohl]: Conjecture 11.4.3. r*(S°(B)) = tanh — in dimension n > 2. With similar reasoning as in the proof of Theorem 11.4.2, we deduce the following result concerning the radius of convexity of $n(S) [Gra-Koh-Kohl]: Theorem 11.4.4. rc($n(S)) = rc($n(S*)) = 2 - v/3. Proof. Let F e *„(£) (or F e $n(S*)). Then F = $„(/) for some / € 5 (or / € 5*). By Theorem 2.2.22, the radius of convexity for S (or for S*) is r• = 2 - \/3. Hence
0> N < r
'
and this quantity can be negative if \z\\ > r. Now if g € K(UP), 0 < p < 1, then 0p € K(U), where pp(C) = -p(pC). C e C7. Hence from Theorem 11.1.2 we deduce that $n(gp) € K(B), which implies that $n(
*n(gp)(z) = -p*n(g)(pz),
z e B.
468
The Roper-Suffridge extension operator
Using this observation, we conclude that F = $ n (/) is convex on Br where r = 2 — V3. Furthermore, taking into account Remark 11.4.1, we deduce that F may fail to be convex in any ball Bri with r\ > r. Therefore rc($n(5)) = rc($n(S*)) = 2 - V3. This completes the proof. Since $n(S) C S°(B), $n(S*) C S*(B) and S*(B) C S°(B), we conclude from Theorem 11.4.4 that rc(5°(B)) < rc(5*(B)) < 2 - V§. However, an example provided by Suffridge shows that in C™, n > 2, the radius of convexity of S* (B) is strictly less than 2 — \/3. Therefore, it remains an open problem to find this radius in several complex variables. Example 11.4.5. Let n = 2 and let / : B C C2 -» C2 be given by
f ( z ) = (zi + az$, z2),
z = (zi, z2) G B,
where a € C, |a| = 3\/3/2. Then / is starlike on B and is convex on Br where r = l/(3\/3)- However / is not convex in any ball centered at zero and of radius greater than r. Proof. Since |a| = 3\/3/2, Problem 6.2.1 implies that / is starlike on B. With a similar argument as in the proof of Example 6.3.13, we can deduce that / is convex on Br where r = 1/(3V3). Indeed, using the criterion of convexity in Theorem 6.3.10 we have to show that Re ( ( D f ( z ) ] - l ( f ( z )
- /(u)), z) > 0,
|H| < \\z\\ < r.
Straightforward computations yield (see the proof of Example 6.3.13) Re {[D/^)]-1^) -/(«)), z> > (||z||2 - Re {*, ti»(l - 2a|2!i|) + |a||zi||zi -
2
Ul\
> 0,
for all z = (zl,z2) e Br, u = (wi,w 2 ) G Br, \\u\\ < \\z\\, when |a| = 3V3/2 and r = 1/(3V3). On the other hand, / is not convex in any ball Bri with r\ > l/(3\/3)- To see this, let z = (zi, z-i) G B and u = (ui, u2) 6 B, where zi = u\, z2 = —u2 G ^ \ {0}, N > 1/(3V3) and Re {azi} > 1/2. Hence ||z|| > 1/(3V3) and Re([Df(z)}-l(f(z)-f(u)),z}
11.5. Lineax-invaxiant families and the operator $„,«
469
= l^ll 2 - Re (z,u) - Re {azi(z2 - u2)2} = \\z\\2 — Re {z\z\ — 22^2} — 4z|Re {dzi} = 2z|{l - 2Re {azi}} < 0. Thus / is not convex in any ball of radius greater than r = l/(3\/3) and centered at 0. This completes the proof. Open Problem 11.4.6. Find rc(S*(B)) and rc(SQ(B)) in dimension n > 2. In connection with this problem, we make the following observation: Remark 11.4.7. There is no positive radius of convexity for the class of normalized starlike mappings on the unit polydisc P of C™ with n > 2. Proof. Let F : P -+ C" be given by
It is not difficult to deduce that F is starlike. On the other hand, it is obvious that F violates the decomposition result in Theorem 6.3.2, and hence F is not convex on Pr for any r € (0, 1]. This completes the proof.
11.5
Linear-invariant families and the operator $n>a
In this section we use the operator $n>a given by (11.3.1) to obtain linearinvariant families on the Euclidean unit ball B of C* and to study the order of these L.I.F.'s. To make the dimension explicit, in this section we shall denote the Euclidean unit ball in Ck by Bk. Also by £S(Bk) we shall denote the set of normalized locally biholomorphic mappings from Bk into Cfc . As we have already seen in Theorem 6.1.23, the automorphisms of Bn, up to multiplication by unitary transformations, are the mappings ha(z) = Ta -
,z€Bn,ae Bn,
where TQ = I and for a =£ 0, Ta is the linear operator given by Ta
=
470
The Roper-Suffridge extension operator
and
(In fact, ha(z) = -(j)a(z), where 4>a(z) is given by (6.1.9).) In other words, Aut(Sn) = {Vha : a e Bn, V € U}, where W denotes the set of unitary transformations in Cn . The following lemmas will be useful in this section. The proof of Lemma 11.5.1 is contained in the first part of the proof of [Pfa-Su4, Theorem 3.3], and Lemma 11.5.2 has recently been obtained in [Gra-Ham-Koh-Su]. Lemma 11.5.1. Let 3= C £S(Bn) and A[f] be the L.I.F. generated by T on Bn. LetaeU and b e Bn~l . Then
where ha = haei and h^ = fyo,&)-
Proof. We first observe that {ha ° hb : a 6 U, b € Bn~1} is a family of automorphisms (p of Bn such that ip(Q) = ( a, \/l — |a|26J . Since this includes all points of Bn as a and 6 vary, we conclude that Aut(B n ) consists of the composition of all unitary mappings with members of this special family. Since the trace is invariant under similarity, it follows that it is sufficient to consider automorphisms of the above type. Since A/16 o A^a = A/la0/lb, the lemma now follows. Lemma 11.5.2. Assume f , g : U —>• C are holomorphic functions such that f is locally univalent, /(O) = 0 and /'(O) = 1 = 0(0). Define F : Bn ->• Cn by F(z) = ( f ( z i ) , g ( Z l ) z ' ) , z = (Zl,z'} e Bn. With G(z) = A.hb(F)(z) we have sup{|trace{D2G'(0)(7,-)}| : IN < 1, IMI = l} = max [n + 1, sup {|trace{D2F(0)(7, -)}| : \\J\\ = l}}. Proof. Without loss of generality, we may assume that the coordinates are chosen so that b = xe% where 0 < x < 1. We write z — (z\,zi,'v) where v € Bn~2 and ||z|| < 1. Of course if n = 2, v will not appear. Then
11.5. Linear-invariant families and the operator
471
Since
-VI - x2 DF(hb(0))Dhb(Q) =
o
-xVl - zY(0) -(1-z 2 )
0
\
0
-VI - x'2I )
0
it follows that G(z) = H(F(hb(z)} - ze2), where 1
L^ xg'(Q) 1-z 2
H=
0
o
o
N
VT-z 2
/
1_ 1-z 2 0
\
Because of the form of G, the trace of D2G(0)(7, •) is
32G2(ohl IM
a^
-
and the entries on the diagonal of D2G(0)(7, •) axe -Vl - x2/"(0hi + X72, -Vl-xV(0)7i + 2x72, -\/l -xV(0)7i + X72, . . . , -\/l -^V(0)7i + X72. The trace is therefore (n - lV(0))7i + (n + I)x72. By elementary calculus, the supremum of this quantity with 0 < x < 1 and ||7|| = 1 is max{n+l, |/"(0) + (n - lV
The Roper-Suffridge extension operator
472
Since trace{D2F(0)(7, •)} = (/"(O) + (n - l)^ the lemma now follows. The main result of this section is a theorem of Graham, Hamada, Kohr and Suffridge [Gra-Ham-Koh-Su] concerning the order of L.I.F.'s generated using the operator $n,aTheorem 11.5.3. Let F be a L.I.F. on U such that ord T — 6 < oo and let a € [0,1/2]. Then ord A[<Jya(:F)] = 77, where 77 = (l + (n-l)a)<5 +
(n - 1)(1 - 2a) z,
Proof. Let / e f and set G ~ $„,«(/). Using Lemmas 11.5.1 and 11.5.2, it follows that
= sup Letting /(f^-V/Ca) \ I — az\ J we have
1— 1— \
Now the diagonal of D 2 A fta (G)(0)(7, •) has
as its first entry and
(i - H2)/» , /'(a)
11.5. Linear- invariant families and the operator $n>
473
in the remaining positions. The trace is therefore
Since we may replace / by a function g € T such that «IK 0. = -(1 - |q|2)/"(a) +, „_
— —
(i.e. g(z\) = f ( z i ; a ) ) , it is clear that we want to find sup
(„ - 1)Q)
This is evidently sup
= (l + (n-l)a)*+^
z,
and the proof is complete. There are some interesting particular cases of the above theorem. When a = 1/2, we obtain the following result for the Roper-Suffridge operator (see [Pfa5,6], [Lic-Stl]): Corollary 11.5.4. Let F be a L.I.F. on U such that ord.F = S < oo. Then ord A[$n(F)] = S(n + l)/2. On the other hand, if f = K in Theorem 11.5.3, we obtain another consequence due to Graham, Hamada, Kohr and Suffridge [Gra-Ham-Koh-Su], which illustrates an interesting phenomenon in several complex variables. Corollary 11.5.5. Let a € [0,1/2]. Then ord A[$n,a(K)] = (n + l)/2. Proof. It suffices to apply Theorem 11.5.3 and then to use the fact that Remark 11.5.6. We have seen in Example 11.3.4 that the operator $„,«, a € [0,1/2], preserves convexity on the unit ball of C71, n > 2, if and only if a = 1/2. Therefore Corollary 11.5.5 shows that in several complex variables there exist L.I.F.'s of minimum order which are not subsets of K(Bn}. (Compare
474
The Roper-Suffridge extension operator
with Theorem 10.3.8.) Indeed, ord A[$nja(K)} = (n + l)/2, but for a ^ 1/2 and n > 2, A[$n,a(K)] £ K(Bn). Finally we shall give simple applications to radius problems for families of mappings generated using the operator $n>a. The result below is a generalization of Theorem 11.4.4 and has recently been obtained in [Gra-Ham-Koh-Su] . Theorem 11.5.7. Let T be a L.I.F. on U such that ord^7 = 7 < oo. Then = 7 - vV2 - 1- In particular, rc($n(K)) = I and rc($n(S)) = Proof. Let F € $n(F). Then F = $„(/) for some / e T. In view of Theorem 5.2.3, we have / € K(Ur) with r = 7 — \/72 — 1, and in fact this number is the radius of convexity of F . Since $ n (/) € K(B?} by Theorem 11.1.2, it follows that rc($n(.F)) = r. This completes the proof. Recall that in dimension n > 1, there is in general no such connection between the order of a L.I.F. .M of Bn and its radius of convexity (see Chapter 10 and also [Pfa-Su2], [Pfa-Su4]). We finish this section with a consequence of Corollary 11.3.2 obtained in [Gra-Ham-Koh-Su]. Theorem 11.5.8. Let T be a L.I.F. on U such that ord T = 7 < oo. Also let a e [0, 1/2]. Then 3>n,a(F) Q S*(B? ), where r = 1/7. Proof. Let Fa € $n,a(F)- Then Fa = $„,<*(/) for some / 6 T . Since ord-F = 7, we deduce from Corollary 5.2.6 that / e S*(Ur) with r = 1/7. We then conclude from Corollary 11.3.2 that Fa € 5*(S"), as desired. Problems 11.5.1. Let a. > 0 and (3 > 0. Also let ^n,a,/3 denote the operator
where / is a locally univalent function on U, normalized by /(O) = /'(O) — 1 = 0 and such that f ( z i ) ^ 0 for z\ € U \ {0}. The branches of the power functions are chosen such that (—-—- J following assertions:
= 1 and (f'(zi))P\zl=o
= 1. Prove the
11.5. Linear-invariant families and the operator 3>n,a
475
(i) Vn,a,(3(S) £ S°(B) for a £ [0, 1], 0 € [0, 1/2] and a (ii) *n)Q)/3(5'*) C S*(B) for a € [0, 1], (3 € [0, 1/2] and a + 0 < 1. (iii) Vn,a,0(K) C Jif(B) for n > 2, if and only if (a,/3) = (0, 1/2). (Graham, Hamada, Kohr, Suffridge, 2002 [Gra-Ham-Koh-Su].) 11.5.2. Show that r*(* n>a>/ j(S)) = tanh(7r/4) for all a € [0,1] and (3 £ [0, 1/2] with a+(3 < 1, where ^n,a,0 is the operator defined in Problem 11.5.1. (Graham, Hamada, Kohr, Suffridge, 2002 [Gra-Ham-Koh-Su].)
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List of Symbols Symbols in Chapter 1 C R Re z Im z U(zQ,r) Ur U A H(G] HU(D) S E Moo(r,/)
the complex plane the field of real numbers the real part of z the imaginary part of z the open disc centered at ZQ and of radius r the open disc centered at 0 and of radius r the open unit disc the exterior of the closed unit disc the set of holomorphic functions on an open subset G of C the set of univalent functions on a domain D of C the class of normalized univalent functions on U the class of univalent functions (p on A with a simple pole at oo and normalized so that <£>(£) = C, + otQ + ai/C H = max|/(z)| for / e H(U) and 0 < r < 1
dh(a, 6)
the distance function induced by the hyperbolic metric
|zj=r
Symbols in Chapter 2 "P X V S*
the Caratheodory class subordination the class of Schwarz functions the class of normalized starlike functions on U
521
522
List of Symbols
K S*(a) K((3) Ma C Sa {/; z} fi r* (F] rc(F)
the class of normalized convex functions on U the class of normalized starlike functions of order a the class of normalized convex functions of order (3 the class of a-convex functions the class of normalized close-to-convex functions the class of normalized spirallike functions of type a the Schwarzian derivative of / G H(U] at z € U the convex hull of a set 17 C C the radius of starlikeness of T the radius of convexity of T
Symbols in Chapter 3 Gn —t G f(z,t) v(z, s, t)
kernel convergence subordination chain, Loewner chain transition functions for a Loewner chain
ft(z) p(z, t)
= flz,t) a function which belongs to P as a function of z € U and is measurable in t € [0, oo) the extended complex plane
C
Symbols in Chapter 4 Aut(f7) B BI B BQ A(l,r) A(l, r)
the set of automorphisms of U the set of Bloch functions the set of normalized locally univalent Bloch functions of Bloch seminorm 1 the Bloch constant the locally univalent Bloch constant horodisc internally tangent to dU at 1 closure of A(l, r) relative to U
II • II
the Bloch seminorm
List of Symbols || • \\B /* r(a, /) L A
523
the Bloch norm the spherical derivative of / the radius of the largest schlicht disc centered at /(a) the Landau constant the univalent Bloch constant
Symbols in Chapter 5 CS A^,(/) /(z; £) L.I.F. ordJ" p(z, f) tf(a) h[Q]
the set of normalized locally univalent functions on U the Koebe transform of / with respect to 0 € Aut(t7) the Koebe transform of / with respect to the disc automorphism <j)(z) = (z + C)/(l + £*) linear-invariant family the order of a L.I.F. T the hyperbolic radius of the largest hyperbolic disc in U centered at z in which / is univalent the universal L.I.F. of order a the L.I.F. generated by a nonempty subset Q of CS
Symbols in Chapter 6 C" {•, •) B(a,r) B B(p) P(a, R) P doP(a, R) A* A A*
the space of n-complex variables the Euclidean inner product in C1 open ball centered at a and of radius r the Euclidean unit ball of C"; the unit ball of C1 with respect to an arbitrary norm; the unit ball of a Banach space the unit ball of C" with respect to a p-norm open polydisc of center a and polyradius R the unit polydisc of C" the distinguished boundary of the polydisc P(a, R) the transpose of a matrix A the conjugate of a matrix A the conjugate transpose of a matrix A
524
List of Symbols
U. H(Q, fi') H (17) Jf(z) Jrf(z] || • ||p Aut(17) L(X, Y) \\A\\ I X* T(z] M0 AT M S(B) S*(B) K(B) V(h) \V(h)\ G T
the set of unitary transformations of C"1 the set of holomorphic mappings from a domain 17 C Cn into another domain 0' C C71 the set of holomorphic mappings from a domain 17 e Cn into Cn the complex Jacobian determinant of / at z the real Jacobian determinant of / at z p-norm the set of biholomorphic automorphisms of a domain ft C Cn the space of continuous complex-linear operators from a complex Banach space X into another complex Banach space Y the norm of the operator A £ L(X, Y) the identity in L(X, X) the dual of the complex Banach space X = {lzeX*:lz(z) = \\z\\,\\lz\\ = l} = {f : B -> X : f e H(B)J(0) = 0,Re [/,(/(*))] > 0,z 6 3\{0},l,eT(z)} = {/ e A^o : Re (lz(f(z))} >Q,zeB\ {0},lz € T(z)} ={f£tf: D/(0) = /} the set of normalized biholomorphic mappings on B the set of normalized starlike mappings on B the set of normalized convex mappings on B the numerical range of h the numerical radius of h the set of quasi-convex mappings of type A the set of quasi-convex mappings of type B
Symbols in Chapter 7 ?ru fi ) )
the orthogonal projection of C" onto the subspace Cu the convex hull of a subset 17 of C71 the infinitesimal Caratheodory metric the infinitesimal Kobayashi-Royden metric =(l-\\z\\*}\\(Df(z}}-i\\-\ztB
List of Symbols
525
Symbols in Chapter 8 -< f(z,t)
subordination subordination chain, Loewner chain
/«(*) Df(z, t) v(z, s, t) h(z, t)
= /(*,*) differential in z of f ( z , i) transition mapping for a Loewner chain a mapping which belongs to M. as a function of z € B and is measurable in £ € [0, oo) = {/i € #(B) : fc(0) = 0,
Mg
Sg(B) S°(B) Sl(B) ACL
the set of mappings in H(B) which have ^-parametric representation the set of mappings in H(B) which have parametric representation the set of mappings in H(B) which can be embedded as the first element of a Loewner chain absolutely continuous on lines
Symbols in Chapter 9 B m(/) ||/||o VB V^0 Bn(k) Bn,0(fc) B(^r) B(5*(B)) B(K(B))
the set of Bloch mappings from the unit ball B of C" into C71 =sup{(l-\\z\f )\\Df(z)\\:zzB} = sup{| J/o^O)!1/* : V 6 Aut(B)} = {/ 6 ff (B) : H/Ho < 00} = {/ e VB : H/Ho = 1 and J,(0) = 1} = {/ : B -> e : / G B(B), ||/|| < fc} = {/ € ft,(*) : J/(0) = 1} the Bloch constant for the set T the Bloch constant for 5*(B) the Bloch constant for K(B)
List of Symbols
526
Symbols in Chapter 10 L.I.F. Aut(B) Aut(P)
linear-invariant family the group of biholomorphic automorphisms of the Euclidean unit ball B the group of biholomorphic automorphisms of the unit polydisc
P CS(B) £S(P)
the set of normalized locally biholomorphic mappings on Euclidean unit ball B the set of normalized locally biholomorphic mappings on unit polydisc P the Koebe transform of / with respect to 0 e Aut(B) $ € Aut(P)) the L.I.F. generated by a nonempty subset Q of £S(B)
the the (or (or
CS(P)) ordJF ||ord|| JF
the the the the
trace order of a L.I.F. jF norm order of a L.I.F. T radius of nonvanishing of a L.I.F. T radius of univalence of a L.I.F. T
Symbols in Chapter 11 $„ r*(jF] TC^}
the Roper-Suffridge extension operator the radius of starlikeness of a nonempty subset T of CS(B] the radius of convexity of a nonempty subset T of £S(B)
Index .K"-quasiconformal, 354 .KT-quasiregular, 354 $-like function, 79 $-like mapping, 245 $-like region, 80, 245 a-convex Koebe function, 61 a-spiral Koebe function, 78 a-spirallike, 73 &-quasiconvex, 437 c-chain, 346 g-Loewner chain, 332 p-parametric representation, 332 /e-fold symmetric, 44, 258 kth-polai form, 283 nth-root transform, 7 p-norm, 202 .M-invariant family, 397
Banach space, 197 Becker's univalence criterion, 130 Bieberbach conjecture, 13, 113 biholomorphic, 192, 201 biholomorphically equivalent, 192 Bloch constant, 152, 378 Bloch function, 145, 378 Bloch mapping, 378 Bloch seminorm, 145 boundary point, 382 Caratheodory class, 27 Caratheodory convergence theorem, 88 Caratheodory distance, 292 Caratheodory metric, 287 Cartan's conjecture, 211 Cauchy estimates, 188 Cauchy's integral formula on the polydisc, 187 Cayley transform, 414 circular domain, 189 close-to-convex function, 64 close-to-convex mapping, 324 close-to-starlike, 325 contractive property, 287 convex circular domain, 227
absolutely continuous on lines, 354 acts transitively, 195 Ahlfors and Becker's univalence criterion, 132 Alexander's theorem, 42 alpha convex function, 58 area theorem, 9 balanced domain, 189 527
Index
528
convex domain, 36 convex function, 36 convex function of order a, 54 convex hull, 44, 278 convex mapping, 224 critical point, 382 critical value, 382 critical-slice symmetry, 279 de Branges' theorem, 13, 113 dilation of /, 6 distinguished boundary, 187
homogeneous polynomial, 208 horodisc, 158 Hurwitz's theorem, 18, 193 hyperbolic metric, 21 identity theorem for holomorphic functions, 109 initial value problem, 100, 298 inner dilatation, 354 invariant distortion theorem, 22 inverse mapping theorem, 191 Julia's lemma, 158
entire biholomorphic mapping, 212 entire univalent function, 212 Euclidean inner product, 185 Euclidean norm, 185 Euclidean unit ball, 185 Fatou-Bieberbach maps, 212 Frechet derivative, 198 generalized Koebe function, 8 generalized Schwarz lemma, 35 growth theorem, 15 Hahn-Banach theorem, 202 harmonic function, 37 Hayman index, 19 Helly selection theorem, 30 Herglotz representation formula, 29 Hilbert space, 185 holomorphic function, 4 holomorphic mapping, 191 homogeneous domain, 195
kernel, 88 Kobayashi-Royden metric, 287 Koebe 1/4-theorem, 13 Koebe distortion theorem, 15 Koebe function, 8 Koebe function of order a, 61 Koebe transform, 7, 396 Landau constant, 153 Landau's reduction, 152 Laurent series, 5 Lebesgue's dominated convergence theorem, 102 linear operators, 186 linear-invariant family, 166, 397 linearly accessible domain, 71 linearly accessible function, 71 Lipschitz, 101, 298 local magnification factor, 4 local rotation factor, 4
Index locally absolutely continuous, 101, 298 locally biholomorphic, 192, 201 locally Lipschitz, 97, 304 locally univalent Bloch constant, 153 locally univalent function, 4 Loewner chain, 94, 296 Loewner differential equation, 110, 297 logarithmic a-spiral, 73 Marx-Strohhacker theorem, 54 maximal dilatation, 354 maximum modulus theorem, 188 maximum norm, 185 Milin's conjecture, 113 minimum principle for harmonic functions, 37 Minkowski function, 189 Mocanu functions, 59 Montel's theorem, 70, 320 norm order, 430 normal family, 210 normal function, 147 normal linear operator, 249 normalized convex function, 36 normalized convex mapping, 224 normalized holomorphic function, 5 normalized holomorphic mapping, 191, 201 normalized starlike function, 36 normalized starlike mapping, 213
529
numerical radius, 208 numerical range, 208 orderofaL.I.F., 166, 397 order of the normal function /, 147 outer dilatation, 354 parametric representation, 332 plurisubharmonic function, 189 Poincare metric, 21 Poisson integral, 68 Poisson kernel, 68 polydisc, 185 principle of univalence on the boundary, 38 pseudoconvex domain, 190 pseudometrics, 287 quasi-convex mappings of type .A, 239 quasi-convex mappings of type B, 241 quasiconformal, 354 quasiconformal extension, 320 quasiregular, 353 radius of close-to-convexity, 122 radius of convexity, 51, 466 radius of starlikeness, 51, 466 Reinhardt domain, 189 Riemann mapping theorem, 5 Robertson's conjecture, 113 Roper-Suffridge extension operator, 444 rotation of /, 6
Index
530
rotation theorem, 123 Rouche's theorem, 89 schlicht ball, 377 schlicht disc, 152 Schwarz function, 28 Schwarz mapping, 201 Schwarzian derivative, 49 second fundamental form, 229 semigroup property, 95, 297 set of uniqueness, 193 single-slit mapping, 92 spherical derivative, 147 spirallike of type a, 73, 251 spirallike relative to A, 250 starlike domain, 36 starlike function, 36 starlike function of order a, 54 starlike mapping, 213 starlike mapping of order 7, 222 strongly spirallike mapping of type a, 372 strongly starlike mapping, 346 subharmonic function, 189 subordinate, 28, 296 subordination chain, 94, 296 subordination principle, 28 Taylor series, 36 trace formula, 404 trace order, 398 transition functions, 94 transition mapping, 296 translation theorem, 281
колхоз 7/31/06
uniformly convex function, 63 uniformly locally convex, 177 uniformly locally univalent, 177 uniformly starlike function, 62 unitary transformation, 195 univalent Bloch constant, 153 univalent function, 4 univalent mapping, 192 univalent subordination chain, 94, 296 universal L.I.F., 167 upper semicontinuous function, 189 Vitali's theorem, 91, 193 Weierstrass' convergence theorem, 93
