Semigroups in Geometrical Function Theory

cp(t) = p((1 - t)z + tu, (1 - t)w + tv)

(2.3.1)

for given points z, w, u, v in A. Using the equality 1 - I m_z(w)12 = a(z,w), where a(z

w) = (1 - Iz12)(1

(2.3.2)

11 - zw1a Iw12)

(see section 2.1) we can write p(z, w) in the form

p(z, w) = tanh-1(1 - a(z, w)) 2

.

(2.3.3)

Therefore, p(z, w) < p(u, v) if and only if a(z, w) > a(u, v) and a(z, w) = 1 if and only if z = w. Also, defining the function V) : [0, 1] H [0, 1] by the formula: 0(t) = or ((1 - t)z + tu, (1 - t)w + tv)

(2.3.4)

HYPERBOLIC GEOMETRY

49

we have by (2.3.3)

cp(t) = tanh-1

1 - O(t).

(2.3.5)

It is clear that 0 is a continuous function on [0, 1] and V(0) = o ,(z, w). We will show that 0 is right-differentiable at the origin. Indeed, for t # 0 we have: (1

I(1 - t)z + tu12)(1- I(1- t)w + tv12) = a(t)

_

2

1 - [(1 - t)z + tu] [(1 - t)w + tv]

- b(t)

(2.3.6)

where

a(t)

- t)2IzI2 + t2Iu12 + 2 Re(1 - t)tzu)]

=

[1 - ((1

x

[1

- ((1 - t)21wI2 + tIvI2 + 2 Re(1 - t)twv)]

=

[1

- Iz12 - (t2lzl2

x

[1 - Iwi2 - (t2Iw12 - 2tlw12 + t2lul2 + 2 Re(1 - t)twv)]

- 2tlzl2 +t2lul2 + 2 Re(1 - t)tzu)]

(1 - Iz12)(1 - lwl2) . [1 - tc(t)] [1 - td(t)], b(t)

=

c(t)

=

d(t)

11

(2.3.7)

- [(1 - t)z + tu] [(1 - t)w + tv] I2

(2.3.8)

(t - 2)Iz12 + tlul2 + 2(1 - t) Re zu

(1-

(2.3.9)

Iz12)

(t - 2) wl2 + tlvl2 + 2(1 - t)

Rev

(2.3.10)

(1-Iwl2)

Similarly, we calculate b(t) = 11 - zwl2 11 - te(t)12,

(2.3.11)

where

e(t) -

(t - 2)zw + (1 - t) (zv + uw) + tuv

(2.3.12)

1-zw

So, by (2.3.6)-(2.3.8) we obtain: (1 - zl2)(1 - Iwl2)

(1 - tc(t))(1 - td(t))

11 - te(t)l2 - 2wl2 tc(t))(1 - td(t)) Q(z W) (1 I1- te(t)IZ 11

This implies 01(0+)

clli

t

(0(t) - 0, (Z' w)) _ -a (z, w)

cllim

t p(t),

Chapter II

50

where

p(t)

= 1 _ (1 - tc(t)) (1 - 2td(t)) I1- te(t)I

-

1 - 2t Re e(t) + t2le(t)12 - (1 - tc(t))(1 - td(t))

11 - te(t)12

1 - 2t Re e(t) + t2le(t)12 - (1 - t(c(t) + d(t)) + t2c(t) d(t)) 11 - te(t)I2 1 - 2t Re e(t) + t2I e(t) I2 - 1 + t(c(t) + d(t)) - t2c(t) d(t) 11- te(t)12 or

V)'(0+) _ -o (z, w) lim [-2 Re e(t) + c(t) + d(t)]

_ -o (z, w)

lim [c(t) + d(t) - 2 Re e(t)].

to+

Using (2.3.8), (2.3.9) and (2.3.11) we obtain:

lim c(t) =

t-.o+

2(Re2u - Iz12) 1

lim d(t) =

t-.o+

- Iz12

2(Rewv -2wi2 1- IwI

and

lim Re e(t) = Re

to+

[(zv + uw) - 2zw] 1 - zw

Thus we obtain: zu - Iz12 + Re wv - Iwi2 - Re [(iv +,&w) - 2zw] V)'(0+) = -2o,(z, w) 1Re 1-Iz12 1-IwI2 1-zw (2.3.13)

Denoting the expression inside the square brackets of (2.3.13) by µ we see that if z = w, then µ = 0. Hence, in this case:

,0'(0+) = -2jo(z, w) also equals to zero. If z w, then by using (2.3.5) and (2.3.13) we obtain: Y

(0+) 2

1 - bi/ (0) ) 0(0)

m_Z(w)I

.

(2.3.14)

We are now ready to formulate our main result in this section.

Proposition 2.3.2 (cf., [116], Lemma 2.2) For given four points z, w, u and v in 0 the following statements are equivalent:

HYPERBOLIC GEOMETRY

cp(t) = p((1 - t)z + tu, (1 - t)w + tv), is not

(a) the function cp : [0, 1] --* R+, decreasing on [0, 1]; (b) cp(0) < W(t), t e [0,1]; (c) cp'(0+) > 0;

(d) Re

z(z - u) + w(w-v) 1-Iz12

51


1-zw

1-J"w12

Proof. It is clear that (a) implies (b) which, in turn, implies (c). The equivalence (c) and (d) follows from (2.3.12) and (2.3.13), because µ > 0 if and only if (d) holds. Now, we will show that (c) implies (a). Assume that cp'(0+) > 0. We claim that cp is a monotone function on [0, 1]. Indeed, suppose that cp has a local maximum,

which is achieved at the point to E (0, 1). Then there are two points t1 and t2 E (0, 1) such that to = 2(t1 + t2) and W(to) > max{cp(tt), cp(t2)}. Setting

wt = (1 - tt)w + tiv,

z1 = (1 - tt)z + t1u, z2 = (1 - t2)z + t2u,

w2 = (1 - t2)w + t2v,

we have p(z1iw1) = p(t1) and p(z2iw2) = cp(t2). At the same time: 1

1

2 z1 + 2 z2 = (1 - to)z + tou, 1

2w1 + 2w2 = (1 - to)w + tov.

Thus we have by property (iii) of Proposition 2.3.1 0(to)

1

1

1

1

P 2z1+2z2, 2w1+2w2 < max {p(zt, w1), p(z2i w2)}

= max{cp(tt), cp(t2)}. Contradiction. Since w'(0+) > 0 this implies also that cp has no local minimum on (0, 1) and therefore cp is monotone. Hence it is not a decreasing function.

Chapter II

52

2.4

Fixed points of p-nonexpansive mappings on the unit disk

Definition 2.4.1 A mapping F : A '-+ A, is said to be p-nonexpansive if for each pair z, w p(F(z), F(w)) < p(z, w). (2.4.1) If for all z, w in A

p(F(z), F(w)) = p(z, w),

(2.4.2)

then F is called a p-isometry of A.

It follows by Proposition 2.1.3 that each F E Hol(A) is p-nonexpansive and F E Aut(o) is a p-isometry. However, the class of p-nonexpansive mappings is essentially wider than Hol(A). Indeed, it is not difficult to see that if F E Hol(A), then F is also p-nonexpansive. Therefore, it follows by the Proposition 2.3.1(iii) that each convex combination of F and F is also a p-nonexpansive. In particular, for F E Hol(A), Re F = 1(F+F) is a p-nonexpansive mapping. Generally, denoting the class of p-nonexpansive mappings by Np(A) we conclude: Np(A) is a closed convex subset of the space of all continuous mappings on A, Np(A) is a semigroup with respect to composition, and Hol(A) is a proper subset of NP(A). At the same time it can be shown that the set of all p-isometries of A, usually

denoting by Isom(A) is the joint of Aut(o) and Aut(o), i.e., F E Isom(A) if and only if either F E Aut(o) and F C Aut(o). We will study here fixed points of p-nonexpansive mappings. It will be shown that the Wolff-Denjoy Theory can be extended for the class N,(A). First we note that if F has a fixed point a in A, then each p-ball B(a, R) = {z E A, p(z, a) < R} is F-invariant. This fact contains Proposition 1.2.1 for holomorphic mappings. To study more deep properties of fixed points of p-nonexpansive mappings we need the following observation. If F E Np(A), and it is a fixed element of A, then the mapping Gt defined by the formula:

Gt(z) _ (1 - t)u + tF(z),

0 < t < 1,

(2.4.3)

also belongs to Np(A). Moreover, it follows by Proposition 2.3.1(ii) that all z, w E A, z w, and t C [0, 1)

p(Gt(z), Gt('w))
(2.4.4)

where q = (1 - t) Jul + t < 1, i.e., G is a strict contraction of (A, p). Therefore by the Banach Fixed Point Theorem Gt has a unique fixed point zt = zt(u) in A, and

zt = lim G(n) n-.oo

(2.4.5;

HYPERBOLIC GEOMETRY

53

uniformly on all compact subsets of A. In addition, it is clear that if for some sequence {tn}, 0 < t,l < 1 there is the limit

a = lim zt ,

(2.4.6)

t,._1-

in A, then a must be a fixed point of F. Indeed, since a E A and F is continuous we have:

F(a) -a = t lim 1- [F(zt,) (1 - t,,)u -

lim

tlim (1 1-

This implies F(a) = a. Definition 2.4.2 ([52]) For a fixed element u E A the curve zt = zt (u) : [0,1) F---* A defined by formulas (2.4.3) and (2.4.5) is called an approximating curve for F. From a different perspective, fixing t E [0, 1) and varying u E A we can consider

zt as a mapping from A into itself which will be denoted by it = Jt(u), that is Jt : A " A is a mapping implicitly defined by the equation:

Jt(u) = (1 - t)u + tF(Jt(u))

(2.4.7)

or, in operator form,

it = (1-t)I+tFoJt.

(2.4.7')

Proposition 2.4.1 If F E N,(O), then for each t E [0,1) the mapping it is also in N,(O) and for t E (0, 1), the mapping Jt has the same fixed points as F. Proof. Let u and v be different points in A. For iterates Gi"), G(,°) = I we have: P(G,n+l)

(u),

Gen+1)

(v))

= P((1 - t)u + tF(Gt") (u)), (1 - t)v + tn)

max{P(u, v), P(G(

(u),

Gin)

tF(G(t")

(v)))

(v))} = p(u, v)

by induction. Hence, from (2.4.5) and (2.4.7) we have:

P(Jt(u),

Jt(v)) < P(u,v),

(2.4.8)

i.e., Jt E N for all t E [0, 1). Now it is clear that if a E Fix Jt, i.e. a is a fixed point of Jt, then by (2.4.7) we obtain: a = (1 - t)a + tF(a),

t E (0, 1).

Assume now that F has a b, which is not a fixed point of Jt for some t E (0, 1). Then Jt(b) = (1 - t)b + tF(Jt(b)) b.

Chapter II

54

At the same time b is a solution of the equation:

zt = (1 - t)b + tF(zt). Since the latter equation has a unique solution for each 0 < t < 1, it follows that Jt(b) must be equal to b.

Remark 2.4.1 Since the set Hol(A) is closed with respect to the topology of uniform convergence on each compact subset on A, we have by the constructions of Proposition 2.4.1, that if F belongs to Hol(A) then so does Jt for each t > 0. We can also obtain the following consequence of this proposition.

Proposition 2.4.2 If F is a fixed point free p-nonexpansive mapping of A, then for each u E A we have: I Jt(u)I -> 1,

as

t - 1-.

(2.4.9)

Conversely, if (2.4.9) holds for at least one u E A, then F is fixed point free.

In other words, F E N has a fixed point in A if and only if there are u E A and a subsequence t.n -> 1- such that the approximating sequence zt,. = bounded away from the boundary of A.

is

Proof. Indeed if for some u c A the sequence {J,, (u)} lies strictly inside A, then there is a subsequence which converges to a point a E A. This point must be a fixed point of F, as we saw above.

Now, if F has a fixed point a c A then Jt(a) = a for all t E (0,1] and we obtain by (2.4.8)

p(Jt(u),

Jt(a)) < p(u,a) < oo,

u E A.

Thus Jt(u) is strictly inside A for all t E (0, 1) contradicting (2.4.9). Actually, we can show somewhat more: If F is fixed point free, then there is a unique point e E 8A, such that for each u c A, the net { Jt (u) } converges to e, as

t tends to 1-. The following assertion contains an extension of Wolff's Lemma.

Proposition 2.4.3 ([52], p. 131) Let F be a p-nonexpansive mapping of the unit disk A, and suppose that F has no fixed point in A. Then there is a unique point e c DA such that: (a) each horocycle

D(e,K)= zEA:

11

-

0}l

1- Iz1JJJ ze122

internally tangent to 8A ate is F-invariant, i.e., F (D(e, K)) C_ D(e, K). (b) For each t E (0,1) Jt(D(e, K)) C D(e,K).

HYPERBOLIC GEOMETRY

55

(c) For each u E A the approximating curve zt(u) = Jt(u) converges to e, as t tends to 1-, i.e., lim zt(u) = e. t

1

Proof. We already know that for each u E A:

lim IJt(u)l = 1. t-.1-

(2.4.10)

Take firstly u = 0 and consider a net zt = Jt(0). Since {zt}tE(0,1) is precompact

there is a sequence Zn = Jt (0) which converges to a point e E 0. It follows by (2.4.10) that jej = 1. Since F is p-nonexpansive we have: o (F(zn), F(z)) > Q(zn, z)

(2.4.11)

for each z E A. Recall also that (2.4.12)

F(zn) = 1tnzn, and using (2.4.11) and (2.4.12) we obtain: 11- F(zn)F(z)12

1-IF(z)I2

<

-

1 - IF(Zn) 12

1-lzn12 1 - tl IznI2 1 - Jzn12

<

11

- znz12

1-Iz12 11- znz12 1 - Iz12

(2.4.13)

11 - z,,z12

1-IZ12.

In addition (2.4.12) implies that: lim F(z,1) = e, n-oo

and letting n -, oo in (2.4.13) we obtain: 11 -eF(z)12

11-ez12

1 - JF(z) 12

- 1- Iz12

This means that

F (D(e, K)) C D(e,K)

for all K > 0. So e is indeed a sink point of F. Its uniqueness can be proved exactly as in Proposition 1.4.2. Now, since D(e, K) is convex it follows that for each t E [0, 1) and u E D(e, K) the mapping Gt, defined by the formula

Gt(z) = (1 - t)u + tF(z) maps D(e, K) into itself, and so does Jt = linl,l-,,, Gtt1)

Chapter II

56

Thus for each t E (0, 1) the horocycle D(e, K) is Jt-invariant. Hence for a given z E A one can find K > 0 such that z E D(e, K) and 11 - eJt(z)P2 1 - IJt(z)12 < K.

But together with (2.4.10) this implies that Jt(z) converges to e as t tends to 1-. Finally, if F is contractive then it follows by the general theory of compact mappings on metric spaces (see, for example, [84]), that F(") converges to a point a E 0. Arguments similar to that above show that a must be equal to e E 80, and we have completed the proof.

Remark 2.4.2 Note that the situation is simple when F E Isom(A) and it is fixed point free. In this case either F or F is an automorphism which is power convergent to a sink point of F on the boundary of A. The case of P") iterations when F is a fixed point free p-nonexpansive mapping neither holomorphic nor antiholomorphic, seems to be rather complicated in the framework of the pure one-dimensional metric theory.

The full answer to this question was given by R. Sine [135] in 1989: if F is a fixed point free p-nonexpansive mapping, then it is power convergent. The result for contractive F (i.e., p(F(z), F(w)) < p(z, w)) has established earlier by K. Goebel and S. Reich (see [52], p.138). Now we turn to the situation when F E Np(A) has a fixed point in A. Generally, such a situation differs form the holomorphic case. Namely, we mean that there are p-nonexpansive mappings which have more than one interior fixed point. The following simple example F(z) = 2(z + z) shows that in fact F may have infinite number of fixed points. Actually, in this example F is a p-nonexpansive retraction (F o F = F) of the unit disk onto the open interval (-1, 1). In general, the iterates of a p-nonexpansive mapping with interior fixed points may not be convergent. A rather complete discussion on their behavior can also be found in [135].

Thus the following question arises: does the approximating curve Jt(u) for a fixed u E A converge to a fixed point of F, when t tends to 1- ? The following assertion states that the answer to the above question is affirmative.

Proposition 2.4.4 ([52], Theorem 28.3, p. 134) Let F be a p-nonexpansive mapping on A with the nonempty fixed point set. Then there exists the limit limn Jt

t-1which is a retraction onto this set.

This assertion together with assertions (h) and (c) of Proposition 2.4.3 can he considered as an implicit continuous analog of the Wolff-Denjoy Theory for p-nonexpansive mappings.

HYPERBOLIC GEOMETRY

57

Since Jt for each t E (0, 1) has the same fixed point set as F, one may ask what happens with the discrete iterations Jt') for a fixed t E (0, 1). The following assertion is a special case of Theorems 30.5 and 30.8 in [52]. For the holomorphic case, see, also [51].

Proposition 2.4.5 Let F be a p-nonexpansive mapping on the unit disk A. Then for each t E (0, 1) the mapping Jt defined by (2.4.3) and (2.4.5) is power conver-

gent. In particular, if F has no fixed point in A, then the iterates Jt n) (z), n = 1, 2 ..., converge to the sink point e E aO of F as n - oo for each t E (0,1) and each zEA. This assertion is quite simple if F is holomorphic. Indeed, it follows by algorithm (2.4.5) that for each t E (0, 1) the mapping Jt belongs to Hol(A). Thus to prove our assertion we need only to show that for every F E Hol(O) and t E (0, 1), the mapping Jt cannot be an elliptic automorphism. In fact, if a E A is the fixed point of F then it follows by the chain rule that

(Jt)'(a) = (1 - t) + tF'(a)(Jt)'(a), (Jt)'(a) =

1

t

1 - tF'(a) If F(a) = 1 then F, hence Jt, is the identity for each t E (0, 1) and we have completed the proof. If F(a) J 1, then it follows by the Schwarz-Pick Lemma (Proposition 1.1.3(ii)) that IF'(a)I < 1 and we have: I (Jt(a) I < 1.

This means that Jt is not an automorphism of A. For a complete exposition of the Wolff-Denjoy Theory under the hyperbolic geometry approach it would be appropriate to mention the results due to P. R. Mercer [95, 97, 98], R. Sine [135] and P. Yang [160] for the one-dimensional case, M. Abate [1]-[5], G.-N.Chen [24], Y. Kubota [80] and P. R. Mercer [96] for the finitedimensional case, T. Kuczumov and A. Stachura [82], [83], P. Mazet and J. P. Vigue [92], [93], S. Reich [111]-[113], S. Reich and I. Shafrir [114], I. Shafrir [129, 130] for the Hilbert ball and its product, and P. Mellon [94] and K. Wlodarczyk [152]-[153] for J*-algebras (see also a survey of S. Reich and D. Shoikhet [118]). P. Yang in [160] suggested a hyperbolic metric characterization of the horocycles D(e, K) in A as follows:

D(e, K) _ (z E A : lim [p(z, w) - p(0, w)] < 2 log K

z-e

}.

(2.4.14)

Even for holomorphic case this characterization is a very useful tool in the study of the boundary behavior of self-mappings of A. By using (2.4.14) and a strengthened Schwarz-Pick inequality, P. R. Mercer [98] has recently obtained a sharper description of Julia's Lemma which makes the Denjoy-Wolff Theorem more transparent. We state his result here.

Chapter II

58

Proposition 2.4.6 Let F E Hol(0) be not a constant. Suppose that there exists a unimodular point e E 8i such that lim inf z-+e

1- IF(z)l

1-IzI

= of < oo.

Then there is a point 71 on the boundary 80, such that for any z E 8D (e, K), z e, the following inclusion holds:

e, and w E D(e, K), w

F(z) E D(rj, where

aaK),

1

(2.4.15)

2 b+ c

Iw12) a- 1 + be b- IF'(w)I(11 - IF(w)I2 '

c= z-w 1-wz

Note that it follows by Corollary 1.1.1 that b < 1. Moreover, the Schwarz-Pick Lemma implies that b (hence a) is equal to 1 if and only if F is an automorphism of A. If this is not the case (i.e., F is not an automorphism of 0), then a in (2.2.15) is strictly less than 1, whence this inclusion is stronger that the original one in Julia's Lemma.

Chapter 3

Generation theory on the unit disk For physical, chemical, and biological applications it is sometimes preferable to study a great variety of iterative processes, including the processes of continuous time. In spite of their simplicity these processes have been applicable in many fields, involving mathematical areas such as geometry, theory of stochastic branching processes, operator theory on Hardy spaces, and optimizations methods. A problem that has interested mathematicians since the time of Abel is how to define n-th iterate of function when n is not integer.

3.1

One-parameter continuous semigroup of holomorphic and p-nonexpansive self-mappings

Let A be a topological Abelian (additive) semigroup with zero and let there exist a natural ordering of A, i.e., r > t if and only if there is s E A such that r = t + s. A mapping S : A f--3 Hol(A) (respectively, Np(A)) which preserves the additive structure of A with respect to composition operation on Hol(A) (respectively, Np(A), i.e.,

(i) S(t + s) = S(t) o S(s), whenever s, t and s + t belong to A; (ii) S(O) = I, the identity embedding of A, will be referred to its holomorphic (respectively, p-nonexpansive) action on A. If A = N U{0} = 10, 1, 2, ...} then S = {Fo, F1 i F2, ... , F,,, ...}, F. E Hol(A), 59

Chapter III

60

(respectively, NP(A)) is called a one-parameter discrete semigroup. Actually such a semigroup consists of iterates of a holomorphic (respectively, p-nonexpansive)

self-mapping F = F1, because of conditions (i) and (ii), i.e., FO = I, F, F("`),

n=1,21....

If A is an interval of R, containing zero and action S : A f-4 Hol(0) (Np(A)) is continuous with respect to the topology of pointwise convergence on 0, we will say that S is a one-parameter continuous semigroup of holomorphic (p-nonexpansive) self-mappings Ft, t E A. In other words, a one-parameter continuous semigroup of holomorphic selfmappings of 0 is a family S = {Ft}teA C Hol(A) (NP(A)) such that (i)' Ft+3(z) = Ft (F., (z)), z E 0, t, s and t + s E A,

(ii)' Fo(z) = z for all z E 0 and limFt(z) = F, (z), t-S

(3.1.1)

whenever t, s E A. Mainly we will concentrate on two cases:

(a) A = [0,T), T > 0; (b) A (-T,T), T > 0. If T = no we will say that S = {Ft}t>o is a flow on A. In the case (b) conditions (i)' and (ii)' imply that Ft o F_t = FO = I, hence Ft E Aut(o) (Isom(A)) and S is actually a group which will be called a oneparameter group of holomorphic automorphisms (p-isometries) of A.

Exercise 1. Prove the following: if A = [0, T) and at least one element Ft of S belongs to Aut(o), then so does each element of S, and S can be extended to a one-parameter group:

S = {Ft}tC(-T,T) C Aut(A).

Example 1. Let a be a complex number such that Re a > 0 and let t E [0,T), T > 0. Define Ft E Hol(A) as follows: Ft(z) = e-°'t z,

z E A.

(3.1.2)

It is clear that Fo(z) = z,

z E 0

and

Ft+9(z) = Ft (F., (z)),

zc

whenever 0 < s, t < T and t + s < T. Since T > 0 is arbitrary we have that such a semigroup can be continuously extended to a flow on A. As a matter of fact, we will show below that each one-parameter continuous semigroup of holomorphic self-mappings of A can be continuously extended to a holomorphic flow on A.

GENERATION THEORY

61

Note also that the semigroup (flow) considered in this example is a linear action

on A. In fact, each one-parameter semigroup which is a linear action on A has the form (3.1.2) with some a E C, Re a > 0. Example 2. Define the family {Ft} C Hol(A), t E R, by the formula: Ft

_ z + tank t ztanht+1

It is easy to see that for each t E R, Ft E Aut(o), and that conditions (i) and (ii) are satisfied. Hence, Ft is a flow of automorphisms on A. Exercise 2. Show that each of the automorphisms in Example 2 is hyperbolic.

Example 3. Set 1 + it

Ft (z) =

z+

1-it

t

+it i

1+1titz.

1 + it

has modulus 1 and at = for all t E R, we have that Ft belongs to Aut(o). Since Ft(z) = Atm,,, (z), where At =

1-it

t eA 1+it

Exercise 3. Let S = {Ft}tER be defined as in Example 3. Show that the following statements hold: (i) S = {Ft}tER is a flow of automorphisms of A; (ii) for each t :A 0 the mapping Ft in Example 3 is a parabolic automorphism.

Exercise 4. Consider the family S = {Ft} defined as follows: Ft

Show that (i) for each t > 0, jFt(z)j < 1, i.e., Ft E Hol(O); (ii) the family S = {Ft} is a one-parameter continuous semigroup on A and every element of S is not an automorphism of A; (iii) for each Ft, t > 0, only elements {0, 1} are solutions of the equation Ft (z) = z;

(iv) find limt.,,. Ft (z),

z c A.

Exercise 5. ([115]) Consider the family S = {Ft}, where

Ft(z)=1- [1-e-Zt+e-Zt 1-zJ

z .

(i) Show that Ft E Hol(O), and S = {Ft} is a one-parameter semigroup; (ii) Find lim Ft(z). t-.oo

Remark 3.1.1 The semigroups in Examples 2 and 3 consist of automorphisms of A and consequently are one-parameter groups, while the semigroups in Exercises

Chapter III

62

4 and 5 cannot be extended to subgroups of Aut(A). In fact, Ft in Exercise 4 is defined also for t < 0, but it does not map the unit disk A into itself. Therefore for t > 0 the mapping Ft is biholomorphic on A, but Ft (A) is a proper subset of A. This fact holds in general.

Proposition 3.1.1 Let the family S = {Ft}tE[o,T), T > 0 be a one-parameter continuous semigroup on A, S C Hol(t ). Then each element Ft E S is a local biholomorphism on A.

Proof. Let U be any convex open subset strictly inside A. Since S is a normal family on A the net {Ft} converges to the identity uniformly on U, as t goes to zero. Therefore for 6 =dist(U, d0) and E E (0, 1 - 6) one can find to E (0, T) such that Iz - Ft(z)I < E for all z E U and t E (0, to). It follows by the Cauchy inequality that

1 - (Ft)'(z) I <

E 1

a

< 1,

hence (Ft)'(z) # 0 for all z c U. Consequently, for each pair z and w in U we have by the Lagrange formula: IFt(z) - Ft(w)I _ I(Ft)'(z + O(w - z))I . Iz - WI,

0 < 0 < 1, equals to zero if and only if z = w. So, Ft is injective (hence biholomorphic) on U for all t E (0, T) small enough. Now it follows by continuity that if Fto is locally injective on A then so is Ft for all t close enough to to. At the same time, if {Ft} is a net of locally injective mappings which converges to F9 when t goes to s E (0, T), then FS must be either also locally injective or zero. But the latter is impossible. Hence the set of t c (0, T) such that Ft is locally injective is a nonempty open and closed subset of the interval [0, T). Therefore it must be [0, T).

3.2

Infinitesimal generator of a one-parameter continuous semigroup

As we mentioned above each one-parameter continuous semigroup of holomorphic self-mappings of A defined on an interval [0, T) can be extended to the flow on A, i.e., we may always assume that T equals to oo. Moreover, only the right continuity at zero of a semigroup implies continuity (right and left) on all of I[8+ = (0, oo). These facts can be shown by different approaches, but here we

GENERATION THEORY

63

will establish them by using a very strong property of a continuous one-parameter semigroup of holomorphic self-mappings of A to be differentiable with respect to

a parameter at each point t on the interval of definition. This nice result for the one-dimensional case is due to E. Berkson and H. Porta [17]. We will give here another proof which can easily be extended to a higher dimension (see, for example, [115, 118], cf., also [5]).

Proposition 3.2.1 Let S = {Ft, t E [0,T)} be a one-parameter semigroup of holomorphic self-mappings of A, such that for each z E A: t

lim Ft(z) = z.

(3.2.1)

0+

Then for each z E A there exists the limit:

z - Ft(z) = f ) (z

(3.2.2) t which is a holomorphic mapping of A into C. Moreover, the convergence in (3.2.2) is uniform on each subset strictly inside A.

lim

t0+

In other words, if a semigroup S = {Ft}, t E [0,T), is right continuous at zero, it is also right-differentiable at zero.

be its iteration family. Let Ar Proof. Step 1. Let 0 E Hol(A) and denote the disk centered at zero with radius r. Suppose that there are positive rl < r2 < 1 and 0 < µ < r2 - rl and an integer p > 1 such that: I Z - 0(k) (Z)

<µ

(3.2.3)

forallk=1,2,...,pandzEAr2. Thus for all z E Or, the following inequality holds µ (p z _0(l)

- p(z - O(z)) <

r2-r1_µ

(3.2.4)

O(z)I.

Then the disk Indeed, let z E Or, and w E Are be such that Iz - wl Ore-r1-µ(z) centered at z with radius r2 - r1 - µ lies in Ore. Hence it follows from (3.2.3) and the Cauchy inequalities that: µ

1 - (la(k)),(z)I <

(3.2.5)

r2-r1-µ

1

Therefore, for Z E Or, and w E A12 such that Iz - wl < µ we have by (3.2.5): µ (3.2.6) Iz - wI. Iz - 0(k) (Z) - (w - 0(k)(w)I <

r2-r1-

Setting w = O(z) and using the triangle inequality we obtain: p-110(k) Iz

-

01) (z) - p(z - O(z)) I

= :

(Z) -

0(k+1)(z)

- (z - O(z))]

kP-1

0(k) (Z)

<

- z - [O(k) Wz))

k=1

<

µ

r2-r1-µ (p - 1)I0(z) - zl,

-

O(z)]

Chapter III

64

and we are done.

Step 2. Let a semigroup S = {F } , t E [0, T) satisfy condition (1). For s > 0 define the mapping

f8 = (1

Fs) E Hol(A, C).

We claim that the net {fs}s>o sis uniformly bounded on each disk 0,, with 0 <

r<1.

Indeed, set rl = r and choose r2 E (rl,1). Take any µ > 0, such that y < r2-rl µ and < 1.

r2-r1-µ

2

By (3.2.1) one can find a E (0, T) such that for all r c (0, v) and all z E A'2 Iz - FT(z)I < ii.

(3.2.7)

Setting p = [U/s] we see that for all s E (0, o,/2), the following relations hold: p > 2 and ps > v/2. Also ks < a for all k = 1, 2, ... , p. Hence by (3.2.7) and the semigroup property for all s E (0, v/2) we have:

Iz - F(k)(z)I < µ, whenever z E A,2 and k = 1, 2, ... , p(= p(s)). From Step 1 we obtain for all z E Zr: pIz - Fs(z)I - Iz - Fsp)(z)I

Ip(z - F., (z)) - (z - F9n)(z)I

<

1

plz - F8(z)I

2 or

Iz-F.,(z)I < p

z - F(p)(z)

,

(3.2.8)

1

whenever s E (0, a/2) and p = [v/s]. So, by (3.2.7) and (3.2.8) we obtain: 2

Ifs(z)1

Ps

2

4/1

Iz - Fsp(z)I < ps < 01- < oo,

as claimed.

Step 3. Now we will show that the net {fs}, fs = 1/s(I -F3), s E (0, T), converges to a holomorphic mapping f on A, when s goes to 0+. Set n = n(s) = [1/s2]

and consider the sequence {f}. Since by Step 2 this sequence is uniformly bounded for all large n one can find a subsequence {fi/nk } which converges to a mapping f E Hol(A, C) uniformly on each compact subset of A. That is for each r E (0, 1) and all z E O, and for a given E > 0 we can choose k large enough such that: (3.2.9) Ifi/nk(z) f(z)I < E. In addition, for such nk and s E (0, 1) we have:

-

fs - f-L-k

=

1(I-Fs)-nk(I-Fnk) CF(lknk)] S

- Fs/ +

[(I s

- F([knk])) - nks(I - FL)J

GENERATION THEORY

65

Observe, that since, n = [1/s2], we have ns , oo and [ns]/ns -p 1 as s - 0+. Moreover, for a given net s -; 0+ we can find (if necessary) a sequence Sk -> 0+ such that Sk/S < 1. Then, setting nk = [11S4], we have:

2 < Ski

rl

S sic

s

Sk

Since, S/sk > 1 we obtain: S

Sk

>--goo Sk

as

Sk -* 0+.

Therefore nks , oo as s 0+ and [nks] /nks 1 as s 0+. Thus we can find 8 > 0 such that 1 - [nks] /nks < E and F[nks]/nk(z) C Or, C A, whenever s E (0, 8) and z E Or, r < r2 < 1. Then we obtain by step 1 and the semigroup property: 1

F[^(z) - F., (z)

S

oF_ nk')(z)

= 1 ^k

<

S

M z-F jj(z) S

4µ

nk

[nks]

Qs((s

(3.2.10)

n

where M = sup I(Fs)'(z)I. Finally we obtain by Step 1: 1

z S

-

F((snk])(z) ..k

- nks (z -

F^k, (z))

- F(lknkl)(z) - [nks]

<s

Iz

<

([nks] + Inks - ns]I) z - F ik (z)

S

C I F [nkS]

\

<

nks 8e.

+

[nkS]

nks

-1

J

(z - Fnk (z))

+S

I

nk I z- F, (z) l "k

(3.2.11)

Thus for z E Or and s c (0, 8) we obtain from (3.2.10) and (3.2.11): Ifs(z) - f(z)I

<-

If 1 (z) - f(z)

<

16µe

^k

and we have completed the proof.

Q

+

i (z) fs(z) - f"k

Chapter III

66

Further we will show that even for the wider class of semigroups of the pnonexpansive mappings the property of the right-differentiability at zero, implies the continuous extension of the semigroup to a flow in NN(A) and the differentiability of this flow at each point t on (0, 00).

Definition 3.2.1 Let S = {Ft} c NN(A), t c [0,T), T > 0, be a one-parameter semigroup of p-nonexpansive self-mappings of A. If there exists the pointwise limit

f : A H C defined by

f _ lim t-.o+

I - Ft

(3.2.12)

t

we will say that S is generated by f and f is called the infinitesimal generator of the semigroup S = {Ft}te[o,T).

If this will not imply a confusion we will often simply say `generator of a semigroup', omitting the word `infinitesimal'. Thus Proposition 3.2.1 states the existence and holomorphity of generators for every one-parameter continuous (even only right-continuous at 0+) semigroup of holomorphic self-mapping of A.

Proposition 3.2.2 Let S = {Ft}te[0T) be a one-parameter semigroup of p-nonexpan: self-mappings on A, which is generated by f : A f--+ C and suppose that its generator f, defined by (3.2.12), is a continuous function on A. Then the complex valued function u(., ) defined on [0, T) x A by u(t, z) = Ft(z) is the solution of the following Cauchy problem:

au(t , z) + f (u(t, z)) = 0, at u(0,z) = z,

t E (0, T),

(*)

z E A.

Proof. Fix s E [0,T) and set F, (z) = w. We have: a+u(t, z) at t=s

lim h-0+

_

Fs+h(z) - F3(z)

-

h

Fh(w) - w

hll.o

lim Fh(F9(z)) - F,(z)

h-,0+

h

-f(w) = -f (u(s, z))

h

This means that the right-hand partial derivative of u(t, z) = Ft (z) in t = s E [0, T) exists and is exactly equal to -f (u(s, z)). Now for fixed s E (0, T) we denote wh = F,-h(z) for h E [0, e) with a small enough. Since z is an interior point of A we can find a small enough such that the set {wIL}hE[O,,) lies in a compact subset S2 of A. Then we obtain by the uniform continuity of f on SZ

a-u(t, z) at

lim t=s

IL

0+

F,,

(z) - F,, (z) h

Wh - F(w1L) lim 0+ h h

lira fh(wh) = -f (w),

h-.0+

GENERATION THEORY

67

where

w = lira wh = F, W.

ho+

So u(t, z) is also left-differentiable on the interval (0, T) and therefore satisfies the Cauchy problem (*).

Remark 3.2.1 As a matter of fact, it is known (see, for example, [159]) that if a function has a right-hand derivative which is continuous, then it also has a left-hand derivative and they coincide.

Corollary 3.2.1 If S = {Ft}, t E [0, T) is a one-parameter semigroup of holomorphic self-mappings, which is right-continuous at zero, then it is continuous at each point of [0, T) and, moreover, it is differentiable on this interval.

Remark 3.2.2 Also we note that since the generator f of a semigroup of holomorphic mappings is holomorphic, it is locally Lipschitzian. Therefore the Cauchy problem (*) has a unique local solution in a neighborhood of t = 0 for each initial value z E A. This fact and the analytic continuation principle can be successfully used to prove the extension of the semigroup to a flow of holomorphic mappings on A (see [115]).

However, to prove this for the class of p-nonexpansive mappings we will use another approach which is based on the so called resolvent method. We will discuss

it in the next section. At the end of this section we present a property which is specific for semigroups of holomorphic self-mappings.

Exercise 1. Prove that if u(., ) is a jointly continuous function on [0, T) x A which satisfies (*) with f E Hol(A, C), then it also satisfies the linear partial differential equation:

au(t,z) a +auaz,z)f(z)0.

Exercise 2. Find the (infinitesimal) generators in Example 1-5 of Section 3.1.

3.3

Nonlinear resolvent and the exponential formula

In this section we consider the notion of the nonlinear resolvent of a continuous (or holomorphic) function and establish a result in the spirit of the Hille-Yosida theory

Chapter III

68

(see, for example, [159]). More precisely, we will show that the global solvability of the Cauchy problem (*) (see Section 3.2) is equivalent to the solvability of a functional equation (see [115, 116, 118]). To make the ideas more transparent we first explain them for the simplest linear case. We already mentioned that the only linear (in the complex sense) semigroup S on C is of the form: S = {Ft}t>0 , Ft(z) = exp(-ta)z, a E C, with the infinitesimal generator f : C --* C defined by

f(z) := lim

to+

z - Ft(z)

t

= az.

Conversely, for a given number a E C two following equivalent definitions of the exponential function u (t, z) = exp (-ta) z usually are employed in the classical analysis. The first one is based on the solution of the linear Cauchy problem:

+au(t , z) = 0 (3.3.1)

z E A,

a c C, while the second one uses the exponential formula:

\n u(t, z) = lim n

t

C1 + -a I n

Z.

(3.3.2)

On the other hand, if for given z E C, a E C, and r > 0, we solve the linear equation

w + raw = z,

(3.3.3)

we have that its solution

w= Jr(z)=(1+ra)-lz can be considered as a linear mapping defined on C. Then the exponential formula (3.3.2) can be rewritten in the form

u(t, z) = li n J(,n(z),

(3.3.4)

where J(n) denotes the n-fold iterate of the mapping Jr., r > 0. In addition, it is clear that S = {exp(-ta)} is a p-nonexpansive action on the unit disk A if and only if

Rea>0. This is, in fact, equivalent to the property that for each r > 0 the mapping J,.: C C is a self-mapping of A, hence is p-nonexpansive.

GENERATION THEORY

69

Now, if we consider, for example, a (nonlinear) analytic function f : A F-+ C defined by

f(z) = z - z2, we see that the solution of the Cauchy problem

I

8u(t, z) + f (u(t, z)) = 0, u(O, z) = z E A,

(3.3.5)

is a well defined semigroup (even a flow) of holomorphic (hence p-nonexpansive) mappings of A: Z u (t, z) (= Ft (z))

(see Exercise 4 in Section 3.1).

But in this case the similar formula (3.3.2) makes no sense with respect to (3.3.5).

At the same time, it turns out, that formula (3.3.4) continues to hold with Jr : A H A, r > 0 defined as the solution w of the equation

w+r(w-w2)=z

(3.3.6)

(compare with equation (3.3.3)).

Exercise 1. Show directly that for each z E A and r > 0 the quadratic equation (3.3.6) has a unique solution w = Jr(z) E A which holomorphically depends on z E A. Show that lim J(' t/n(z) = Ft(z). n-oo

In general such an approach, which we call the resolvent method, is very useful in the study of generated p-nonexpansive semigroups and their properties. In addition, by using this method one can give different characterizations of the class of functions which generate a flow of p-nonexpansive or holomorphic mappings, or in other words, to give sufficient and necessary conditions for the global solvability of the Cauchy problem. To continue we need the following definition.

Definition 3.3.1 Let f : A H C be a continuous function. We will say that f satisfies the range condition (RC) if for each r > 0 the nonlinear resolvent Jr = (I + r f)-1 is well defined on A and belongs to Np(O). In other words, f satisfies the range condition (RC) if for each r > 0 and z E A the equation

w+rf(w)=z

has a unique solution w = Jr(z) in A, such that P(Jr(W1), Jr(W2)) < P(w1,w2), whenever w1i W2 belong to A.

(3.3.7)

Chapter III

70

Proposition 3.3.1 A continuous (holomorphic) function f on 0 is an infinitesimal generator of a one-parameter semigroup S = {Ft} C Np(0) (respectively, Hol(0)) defined on [0, T), T > 0 if and only if it satisfies the range condition (RC).

Moreover, the semigroup S = {Ft} is unique and can be continuously extended to a flow of p-nonexpansive (holomorphic) self-mappings defined on 1R+ = [0, oo) by the following exponential formula

Ft(z) = n-+oo lim J(c )(z) z E 0, t > 0. n

(3.3.8)

Since the proof of this assertion is rather long we will give it step by step using several lemmata which will also be needed independently in the sequel.

Lemma 3.3.1 Let p be the Poincare metric on 0 and let {Gt}, t E [0, T), T > 0 be a family of p-nonexpansive mappings of 0, i.e.,

p(Gt(z), Gt(w))

p(z,w)

(3.3.9)

for all z,w E 0, t E [0,T). Suppose that for each z E 0 there exists the limit:

f (z) = tllio t (z - Gt(z))

(3.3.10)

and assume that f is continuous on each compact subset of A. Then for each r > 0 and each w c 0 the equation

w+rf(w)=z

(3.3.11)

has a unique solution z = Jr.(w) and J,.: 0 H 0 is also p-nonexpansive.

Proof. Given t E (0, T) denote:

ft= 1(I-Gt)

(3.3.12)

w+rft(w)=z, zEA, r>0.

(3.3.13)

and consider the equation:

This equation can be rewritten in the form:

w=

r Gt(w) +

r+t

t

r+t

(3.3.14)

Z.

For fixed z E A the mapping defined by the right-hand side of (3.3.14) is a strict contraction with respect to the metric p (because of (3.3.9) and Proposition 2.3.1(ii)). Therefore, for each z E A and r > 0, this equation has a unique

solution wt = Jr.,t(z) E A. In addition, this solution can be obtained by the iteration method:

wn+i(= wn+i(z)) = r+tGt(wn) +

r+tz,

(3.3.15)

GENERATION THEORY

71

where zo is an arbitrary element in A. Setting wo(z) = z we have by induction and Proposition 2.3.1(iii) that

5 max{p(Gt(wn(zl)),Gt(wn(z2))),P(z1,z2)}

P('Wn+l(zl),Wn.+1(z2))

< max {p(wn(z1), wn(z2)), P(z1, Z2)1

P(z1, Z2)

limn_.oo Wn( ). It means that all wn(.) are p-nonexpansive and so is Note, in passing, that if Gt E Hol(O) then Jr,t : A H A defined as the solution of

t3.3.14) is also holomorphic on A.

Now we want to show that for some r > 0 and z E A the net {Jr,t(z)}tE(o,T) converges to Jr(z), as t tends to 0+ and its limit is a solution of equation (3.3.11).

To do this we first claim that this net lies strictly inside A. The latter is equivalent to the inequality:

p(z, J,.,t(z)) < M (= M(z)) < oo, as t -> 0+.

(3.3.16)

Indeed, since

Jr,t(z)

r+tGt(Jr,t(z)) +

rt

Z'

we have by Proposition 3.3.1(ii): P(Jr(z) 'c

r

t

r+tz+r+tz (r r r r

\

+

r t IzI) [p(Gt(Jr,t(z), Gt(z)) + p(Gt(z), z))]

t

(r + t + r

t

t IzI) [P(Jr,t(z), z) + p(Gt (z), z)]

This inequality implies that

P(Jr,t(z), z)

t(1

Iz ±IzI) Cr + t + r t t

p(Gt(z), z)

Since for t E (0, T) small enough the element Gt (z) is close to z we have that there exists a positive number Ml < oc such that limsup 1 p(Gt(z), z) t--,o+ t

<

limsup M1 IGt(z) - zI t t-,o+

= MiIf(z)I Consequently, we have the estimate:

limsuP P(Jr,t(z),z) < t--o+

which implies (3.3.16).

rizl 1

MiIf(z)I =: M,

Chapter III

72

Thus this lemma proves the necessary assertion of Proposition 3.3.1 if we set

Gt=Ft, tE [0,T), T>0. To accomplish this matter we note that it follows by the uniqueness of the local solution of the Cauchy problem (see also remark at the end of this section) that if f is holomorphic in 0, then so is Ft for each t E [0, T). Then our constructions in Lemma 3.3.1 show that the resolvent Jr : 0 - 0 is a holomorphic mapping for each r > 0. This fact can be shown also by using the local Implicit Function Theorem (see, for example, [115]). Let as above p be the hyperbolic Poincare metric on 0 and B(a, R) = {w E O : p(a, w) < R}, a E A, R > 0.

Lemma 3.3.2 Let f be a continuous function which satisfies the range condition

(RC). Then for each a E 0 and R > 0 there are r = r(a, R), 0 < r < 1 and L = L(a, R) < oo such that p(J(k) (z), z) < rkL

for all r E (0, r) and k = 0, 1, 2, ... .

Proof. Since each p-ball is bounded away from the boundary of 0 for given

a ED and R >0 we can find 0< s <1 such that B(a,R) C

= {z E i: Izi <

s}.

Denote M = max{ If (w) I, w E B(a, R) } and set r = d/M, where d =dist{aB(a, R), az } > 0. Then for each r E (0, r) and w E B(a, R) we have

z = w+ r f (w) Ci

(3.3.17)

and w = Jr(z) E B(a, R). Hence, for such r and all w E B(a, R) we obtain by (3.3.17) p(Jr('W), w) = p(Jr(w), Jr(z))

< tanh-1

p(z, w)

_z - wl

1-s2 = arctanh rM

1-s2.

(3.3.18)

Further, it follows by the Lagrange mean value theorem that for each t E [0, t°], to < 1,

tanh-1 t < t

-

1

1-t2

Then setting to

rAl 1 - s'-

d

and L -

1 - s2

we obtain by using (3.3.18):

p(Jr(w), w) < rL for all w E B(a, R) and r c [0,,r).

AI(1 - s2) (1 - s2)2 - d2

GENERATION THEORY

73

Now using the triangle inequality we have k

P(J(k)(Z) z) < > p (J(j)(z) J(i-1)(z)) < kp(Jr(z), z). i=1 Hence

p(J,((k) (z), z) < rkL

and the Lemma is proved.

Lemma 3.3.3 Let f be a continuous function in 0 which satisfies the range condition (RC). Then for each a c A, R > 0, and E > 0, there is µ =µ(a, R, E) > 0 such that for all r E [0, µ) and each p = 0, 1, 2.... the following inequalities hold

f(z) -

z - Jr/P(z)

<E

(3.3.19)

< 2rs,

(3.3.20)

r

and Jr(z) - J(/P(z)

whenever z E B(a, R).

Proof. Since f is continuous in 0, for each a E 0, R > 0 and E > 0 one can find b > 0 such that If (z) - f (w) I < E, whenever z and w belong to B(a, R) and p(z, w) < d.

Let T = T(a, R) and L = L(a, R) be found as in Lemma 3.3.2, and set µ = min{-r, 6/L}. Then for all r E (0, ft) and each p = 1, 2,... we obtain by this Lemma

Ur

p(z,

<

z E B(a, R).

-b,

p

P

Hence, for all k = 1, ... , p we have

If (z) - f (J(kp(z) < E,

z c B(a, R).

In addition, it follows by the definition of the resolvent that for all w E 0

J,/P(w) - w =

rf

(Jr/P(w)).

Now using the triangle inequality we estimate Z - J(P)(z)

f (z) -

1 rf(z) - z + J(P) (z)I

r 1

P

rE k=1 <

1

V

P

E

r pf(z) - Jr/P 1) (z) + J*/P(z) r

k=1 p P r 1:

T

k=1

p

1)(z))

IP

- Jr/P(z)

f(z) - f(Jr/p(z)) < E ,

Chapter III

74

whenever z E B(a, R). So, (3.3.19) is proved. In turn (3.3.19) implies (3.3.20): Jr(z) - J /p(z)I

< <

z - Jr/)(z) - rf(z) + Irf(z) - z + Jr(z)I

r f(z) -

r r/P + rIf(z)

- f(Jr(z))I < 2re,

and we have completed the proof.

Lemma 3.3.4 (The resolvent identity) Let f be a continuous function in A which satisfies the range condition (RC). Then for 0 < s < t the following resolvent identity holds: Jt(z) J9(tz+(1-t)it (z)), zE

Proof. Since for each z E A and t > 0 the element Jt(z) E A we have

w=sz+(1-t)Jt(z) ED

(3.3.21)

by the convexity of A. It follows by the definition of the resolvent that

z - Jt(z) = tf(Jt(z))

(3.3.22)

J9(w) + s f (J9(w)) = w.

(3.3.23)

and

On the other hand (3.3.21) and (3.3.22) imply

w = Jt(z) + t (z - Jt(z)) = Jt(z) + sf (Jt(z)) Since equation (3.3.23) has a unique solution we have the equality Jt(z) = J9 ('w),

which is equivalent to the resolvent identity. Now we are able to complete the proof of Proposition 3.3.1. As we mentioned above the necessity of the assertion of this Proposition follows from Lemma 3.3.1.

To prove the sufficiency we first show that for all t > 0 and z E A the limit in (3.3.8) exists. In fact, it is enough to prove that for each t > 0 and z E A the sequence {Jt7n(z)} is a Cauchy sequence in the Poincare metric on A. Indeed, fix any t > 0 a E A and consider the sequence {zj}l=1 C A defined as follows

zt =

(a),

k(l) < 1.

(3.3.24)

For an arbitrary s E (0, 1) close to 1, one can find to such that elements

wt=a+if(a)EO,={IzI<s<1},

GENERATION THEORY

75

whenever l > 1. Also, for such 1 we have Jt/l(wt) = a. Then as in Lemma 3.3.2 we obtain that there is d = d(s) > 0 such that for all 1 > to p

(Jtk('))(a),a)

<

k(1)P (Jt/t(a), a) = k(l)p (Jt/i(a), Jt/t(wj))

<

k(1)P(a,w1) < k(l)Iwt -aI .

l)tIf(a)l.

k

1

d

1-s

ds2
2

ds2

1

Setting now

l-1 2 ..

R=max{ a z we obtain that the sequence {zj}1

1

tIf(a)j1

d'2

}

defined by (3.3.24) is contained in B(a, R).

Using this fact we will show now that the sequence {J(t/n ')(a)} is a Cauchy

sequence in Poincare metric on A. Taking any e > 0 and z E B(a, R), define u > 0 as in Lemma 3.3.3. Then we

have by this lemma that there are no > 0, mo > 0 and L > 0 such that for all

n>no and m>mo p

(JJ/n(z), Jt/,,.L(z)) <

P

(Jt/m(z), Jt/n m.(z)) <

2Lst Ti

and

2L et n

whenever z E B(a, R). Using these inequalities we obtain after several manipulations with the triangle inequality that P

(Jt/n(a), Jt/.(a)) < 4Lte,

as required. Since a is an arbitrary element of A we have that the limit in (3.3.8) exists and the mapping Ft : A H A defined by this formula is a p-nonexpansive mapping on A.

Now we have to establish the continuity of {Ft(z)}t>o, for each z E A. By using (3.3.8) it is enough to prove continuity of the resolvent Jr(z) for r sufficiently small.

To this end let us choose 0 < t < r < r for any a E A and R > 0 as in Lemma 3.3.1. Then we have for such r, that p(JJ(a), a) < R and by the resolvent identity (Lemma 3.3.4) we obtain

(Jt(a+ P(Jt( a),Jr(a)) = pr
a+

(r as r - t -+ 0.

r

r

t

Jr (a)), Jt (a)

\ I

t Jr(a), a ) < p(a, a - (r - t) f (Jr.(a)) -+ 0

Chapter III

76

The semigroup property of S = {Ft}t>o can be proved in a standard way (see,

for example, [29]), using (3.3.8) and passing from rational t > 0, s > 0 to real numbers by the continuity of {Ft}t>o. Now using (3.3.19) (see Lemma 3.3.3) and again (3.3.8) one can easily show that for each z E A z - Fc(z) lim = f (z) to+ t Thus f : A H C is the infinitesimal generator of the flow S = {Ft}t>o, defined by (3.3.8). Finally, the uniqueness of this flow follows by Proposition 3.2.2, and the proof is complete.

Remark 3.3.1 By QNp(0) (respectively, Q Hol(0)) we will denote the set of all continuous (respectively, holomorphic) functions on A which are generators of one-parameter semigroups (flows) of p-nonexpansive (respectively, holomorphic) self-mappings of A. A direct consequence of the above Proposition and Lemma 3.3.1 is that these sets are real cones, i.e., if f and g belong to QNp(A) (respectively, 9Hol(0)) then so does the function

h = of +13g for each pair of nonnegative numbers a and,3.

Indeed, let {Ft}t>o, {Gt}t>o be the flows generated by f and g respectively. Then if we define the family {Ht}t>o by

Ht(z) = FFt(Got(z)) we have that lim h(z) = of (z) + Og(z) = t-.o+

z - Ht (z)

t

for each z E A, and we are done. This fact can be also established by using representation theory of generators or the so called flow invariance conditions (see following sections). Moreover, we will see below that the sets QNp(A) and 9 Hol(z) are closed with respect to the open compact topology on A.

Exercise 1. Prove directly that if f,, c 9 Hol(A) is a convergent sequence on each compact subset of 0, then its limit function f also belongs to 9 Hol(A). Hint: Use Proposition 3.2.2 and the properties of the solution of the Cauchy problem (*) (see section 3.2). At the end of this section we will give another important consequence of Proposition 3.3.1, which will be used in the sequel.

Corollary 3.3.1 Let F : A H A be a p-nonexpansive (respectively, holomorphic)

self-mapping of A. Then f = I - F (i.e., f (z) = z - F(z)) belongs to 9NP(0) (respectively, 9 Hol(A)).

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77

Proof. We have to show that for each r > 0 and z E A the equation

w + r(w - F(w)) = z

(3.3.25)

has a unique solution w = Jr(z) in A. Indeed, setting t = r/(r + 1) we can rewrite (3.3.25) in the form

w = tF(w) + (1 - t)z.

(3.3.26)

It was shown in Section 2.4 that for each t E [0, 1) equation (3.3.26) has a unique solution w = Gt(z) and that for each t E [0, 1) the mapping Gt : A H A is a p-nonexpansive self-mapping of A (see Proposition 2.4.1). In addition, if F E Hol(A), then so is Gt (Remark 2.4.1). Thus the function f = I - F) satisfies the range condition and its resolvent Jr : A --* A, r > 0 is defined by

Jr(z) = Gr/(1-r)(z).

(3.3.27)

This completes the proof.

Note, by the way, that the family of resolvents {Jr(z)}r>o at the point z E A for f = I - F is, in fact, the rescaling approximating curve of F at this point. Thus the question whether the mapping I - F belongs to CNP(O) whenever F E Np(O) has been answered in the affirmative. In its turn, this fact implies another useful resolvent identity formula for the general case when f E CNP(O).

Lemma 3.3.5 Q 116j) Let f be a continuous function in A which satisfies the range condition (RC), i.e., for each s > 0 the resolvent J., = (I + s f)-1 is well defined on A and belongs to NP(O). Then for each pair s, t > 0 the mapping Gt := (I + t(I - Js))-1 is also well defined on A and belongs to Np(O) and the following identity holds:

J(t+1)s = Js o Gt = J9 o (I + t(I - Js))-1

(3.3.28)

Proof. The existence of the resolvent Gt := (I + t(I - Js))-1 follows directly by Proposition 3.3.1 and its Corollary 3.3.1. The same assertions imply that Ct E NP(O). By definition this mapping satisfies the identity:

(I + t(I - J,))(Gt(z)) = z, z E A.

(3.3.29)

Reminding that I - Js = s f (Js) we obtain Gt(z) + stf (J8(Gt(z)) = z, z E A

for allzEA. At the same time rewriting (3.3.29) in the form:

Ct(z) =

1 Z' z E 0 t J s (Gt(z)) + 1+t 1+t '

(3.3.30)

Chapter III

78

we have by (3.3.30) the following identity

J9 (Gt(z)) + (1 +t)sf (J9 (Gt(z))) = z

(3.3.31)

for all z E A. Since the equation

w+(1+t)sf(w)=z, zEA, has a unique solution w = (I + (1 + t)sf)-1 (z) := J(l+t)9(z), provided that (3.3.31) holds, (3.3.28) results.

Here follows the basic assertion which demonstrates the resolvent method.

Proposition 3.3.2 Let f be a continuous function in A which satisfies the range condition (RC) and let Jr = (I + r f)-1 E NP(O), r > 0, be its resolvent. Then: (i) for each r > 0 the sets Fix(Jr) and Null (f) in A coincide; (ii) for each z c A the net {Jr(z)}r>o is convergent as r - oo. Moreover,

(a) if W = Null(f) in A is not empty, then the limit mapping F = lim Jr is r-oo

a p-nonexpansive retraction on W;

(b) if W = Null(f) in A is empty, then the limit mapping F = limr_ Jr is a unimodular constant ( which is the sink point for each Jr, r > 0. In other words, for each K > 0 the horocycle

D((,K)zEA:W((z):= internally tangent to aA at

1

I1-I I22


is Jr-invariant, i.e.,

Jr (D ((, K)) c D ((, K)

.

Proof. Assertion (i) follows directly by the definition of the resolvent and the uniqueness of the solution of equation (3.3.11). Thus for each r > 0 we have

W := Null(f) = Fix (Jr). Consider now the mapping Gt :_ (I + s(I - Js)-1) that was introduced in the previous lemma. By the construction of this mapping we have, in turn, that for each pair s, t > 0 W = Fix (Js) = Fix (Gt). Moreover, it follows by Propositions 2.4.3 and 2.4.4 that for each z E A there exists the limit

F(z) = lim Gt(z) E S. More precisely, if W = Null(f) in A is not empty, then t o0 by Proposition 2.4.4 F is a p-nonexpansive retraction on W. Otherwise, F is a unimodular constant ( which is the sink point for J3. But in both cases equation (3.3.29) implies:

J., (Gt(z))-Gt(z)-*0ast->oo for all z e A. Setting now r = (1 + t)s and letting t to oo we obtain by Lemma 3.3.5 that F = lim Jr. Finally, condition (b) follows from the resolvent identity r-oo (Lemma 3.3.4) and we are done.

GENERATION THEORY

3.4

79

Monotonicity with respect to the hyperbolic metric

Our main goal in the following considerations is to find analytical characterizations and parametric representations of the classes CN,(A) and 4 Hol(O), respectively. It turns out, that a good tool in these investigations is the notion of monotonicity with respect to the hyperbolic Poincare metric on the unit disk.

To motivate the definition below, we recall that a mapping f : R2 ,

R2 is

said to be monotone with respect to the Euclidean norm of R2 if for each x, y in a domain of definition f (x - y,f(x) - f(y)) ! 0,

(3.4.1)

where by we denote the inner scalar product in R2. Since each complex-valued function f : C --> C can be considered as a mapping from R2 into itself, condition

(3.4.1) can be rewritten in the form:

Re [(f (z) - f (w))(z - w)] > 0

(3.4.2)

for each pair z, w in a domain of definition of f. At the same time it is easy to see that (3.4.2) is equivalent to the following condition:

Iz + rf (z) - (w + rf (w)) I ?Iz - wl

(3.4.3)

for all r > 0. The latter condition is a key to define the notion of monotonicity with respect to the Poincare hyperbolic metric on A (see [116]).

Definition 3.4.1 Let f be a complex valued function on A, and let p be the Poincare metric on A. The function f is called p-monotone (monotone with respect to the metric p) if for each pair z, w E A the following condition holds: p(z + rf (z), w + rf (w)) > p(z, w)

(3.4.4)

for all r > 0 such that z + r f (z) and w + r f (w) belong to A.

Proposition 3.4.1 Let f : A '--+ C be a continuous function in A. Then f is p-monotone if and only if it satisfies the range condition (RC).

Proof. Let f satisfy the range condition (RC), i.e., for all r > 0 the nonlinear resolvent Jr = (I + r f)-' is a well defined p-nonexpansive self-mapping of A:

p ((I + rf)-'(u), (I + rf)-1(v)) < p(u, v)

(3.4.5)

for all u, v E A. Take now any pair z and w in A and let r > 0 be such that z + r f (z) := u and w + r f (w) := v belong to A. Then by definition

z = (I +rf)-1(u) and w = (I +rf)-1(v).

Chapter III

80

It is clear now that (3.4.5) implies (3.4.4), i.e., f is p-monotone. Conversely. For a pair z, w E 0 denote u = z + f (z) and v = w + f (w). Then for r > 0 sufficiently small (3.4.4) implies:

p ((1 - r)z + ru, (1 - r)w + rv) > p(z, w).

(3.4.6)

Denoting the left hand of (3.4.6) by O(r) we can rewrite it as 0 (r) > 0 (0) .

Now it follows by Proposition 2.3.2 that the latter inequality is equivalent to the condition: Re

(z-u)z+(w-v)w >Rez(w-v)+w(z-u) 1-Iz12

1-1w12 ]

-

1-zw

Substituting u = f (z) + z and v = f (w) + w into this inequality we obtain a characterization of a p-monotone function: Re

f (z)z + f (w)w > Re f f ('w) + w f (z) 1 - w12] 1-zw

1-Iz12

-

(3.4.7)

Substituting now z = 0 into (3.4.7) we obtain the condition: Ref (w)w > Re f (0)w(1 - Iw12).

(3.4.8)

for all w E i. Now we will show that condition (3.4.8) implies the solvability of the equation: w + r f (w) = z

(3.4.9)

for each r>0andzEL A. To this end we will establish a more general assertion which we will call the numerical range lower bound (cf., [60] and [62]).

Lemma 3.4.1 Let a :

[0, 1]

--* R be a continuous function on [0, 1] such that

a(0) < 0 and the equation

s + ra(s) = t has a unique solution s = s(t) E [0, 1) for each t e [0, 1) and r > 0. Suppose that f : 0 --> C be a continuous function on 0 which satisfies the following condition: Ref (w)w > a (Iwl) Jwj, w c A.

(3.4.10)

Then for each z E 0 and r > 0 equation (9) has a unique solution w = w(z) such that w(z)I < s(t), (3.4.11) whenever jzj < t < 1.

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81

Proof. Fix t E (0, 1) and z E A, I zI < t < 1, and consider the equation:

s+ra(s) = IzI. It follows from our assumption that this equation has a unique solution so = so (I z1). Then setting y(s) = s+ra(s) - I z I we have y(0) < 0 and y(so) = 0. Note that y must be monotone on [0, 1]. Hence, for an arbitrary 0 < b < 1 - so we can find e > 0 such that y(so + 6) > E. Taking now w E A such that jwj = so + 6 we have by (3.4.10) for those w: Re(w + rf (w) - z)w > 1w12 + ra (IwD) iwi - iwi jzj I wI -Y (Iwi)

- IwIE > 0.

Then it follows by the Bohl-Poincare theorem (see, for example, [79]), that equation (3.4.9) has a unique solution w = w(z) such that lwl < so + J. Since S is an arbitrary sufficiently small number, we must have jwj < so < s(t). To complete the proof of Proposition 3.4.1 we first note that the function

a(s) _ - If(o) I (1- s2) satisfies the conditions of the above Lemma. Therefore inequality (3.4.8) and this Lemma imply that for each r > 0 the resolvent Jr, defined by Jr(z) = w(z) - the solution of (3.4.9) is a single-valued self-mapping of A. It remains to show that

this mapping is p-nonexpansive on A. Indeed, for a pair u and v in A, setting z = Jr(u) and w = Jr(v) we obtain z + r f (z) = u and w + r f (w) = v. Since f is p-monotone we obtain finally p(u, v) = p (z + r f (z), w + r f (w)) >_ p(z, w) = p (Jr(u), Jr(v))

The Lemma is proved. We already mentioned in the proof of Proposition 3.4.1 that, in fact, condition

(3.4.7) is equivalent to the property of a continuous function f on A to be pmonotone. Thus combining these assertions with Proposition 3.3.1 and Lemma 3.3.1 we can formulate a summary assertion for this chapter which characterizes the property of a continuous function on A to be in class ggNP(O).

Proposition 3.4.2 Q116]) Let f be a complex-valued continuous function on A. Then the following conditions are equivalent:

(i) f E gNp(O), i.e., it is a generator of a continuous flow S = {Ft}t>o of p-nonexpansive self-mappings of A;

(ii) there is a family {Gt} 0 < t < T, of p-nonexpansive self-mappings of A such that z - Gt (z) f (z) = lim t-o+ t for each z E A;

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82

(iii) the Cauchy problem at

+ f (u(t , z)) = 0 ,

u(0,z)=zEO has a unique solution u(t, z) E A for all t > 0 and z E 0, such that for each t > 0 u(t, ) E NN(A); (iv) f satisfies the range condition (RC), i.e., for each r > 0 and z E A the equation

w+zf(w) = z has a unique solution w = Jr.(z), such that Jr E Np(0); (v) f is a p-monotone function on A; (vi) f satisfies the condition: Re

f (z)z + f (w)'w > Re z.f (w) + w.f (z) 1- wl2 1-2w

1-Iz12 for each pair z, w in 0;

Remark 3.4.1 Condition (vi) plays a crucial role in our further considerations. Inequalities of such a type (which characterize the classes of generators of flows) are called flow invariance conditions. For the class of holomorphic mappings a simpler inequality (3.4.8) is also a flow invariance condition, since it characterizes the class

of holomorphic generators. Indeed, as we saw in the proof of Proposition 3.4.1 (or in Lemma 3.4.1) this condition is sufficient for the existence of the (nonlinear) resolvent (I+r f)-1 which maps A into itself. In addition, it follows by the Implicit Function Theorem that this mapping is holomorphic (hence, p-nonexpansive) in 0, so f belongs to g Hol(A). The necessity of this condition follows directly from condition (vi). Thus for f c Hol(A, C) inequalities (3.4.7) and (3.4.8) are equivalent.

We will see below that these conditions can be considered as forms of the Schwarz-Pick inequalities for the classes gNP(o) and g Hol(0) of generators of flows of p-nonexpansive and holomorphic mappings, respectively. A geometric nature of these conditions for holomorphic functions will be explained in the next section. Here we note that an immediate consequence of these flow invariance conditions is the following (cf., Remark 3.3.1):

Corollary 3.4.1 The sets gNp(A) and g Hol(A) are closed (with respect to the topology of uniform convergence on compact subsets of A) real cones.

Exercise 1. Show that the set g Hol(A) n (-g Hol(A)) is precisely the set of all generators of one-parameter groups of automorphisms of A. Hence, this set is a real vector space.

Exercise 2. Describe the set gNp(A) n (-gNp(A)).

GENERATION THEORY

3.5

83

Flow invariance conditions for holomorphic functions

In this section we study several flow invariance conditions for the class of holomor-

phic functions. We will use these conditions to obtain parametric representation of functions of the class g Hol(O), to study their dynamic transformations and the asymptotic behavior of the flows generated by them. In the first step we give a simpler explanation of the necessity of the flow invariance condition (3.4.8) for a function f E 9 Hol(O) and later we study it in greater detail. In addition, we will see below that this condition can be improved by a more qualified condition (see Proposition 3.5.3) which has some additional applications. Also, note that in the case of holomorphic functions one uses a special terminology which comes from the theory of bounded symmetric domains (see, for example, [142, 12, 32]).

Definition 3.5.1 A function f E Hol(O, C) is said to be a semi-complete vector field on A if the Cauchy problem (*): at

+ f (u(t z)) = 0 , ,

u(0,z)=zEA,

(*)

has a unique solution u(t, z) E A for all z c A and all nonnegative t, i.e., t E R+ = [0, oc).

If the Cauchy problem (*) has a solution u(t, z) defined for all real t, i.e., t E 1l = (-oo, oo), then f is said to be complete (or integrated) (see, for example, [32, 142). Thus f is semi-complete if and only if it is an infinitesimal generator of a one-parameter semigroup; f is complete if and only if it is a generator of a oneparameter group. Exercise 1. Show that the mapping f : A H C, defined as f (z) = z - z2 is a semi-complete vector field on A.

Exercise 2. Show that f : A H C, defined by f (z) = z - 1 + 1 - z is a semi-complete vector field. Find the flow generated by f.

Exercise 3. Show that for each a E C the mapping f : f (z) = a - az2 is a complete vector field.

We begin with a characterization of all complete vector fields. Note again that if f is a complete vector field, then it generates a one-parameter subgroup S = {Ft}tER of the group Aut(o) of all automorphisrns of A. Consequently, each

Chapter III

84

Ft E S is a fractional linear Mobius transformation. Thus Ft has a holomorphic continuation on a neighborhood of 0 and so does f, because of the equality:

f = lim

to+

I - Ft

(3.5.1)

t

The family of complete vector fields on A will be denoted by aut(o). As we mentioned above, this family is a real vector space, which can be described as follows.

Since I Ft(z)I < 1, for z E 0 we have by (3.5.1) that:

Ref(z)2>0 for allze8E.

(3.5.2)

At the same time the function -f is also a complete vector field on A. Therefore we also obtain that

-Ref(z)z>0 forallzE80.

(3.5.3)

Comparing (3.5.2) and (3.5.3) we obtain the necessary boundary condition for

f E aut(o):

Ref(z)z=0 forall zEBA

(3.5.4)

(see Figure 3.1)

f(z)

Figure 3.1: Boundary condition for f E aut(o).

Actually, this condition is also sufficient for f E Hol(A, C) to be complete. Indeed, suppose that f E Hol(2K, C) and satisfies (3.5.4). Rewriting f in the Taylor series form: f(z) = ao + a1z + a2z2 + ...

we have for zEa

:

Re f(z)z = Reg(z) = 0,

GENERATION THEORY

85

where

g(z) = al + (ao + a2)z + a3z2 + ...E Hol(A, C). It follows by the maxi mum principle for harmonic functions that: R eal

-ao+a2=a3=...=0.

Hence f (z) is actually polynomial of the second order at most:

f (z) = ao + alz + a2z2 with

To-=-a 2, and

Rea l =0 .

3.5.5)

(3 . 5 . 6)

Now it can be shown by direct computation that the Cauchy problem: au(t, z) + f (u(t, z)) = 0, at u(0, z) = z E A, with f satisfying (3.5.5) and (3.5.6) has a unique solution u(t, ) E Hol(A) for all t E R. Moreover, for fixed t E R the mapping Ft = u(t, ) is a Mobius transformation of the unit disk. So, we have proved the following result (see, for example, (16, 8]).

Proposition 3.5.1 (Boundary group invariance condition) A mapping f E Hol(O, C) is a complete vector field on A (i.e., f E aut(o)) if and only if it has a continuous extension to A and

Ref(z)x=0 for allzEas. This is equivalent to the statement that f is a polynomial of the second order at most:

f(z) = ao + alz + a2z2 with coefficients ao, al, a2 which satisfy the conditions:

Real = 0,

'do = -a2-

Corollary 3.5.1 The family aut(o) is a real vector space of entire functions. Moreover, this space has the following decomposition:

aut(o) = auto(s) ® P2, where

auto(s) = If E aut(s) : f (0) = 0} = If E Hol(s, C) : f (z) = bz, Re b = 0} is the subspace of linear functions, and

P2 = If E aut(A), f'(0) = 0} _ {f E Hol(A, C) : f (z) = a - az2, a E (C} is the subspace of so called `transvections'.

Chapter III

86

Now we will turn to the general case when f E Hol(A, C) is a semi-complete vector field on the unit disk. To explain the nature of the condition (3.4.8) we adduce first some heuristic grasps. Assume temporarily that f E G Hol(0) isholomorphic in the neighborhood of A. In this case we will write just f E C). Hol(,_,

Then, again we have the following boundary flow invariance condition:

forall

zEai

(see Figure 3.2).

Figure 3.2: Boundary flow invariance condition.

It implies that

Re (f (z) - f (0))z > -Ref (0) z,

z E ao.

Dividing the left hand side of this inequality by Iz12 = 1 we obtain:

Re(f(z)f(0) I >-Ref(O)z, zEa&. z / Now again it follows by the maximum principle for harmonic functions that the latter inequality holds also for z c A. Multiplying it by Iz12 # 0, z c A, we obtain:

zEA.

(3.5.7)

However, there are holomorphic mappings on the unit disk which are semicomplete, but have no holomorphic extension to A, the closure of A. Consider, for example,

f(z)=z-1+ 1-z.

Nevertheless, we will see that even f E Hol(A, C) does not extend continuously to

condition (3.5.7) is necessary and sufficient for f to be in 9 Hol(A). The

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87

necessity can be shown also by the following simple considerations which are useful, however, to obtain a parametric representation of the class G Hol(O). As we already

know this class is a real cone. Therefore, if we present f E G Hol(A) in the form: f(z) = g(z) + h(z),

(3.5.8)

where g(z) = f(0) - f(O)z2

is a transvection (i.e., g E P2, see Corollary 3.5.1), we have

h(z) = f (z) + (-g(z)) E G Hol(A)

(3.5.9)

h(0) = 0.

(3.5.10)

and

Now, conditions (3.5.9) and (3.5.10) imply that there is a semigroup Sh, = {Ht}t>o of holomorphic self-mappings Ht of A, such that Ht(0) = 0, for all t > 0. Hence by the Schwarz Lemma we have: lHt(z)l < Izl,

for all z E A.

Since

h(z) = lim 1(z - Ht(z)) t-.o+ t

we obtain:

Reh(z)z>0 forallzEA.

(3.5.11)

Reg(z)z = Ref (0)z (1 - Iz12).

(3.5.12)

In addition, note that Then by (3.5.11), (3.5.12), and (3.5.8) we obtain (3.5.7) and the necessity of this condition is proved. The sufficiency of condition (3.5.7) for f E Hol(A, C) to be a semi-complete vector field was established in Lemma 3.4.1 (see, also, Remark 3.4.1). However, for the case of holomorphic functions one can make this lemma more precise.

Lemma 3.5.1 Let a E 9Ho1(O) be such that a(IzI) E 118, z E A, and let f E Hol(O, C) satisfy the following condition:

Ref(z)2 > a (jzj) Izi,

z E A.

Then:

(i) f is a semi-complete vector field on A; (ii) if S f = { Ft }, t > 0, is the semigroup generated by f, then for all t > 0

and xEA IFt(x)I < at (HxH)

where /3t is the solution of the Cauchy problem: d-/t(s)

dt

+ c(at(s)) = 0,

00(s) = s,

s E [0, 1).

Chapter III

88

Proof. It follows by Proposition 3.3.1 that the function a : A ti C satisfies the range condition, i.e., the equation: w + ra(w) = z has a unique solution w = (I + ra)-1 (z) for each r > 0 and z E A. Since on the interval [0, 1) the function a is real-valued, it is not difficult to see that s = (I + ra)-1 (t) E [0, 1) whenever t c [0, 1). It then follows by Lemma 3.4.1 that for a fixed r E (0, 1) and z E A the equation: w + r f (w) = z has a unique solution w = (I + r f)-1 (z) and

(I

+rf)-1

(z) < (I + ra)-1 (IzI).

Using again Proposition 3.3.1 and the exponential formula we obtain our assertion. 11

Now, observing that the function:

a(z) = - If(0)l (1 - z2) satisfies all the conditions of Lemma 3.5.1 we obtain the following result:

Proposition 3.5.2 (see [7]) Let f E Hol(A,C). Then f is a semi-complete vector field if and only if it satisfies condition (3.5.7): Ref (z)z > Re f (0) z(1 - Iz12)

for allzEA. Moreover, if Sf = {Ft},

t > 0, is a semigroup generated by f, then the

following estimate of growth holds: IF,t(z)I

IzI + 1 - e-21f(0)It(1 - II) z Izi + 1 + e-2If(0)It(1 - IzI)

(3.5.13)

Remark 3.5.1 Note that if equality in (3.5.7) holds for at least one value z0 E A then it holds for all z E A, and f (z) is actually a complete vector field. Indeed, by (3.5.8) we have:

f =g+h,

where g c aut(o), and h E Q Hol(A), with h(0) = 0. Therefore if we present h in the form

h(z) = z p(z), we obtain that p E Hol(A, C) and

Rep(z)>0,

zEA.

(3.5.14)

Thus (3.5.7) is equivalent to the following equation f(z) = f(0) + zp(z) - f(0)z2

If now for some zo E A we have equality in (3.5.11) we also have the equality Rep(zo) = 0. It then follows by the maximum principle that p(z) = p = const, hence f(z) has the form (3.4.5) 2 f(z) = f(0) + ipz - f(0)z,

GENERATION THEORY

89

i.e., it is a complete vector field. As a matter of fact (3.5.14) enables us to find a more qualified estimate which also characterizes semi-complete vector fields. To do this we need the following assertion.

Lemma 3.5.2 (Harnack inequality) If p E Hol(O, C) maps A into the right half-plane, i.e.,

zEA,

Rep(z)>0, then it satisfies the strong estimate:

+ 1z' Rep(0)1 - Izl < Rep(z) < Rep(O)

1+jzj

(3.5.15)

IZI

Proof. If Rep(O) = 0, then as above Rep(z) = 0 for all z E 0 and (3.5.15) is obvious. If p E Hol(O, C) satisfies the strong inequality in (3.5.14), then the function pl: p1(z) = Rep(O) [p(Z) - i IM p(0)] belongs to the so called class of Caratheodory:

Repl(z) > 0, z E A

and

pi(o) = 1.

Since the fractional linear transformation: G(w)

(3.5.16)

w+1

maps the right half-plane {w E C : Re w > 0} into the unit disk (see Exercise 4 bellow), we have that the mapping F: F(z) = G(pi(z))

is a holomorphic self-mapping of 0 and F(O) = 0. Then the Schwarz Lemma implies that for each 0 < r < 1 and z E 0 : jzl < r the following inequality holds: pi(z) - 1

< r.

p1(z) + 1

Now it is easy to see that the circle

w

r is symmetric with respect to

w+

the real axis and intersects it at the points 1 + r and 1 - r (see Figure 3.3).

1-r

l+r

Therefore we obtain the following inequality for each function p1 of the class of Caratheodory: 1 - IZI

l + IZI

< Repl(z) < 1 - IzI l + IZI -

Chapter III

90

Figure 3.3: Values of functions of Caratheodory's class. Since

we obtain (3.5.15).

Exercise 4. Prove that for all w such that Re w > 0 (w E R+) the following inequality holds:

w-1 <1. w+1

Exercise 5. Prove the equivalence of the following assertions: (a) a function p E Hol(0, C) has values (for all z E 0) in a compact subset of the open right-half plane {w E C : Re w > 0}; 1 - w(z)

(b) p(z) =

zEO.

1 + w(z)

, where w E Hol(A) is such that lw(z) l < c < 1 for all

Exercise 6. Show that under the conditions of Exercise 5 the function p satisfies the strong Harnack inequality: Re p(0)1 + Cizi > Rep(z) > Rep(0)1

+ cIzI'

0
Finally, observe that from the representation of f c G Hol(0):

A z) = AO) - f(0)z2 + z p(z), we deduce that:

p(O) = f'(0) Thus we have proved the following assertion.

Rep(z) > 0

(3.5.17)

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91

Proposition 3.5.3 ([6]) A function f E Hol(O, C) is a semi-complete vector field on A if and only if Re f'(0) > 0 and the following inequality holds: z1z(1 + llzI)

Re f'(0)

+ Ref (0)2(1 - 1z12) > Ref (z)2 z

> Re f (0)2(1 - 1z12) + Re f'(0) zl 1(+ zizl)

(3.5.18)

Moreover, the equality in (3.5.16) holds if and only if Re f'(0) = 0.

Corollary 3.5.2 ([7]) A mapping f E 9Hol(O) belongs to aut(o) if and only if

Ref'(0)=0. Corollary 3.5.3 ([7]) If f E G Hol(A) is given with f (0) = f'(0) = 0, then f - 0.

This assertion can be considered as a tangential version of the Schwarz Lemma.

Indeed, if F E Hol(A) is a holomorphic self-mapping, then f = I - F is a semicomplete vector field. Therefore, if F(0) = 0 and F'(0) = 1, then f (0) = f'(0) = 0. Therefore, by the Corollary 3.4.2 we obtain that F(z) = z. This is the statement of the second part of the Schwarz Lemma. Recently M. Abate [5] established another condition, which characterizes a semi-complete vector field f by using the estimate for its derivative f'. To establish his condition we first prove the following characterization of the class P := {p E Hol(A, C) : Rep(z) > 0, z c Al.

Lemma 3.5.3 ([7]) Let p E Hol(O, C). Then condition (3.5.14): Rep(z) > 0, holds for all z E A, if and only if there is a positive function 0 : [0, 1) H R+ such that : Re (zp'(z) + '(Izl)p(z)) > 0 (3.5.19)

for allzEO. Proof. Let p E Hol(O, C) satisfy (3.5.14). Define F = (p - 1) (p + 1)-' which maps 0 into itself, F E Hol(A). Applying the Schwarz-Pick Lemma to F we obtain the inequality: lp + 112 - lp - 112 < 11 +p12 - Ip+ 112(1 - Iz12) 2lp'l

Pp

which implies:

2 Rep(z)

1p(z)1- 1 -Iz12

Chapter III

92

Consequently,

Re( -z p( z)) <-

z < _ Iz p()

2I1 Rep(z)

<

1 1

+

2

Izl2

Re p(z).

2

Setting here V)(t) = 1 + t2 we obtain (3.5.19). Conversely. Suppose that (3.5.19) holds with a positive V& : [0,1) --+ R+ Setting z = reie we have: zp (z) = rap Or

and (3.5.19) becomes:

(Re (r p I + 0(r) Re p(z))

> 0,

z = re`o E A.

(3.5.20)

Assume that there exists zo = roeze0 in A such that Rep(zo) < 0.

But (3.5.20) implies that Re p(0) > 0, hence there is rl E [0, ro) such that Rep(rle`B°) = 0 due to continuity. Then one can find r2 E (rl, ro) such that

Rep(r2e'°°) < 0 and

Re ap (r2eie0) < 0.

But these inequalities contradict (3.5.20). Thus it follows that Rep(z) > 0 everywhere and we are done. 2

Now it is easy to verify that condition (3.5.19) with li(r) = 1 + r2 is equivalent to the condition: Re [2f (z)z + f'(z)(1 - Iz12)] > 0. Thus we can summarize the assertions of this section in the following result.

Proposition 3.5.4 Let f e Hol(A, C). Then the following are equivalent: (i) f c 9 Hol(A, C), i.e., f is a semi-complete vector field on A; (ii) Ref (z) z > Re f (0)z(1 - Iz12);

(iii) Re f'(0) > 0 and Re f(0)z(1 -

zI2) + f'(0)IzI21 - IzI

> Ref (z) z Re

[f(o)z(1- I2I2) +f'(o)IzI2 1

(iv) Re [2f (z)z + f'(z)(1 - IzI2)] > 0;

+

II] IzI

GENERATION THEORY

93

(v) f (z) = f (0) - W z' + z p(z), with p c Hol(O, C), Rep(z) > 0. Moreover, if for some zo E A the equality in one of the conditions (ii) or (iv) holds, then it holds for all of these conditions and for all z E A. In this case the function p(z) in (v) is constant and f is actually a complete vector field.

Remark 3.5.3 The class of functions of the form: h(z) = z p(z), where Re p(z) > 0, z E A usually referred to as class Al. The class M consists of all elements of Al which are not linear functions, i.e., M = {h E Hol(O, C) : h(z) = z p(z), Re p(z) > 0, z E A}.

Note that Al = auto(O) ® M. Thus condition (v) of the above proposition means that the class of the semi-complete vector fields on A admits the following decompositions:

G Hol(O) = P2 + N = aut(o) ®M.

(3.5.21)

(see Corollary 3.5.1).

In addition, it is well known (see, for example, [57] and [122]) that for each holomorphic function p on A with values in the closed right half-plane II+ (i.e., Rep(z) > 0) there exists a positive increasing finite function pp on the unit circle aA, such that

P(z) =

fl+Zcz dp (C) + ib

(3.5.22)

1

ao

with some real b. This formula is called the Riesz-Herglotz representation of functions in P = {p E Hol(O, C) : Re p(z) > 0}. It establishes a linear one-to-one correspondence between the set of all positive measures on ao and P. We will call the function µ. : OA i-+ R the measure characteristic function for p E P. Thus by (3.5.21) and (3.5.22) we have the following integral parametric representation for f E G Hol(O) : f (z) = a - az2 + izb + J z 1 + z ao

1-zt;

(3.5.23)

where a E C, b E R and µ is a positive function on OA. Another parametric representation of the class G Hol(O) which is determined by the location of null points of f E G Hol(O) is due to E. Berkson and H. Porta [17]. We will give it in the next chapter.

Exercise 7. Let h E Hol(O, C) belong to class N : h(z) = z p(z), Re p(z) >

0, zEA.

(a) Show directly that the Cauchy problem: au(at, z)

+ h (u(t, z)) = 0,

u(0,z)=zEA,

3.5.24 (

)

Chapter III

94

has a unique solution u(t, z) E 0 for all t > 0, and u(t, 0) = 0, t > 0. (b) Show that lim u(t, z) = 0

t

00

if and only if h E M, i.e., Rep(z) > 0, zEO. (c) Show that if h E M with h'(0) = 1, then the limit lim etu(t, z) := F(z)

(3.5.25)

t-.oo

exists for all z E 0 and F E Hol(0, C). Hint: Define the function pi E M by pi (z) = 1/p(z) and show that the Cauchy problem (3.5.24) is equivalent to the following integral equation (t,z)

In letu(t, z)] = In z -

(pl

U

1)

d(

which implies /'z

lim etu(t, z) = zexp(J (pl(y) - 1) t-.oo

).

(3.5.26)

(d) Show that F E Hol(0, C) defined by (3.5.25) (or (3.5.26)) satisfies the following inequality: Re

[zF(z)]

> 0.

(3.5.27)

(It is well known due to R. Nevanlinna [102] that the latter inequality characterizes all univalent F functions on 0, normalized by F(0) = 0, F'(0) # 0, whose image F(0) is starlike with respect to zero. See Chapter 5.)

Exercise 8. Show that the function f, defined by:

h(z) = z1 - zeie 1

zeie'

0 E 0' 27f ] [

belongs to M, and h'(0) = 1. Find explicitly the function F(z) in (3.5.25). 00

Exercise 9. Show that if the function f defined as: f (z) = ao + al z +

ak zk k=3

is a semi-complete vector field on 0 and Real = 0, then ao = ak = 0 for all k > 3.

GENERATION THEORY

3.6

95

The Berkson-Porta parametric representation of semi-complete vector fields

An important consequence of Proposition 3.4.1 and Corollary 3.4.1 is the following representation of semi-complete vector field, which is originally due to E. Berkson and H. Porta [17] (see, also [6]).

Proposition 3.6.1 A mapping f E Hol(O, C) is a semi-complete vector field on A if and only if there is a point r E 0 and a function p E Hol(A, C) with Re p(z) > 0 everywhere such that:

f(z) = (z - r)(1 - zf)p(z).

(3.6.1)

Moreover, such a representation is unique and r is either a null point of f in A, or the boundary sink point of the resolvent Jr:_ (I + r f)_1 , r > 0.

Proof. Firstly, let f be a semi-complete vector field on A with a null-point r E A. Then it follows by formula (3.4.7) that the following inequality holds: Re f (w)w > (1 - lwI2) Re or,

Ref(w)

w

1-Iw12

f

1

(wwYr-

(3.6.2)

f - ---)

1-wf >0.

(3.6.3)

We calculate w IwI2

f

_ w(1 - Iw12)(1 - wf)

1 - wT

w-T (1 - IwI2)(1 - wf) Iw - TI

1

(1 - w12)

(w - r)(1 - wf)

(3.6.4)

Now by identifying w E A with z e A we obtain from (3.6.3) and (3.6.4): Re

f (z)

>0

(z - r)(1 - zT) -

(3.6.5)

for all z c A. Denoting: f(z)

p(z) =

(3.6.6)

(z -'r)(1 - zf)

we have (3.6.1).

Now suppose that f c C Hol(O) has no null point in A. Then it follows by Proposition 3.3.2 that in this case there is a unique boundary point r, such that for each w E A, the net {zr(w)}r>o defined as the solution of the equation zr.(w) + r f (Z' (w)) = w

(3.6.7)

Chapter III

96

converges to r, as r

oo. (Indeed, for each r > 0 the value Zr (W) is just the value

of the resolvent Jr = (I + r f)-1 at the point w E 0). Fix E > 0 and consider the mapping ff E Hol(0, C) defined as f, (z) = E z + f (z). It is clear that ff converges to f as E goes to zero. Since 9 Hol(0) is a real cone, it follows that f f E 9 Hol(0) for each E > 0. In addition, the equation ff(z) = 0

is a particular case of (3.6.7) with r = 1/E and w = 0. Hence ff has a unique null point Te E 0 and the net {7-,}fro converges to T as E tends to zero. Since ff satisfies the inequality: fe(z) Re (1 - zre)(z - 7-,-) > O '

letting E tend to zero we have the same (inequality (3.6.5)) for f (z), which in turn implies representation (3.6.1). Conversely. Suppose that f E Hol(L) admits representation (3.6.1) with T E 0 and Re p(z) > 0 everywhere. If Rep(z) = 0 for some z E 0, then by the maximum principle it follows that p(z) = im for some m E R. In this case: 2

f(z) = (z - r)(1 - zf)im = (z - T - z f + zTI2)

= -zmr - z2fim + (1 + ITI2)zmz. Denoting -imT := a,

(1 + Ir12)im := b we obtain:

f(z)=a-az2+bz, Reb=O, i.e., f is a complete vector field (see Proposition 3.5.1). Therefore, we have to consider only the case when Rep(z) > 0. Let us present f c Hol(0) in the form:

f (z) = a - az2 + z q(z),

(3.6.8)

where a = f (0). Comparing (3.6.1) with (3.6.8) we have f (0) = -Tq(0) and q(z) _ (1 - fz +

17-12)q(z)

-

q(z) - q(0)

r - zrq(0).

(3.6.9)

Z

To proceed we need the following lemma which will be also useful in the sequel.

Lemma 3.6.1 (cf., [6]) Let r be in 0 and let p and q be those holomorphic functions in 0 which satisfy equation (3.6.9). Then Rep(z) > 0 if and only if Req(z)>0. Moreover, the values of p lie strictly inside II+ _ {Rew > 0} if and only if the values of q lie strictly inside II+.

GENERATION THEORY

97

Proof. First we note that assuming one of the functions p or q to be holomorphic in 0 we have that the second one is holomorphic on 0 too. Observe also that it is enough to prove our assertion under the above stronger assumption. Indeed, for the general case one can use an approximation argument: given p (or q, respectively) with Re p(z) > E > 0, set p,, (z) = p(r,, z) for r E (0, 1), r,,, - 1-. So we assume that both these functions are holomorphic on 2K. Then substituting z = eie, 0 E R, in (3.6.9) we calculate Re q(eie) = Re{(1 + 1r12

- Tei0)p(ete) - Te-iep(eie)}

(1 + I7-12 - 2 Refete) Re p(ete) I1 - Tetel2

Rep(eie).

Since by our assumptions, r E A, and the functions Rep and Re q are harmonic,

we see that Rep(z)>e>_0ifandonly ifReq(z)>6>0. Moreover, if E is positive, then 6 can be chosen positive too, and conversely.

Returning to the proof of the Proposition 3.6.1, we see that if r E i then f defined by (3.6.1) admits representation (3.6.8) with Req(z) > 0, hence it is semi-complete (see Remark 3.5.2).

If r in (3.6.1) belongs to ai we just apply again the following approximation argument. We choose any sequence r,, C A such that n,, -4 T and set f,,,(z) _ (z - r,)(1 - zT,,)p(z). It is obvious that { f,}n=1 converges to f uniformly on each compact subset of A. Since we already know that each fn belongs to g Hol(O), we have that so does f, and we have completed the proof.

Remark 3.6.1 Thus this proposition implies that a semi-complete vector field f on 0 has at most one null point in A. If such a point exists it must be r in representation (3.6.1). For r E A we will denote by g Hol(A, r) the class of functions f E 9 Hol(O) with f (r) = 0. Thus, g Hol(O, T) = If E Hol(O, C) : f (z) = (z - r) (1 - zT) p(z), p E P} .

If T in this representation is a boundary point of A, then f is null point free. Moreover, it follows by the exponential formula that r is, in fact, the sink point of the semigroup {Ft}t>o generated by f, i.e., for each K > 0 the horocycle

D(r,K)zEA:cpr(z):=

1

2

I1-1z12


internally tangent to OA at T, is Ft-invariant, i.e., Ft (D(T, K)) C D (r, K)

.

Also, it can be shown (see Section 4.6) that r c OA is the limit null point of f in the following sense: liln f (rr) = 0.

r1-

Chapter III

98

However, it happens that f E Q Hol(A) may have more than one null point on the boundary of A (consider, for example, f (z) = 1 - z2). So the question is which one of them is the sink point of the semigroup {Ft}t>o generated by f. Another question relates to the asymptotic behavior of such a semigroup. In the next chapter we intend to answer these questions as much as to find the best rates of the exponential convergence.

At the end of the section we will add some preparatory material concerning this matter. To clarify our further reasoning we first summarize briefly different characterizations of semi-complete vector fields.

Summary Let f c- Hol(A, C). The following are equivalent: (i) f E G Hol(A), i.e., f is a semi-complete vector field;

(ii) for each r > 0 the mapping Jr = (I + r f)-1 is a well defined holomorphic self-mapping of A; (iii) f is p-monotone with respect to the Poincare hyperbolic metric on A, i.e., for each pair z, w c A p(z + rf (z), w + rf (w)) > p(z, w),

whenever z + r f (z) and w + r f (w) belong to A for some positive r; (iv) f admits the following parametric representation f(z) = a - az2 + zq(z)

for some a E C and q c Hol(O, C) with Re q(z) > 0, z c A; (v) f admits the Berkson-Porta parametric representation f(z) = (z - -r) (I - zt)p(z)

for some z E and p E Hol(A, C) with Re p(z) > 0, z E 1 . In addition, different flow invariance conditions given in terms inequalities are presented in sections 3.4 and 3.5. The simplest one can be formulated as follows (vi) there exists a number m c R (in fact, m < 0) such that

Ref (z)z > m(1- 1z12),

z E A.

In the study of asymptotic behavior of the semigroup generated by f E G Hol(A) with an interior null point, a few stronger conditions than (i)-(vi) will be relevant.

Definition 3.6.1 (cf., [39]) A function f : A

, C is said to be strongly p-

monotone if for some E > 0 and for each pair z, w in A there exists S = S(z, w), such that p(z + r f (z), w + r f (w)) > (1 + rE)p(z, w), whenever 0 < r < S.

Of course, a strongly p-monotone holomorphic function in A is semi-complete. Moreover, such a function must have a unique null point in A. Indeed, by definition

GENERATION THEORY

99

we have that (at least for r > 0 small enough) the resolvent Jr E Hol(A) is a strict contraction with respect to the hyperbolic metric p in A: P('Ir(z), Jr(w)) < 1 +1 rE P(z, w).

(3.6.10)

Thus Jr has a unique fixed point T in A because of the Banach Fixed Point Principle. This point is a null point of f. Further, putting w = T in (3.6.10) and differentiating it with respect to r at the point r = 0+ we obtain 17-12

(z

- )(1-) zfr)

>E

or

Rep(z) > E(1 -

rl2)-1 > 0,

where p is the factor in the Berkson-Porta representation (see (3.6.1)). Now Lemma 3.6.2 enables us to conclude that q(z) in representation (3.6.8) has a real part strictly separated from zero, i.e., Re q(z) > E1 for some E1 > 0 and all z E A. In turn, the same formula (3.6.8) implies Ref (z)z > -lal(1 - Iz12) + Iz12E1 > E2 > 0

for all z close enough to 8/, the boundary of A. Thus we have proved the following assertion.

Proposition 3.6.2 Let f E Hol(O, C) be strongly p-monotone in A. Then: (i) f admits the representation

f(z) = (z - T)(1 - z'r)p(z) with T E A and Re p(z) > E for some E > 0; (ii) there exist positive numbers 5 and q such that

Ref(z)z>rl>0 for all z in the annulas {1 - b < zl < 1}. Again from Lemma 3.6.1 it can easily be seen that (ii) implies (i). Thus these conditions are equivalent. As a matter of fact, we will see below (Section 4.5) that (ii) implies the strong p-monotonicity of a function f E HOl(O, C).

Chapter 4

Asymptotic behavior of continuous flows In this chapter we want to trace a connection of the iterating theory of functions in one complex variable and the asymptotic behavior of solutions of ordinary differential equations governed by evolution problems. Therefore our terminology is related to both these topics.

4.1

Stationary points of a flow on 0

Quoting M. Abate [2], note that E. Vesentini seems to be the first person who suggested an analog of the Denjoy-Wolff Theorem for continuous time semigroups. In fact, in 1938 J. Wolff [158] himself initiated the consideration of dynamical systems determined by holomorphic functions. However, the first general continuous version of the Wolff-Denjoy Theory was given by E. Berkson and H. Porta [17] in their study of the eigenvalue problem for composition operators on Hardy spaces.

Definition 4.1.1 A point ( C 0 is said to be a stationary point of a flow S = {Ft}t>o c Hol(A), if

lim Ft (r() _

r-1for all t > 0.

In other words, ( C A is a stationary point of S if it is a common fixed point of all Ft ES. Note that the family S = {Ft}t>o is commuting, that is, FtoF5 = F3oFt = Ft+,, for all t, s > 0. Hence, it follows by the Shield theorem [132] that if each Ft had 101

Chapter IV

102

been continuously extended to 8A the boundary of A, then the stationary point set of S would not be empty. As a matter of fact, it is enough to require the existence of an interior fixed point only for one t > 0 to ensure the existence of such a point for the whole semigroup. Indeed, if for at least one t > 0 the mapping Ft E S has an interior fixed point (E A then it is a unique fixed point for Ft, and for each s > 0 we have:

FS(() = F9(Ft(()) = Ft(F(()) = (, i.e., (is also a fixed point of FS E S, s > 0. Henceforth this fixed point is a unique stationary point of S.

Exercise 1. Show that if ( E A is a stationary point for a semigroup S = {Ft}t>o, Ft E Hol(A), t > 0, then (Ft)'(a) = e-at is a contraction linear semigroup, i.e., Re a > 0. Hint. Use the chain rule and the Schwarz-Pick lemma.

Naturally, the strategy now is to study the convergence of a semigroup to its stationary point. The foregoing result is the first step in the study of the asymptotic behavior of a flow in A.

Proposition 4.1.1 ([81]) Let S = {Ft}t>o c Hol(i) be a flow on A. Then this net converges uniformly on compact subsets of 0 to a holomorphic mapping F E Hol(A, C) if and only if for at least one to the sequence {Ft0n}n°_o converges uniformly on compact subsets of A. Moreover, if Ft. is not the identity then F is a constant with the modulus less or equals to 1.

Proof. The necessity is obvious. To prove the sufficiency we assume that Ft is not the identity for t > 0, otherwise the assertion is trivial. Then the limit: lim FF,,,1 = lim F(og)

n - 00

1L--00

is a constant mapping, say ( (_ ((to)) E 0. If (E 0 then it follows by Corollary 1.3.2 that Ft. (()I < 1.

Consequently, the chain rule and the semigroup property imply that for all t > 0: I Ft(()I < 1

(4.1.2)

(see also Exercise 1). Hence (is an attractive fixed point for each Ft, t > 0, i.e.,

lim F(")(z) = (,

71 -+00

z E A.

(4.1.3)

If ((= lim F,`)) E 9A, then Ft has no fixed point inside A. In this situation, 7700 as we mentioned above, each Ft. t > 0, must be fixed point free on A. Then the Denjoy-Wolff Theorem implies that for each integer m > 0 the sequence of iterates 00 converges uniformly on compact subset of A to a point (m E A.

F('

} -o

ASYMPTOTIC BEHAVIOR

103

But

F(nm)(z) lim to/m = n-00 lim F(n)(z) n-oo 0 and it follows that Cm = C does not depend on m. So, in both cases (either t; E A or (E 80) we have the following equality

lim Fonm(z)

(4.1.4)

for all z E A, and each m E N. Now we will show that (4.1.4) implies that for each z E A the net Ft converges

to (, as t tends to infinity. Indeed, for a given t; > 0 and z E A one can choose S > 0 such that IFt(z)-Ft(w)I < E/2 for all t > 0, whenever w E A and Iw-zl < J. For such S we take m E N so large that IF3(z) - zI < S whenever s E [0, to/m]. Finally, for such m and t > 0, large enough setting s = [tm/to] we have by (4.1.4): IFton/m(z) - S I = Fo/m`z)

-

< 2.

Noting that s = t - ton/m E [0,to/m] and setting w = F, (z), we obtain for such t > 0:

IFt(z) -0 = <_

Fton/m(Fs(Z)) - (I Fton/m(Fln(z)) - Fton/m(z)I

+

Fton/m(z)

E

E

- (I C 2 + 2 = 8,

and we have completed the prof. The result which we have established implies immediately a continuous analog of the Denjoy-Wolff Theorem.

Proposition 4.1.2 ([81]) Let S = {Ft}t>o C Hol(O) be a flow on A. If for at least one to the mapping Ft,, is not the identity and is not an elliptic automorphism oo uniformly on of A, then the net {Ft}t>o converges to a constant (E 0 as t each compact subset of &

Remark 4.1.1 Since every continuous semigroup S = {Ft}t>o of holomorphic self-mappings Ft of A is differentiable (by parameter t > 0), it is natural to describe its asymptotic behavior in terms of the generator f = lim 1 (I - Ft). This becomes t-o t more desirable when such a semigroup is not given explicitly, but is defined as the solution of the Cauchy problem: au(t, z) + f (u(t, z)) = 0,

at

u(0, z) = z E A. We set here Ft (z) = u(t, z).

(4.1.5)

Chapter IV

104

Note also that if f is holomorphic in a neighborhood of the point S E A then it follows by the uniqueness of the solution of the Cauchy problem that f (c) = 0

if and only if ( is a stationary point of S = {Ft}t>o. In particular, an interior null point of a semi-complete vector field is a stationary point of the generated semigroup. However, this fact is no longer true for a boundary null point.The following example shows that even a semi-complete vector field f has a continuous

extension to 0; it may have two null points in 0 (one of them on 80), while the semigroup generated by f has a unique stationary point in 0 (which is the interior null point of f).

Example 1. Set f (z) = z - 1 + 1 _-z. It is clear that f (0) = f (1) = 0. At the same time, solving the Cauchy problem (4.1.5) one can find the solution explicitly: 2

u(t, z)

e-t/2 + e-t/2 1 - z]

.

Setting Ft = u(t, .), it is easy to verify that Ft E Hol(A), hence complete. But for all t > 0:

lim Ft(r) = 1 - 1[ - e- t/2

rl-

I

1

J

f is semi-

2

<1

and therefore ( = 1 is not a stationary point of S = {Ft}t>o. Nevertheless, as we will see below (see Section 4.6), if f has no null point in A then it must have a boundary null point on aA which is an asymptotic limit of the semigroup generated by f.

4.2

Null points of complete vector fields

In this section we deal with a one-parameter group of automorphisms of A. Since the generator of a one-parameter group is a complete vector field, it is a polynomial

at most of the second order, and hence holomorphic in C. Now the assertion follows:

Proposition 4.2.1 The stationary point set of a one-parameter group S = {Ft}teR, Ft E Aut(A), has either one or two points in which are exactly the null points of the function f : f (z) = a + ibz - az, 2 (4.2.1)

ASYMPTOTIC BEHAVIOR where

a

_

OFt(0)

at

and b=

7-

1 a2Ft(z) i

t=o

105

ataz

t=0,z=0

Exercise 1. Show directly that if z0 54 0 is a solution of the equation

az2+ibz-a=0, bER, aEC,

(4.2.2)

then z1 = 1/zo is also a solution of this equation. Thus equation (4.2.2) has at least one solution in A. If one of the solutions of (4.2.2) lies on OA the second one (if it exists) lies on 0A also. Moreover, given a complete vector field f one can characterize the group of automorphisms generated

byf. Proposition 4.2.2 Let f E aut(A) be a complete vector field on A and let S = {Ft}, t E 1l8 be the group of automorphisms generated by f. The following assertions hold: 1) If 21 f (0) 1 < I f'(0) then S has a unique stationary point zo in 0 which is actually in A and S consist of elliptic automorphisms of A. In this case Ft(z) does not converge to zo for each z c A, z # zo. _ 2) If 21f (0)l = f'(0)j then S has a unique stationary point zo in A which actually lies on OA and S consists of parabolic automorphisms of A. In this case for all z E A:

lim Ft(z) = zo E OA. 3) If 21 f (0) I < I f'(0)1 then S has exactly two different null points z1 and z2 in 0. Both of them lie on 8A, and S consists of hyperbolic automorphisms of A.

Consider in more details the group of hyperbolic automorphisms. In this case its generator f has the form: f(z) = a + ibz - az2,

(4.2.3)

2Ial > dbl.

(4.2.4)

with a E A, b E R, such that Using (4.2.4), by direct calculations one obtains that the null points of f are z1 =

2a 4Ia12

and bbl + ib

z2 =

-2a

_-b2 - ib

(4.2.5)

41a12

It is clear that z1 54Z2 andIzil=Iz21=1. Let now S = {Ft}tER be the flow generated by f. Since for a fixed to c R+ the points z1 and z2 are fixed points of Fto, therefore one of them is a sink point for Ftoand thus this point is the limit of the net {Ft} when t goes to infinity.

It is easy to understand that the second null point is the sink point for the mapping Ftol = F_to. Therefore it is the limit of the net {Ft} when t -* -oo.

Chapter IV

106

Write Ft in the form:

z - at

Ft(z) = e

(4.2.6)

1-atz

where at E A, cot E R, t E R. It is easily seen that for each t E R and z E 0: Ft(z) - z1

_

1 - atz2

z - z1

1 - atzl

z -

z2.

Ft(z) - z2

This equation may be written in the form L(Ft(z)) = AtL(z),

(4.2.7)

where L is the fractional linear transformation defined by

L(z) = z - z1 (4.2.8)

z-z2 and At = 1 - atz2

-

z2 - at

Z2 (4.2.9)

1 - atz1 z - at z1 At the same time, since zi, i = 1, 2 are fixed points of Ft we have by (4.2.6):

1-atzi= It follows that At

_ 1 - atz2 1 - atzl

ei°t(zi - at) zi

- z2 - at

z1 - at In addition note, that (4.2.5) implies that z1

z2

Z2

zl

z2 (4.2.10)

z1

(It is also clear because of the equality zizi = 1, i = 1, 2). Comparing (4.2.9) and (4.2.10) we obtain that: At = At,

so the result that At is real is established for all t E R. Furthermore, it follows by the group property and (4.2.7) that At+9L(z) = L(Ft+3(z)) = L(Ft(F3(z))) = AtL(F9(z)) = At A8L(z),

At+s = At A

,

Ao = 1.

So At = ekt for some k E R. If k > 0 we have again by (4.2.7): Ft(z) = L-1(e ktL(z)),

ASYMPTOTIC BEHAVIOR or

107

ekt=1 F t(z) -

Z-z2

(4.2.11)

1-ekt -21 Z-Z2

lim Ft(z) = z2 for all z E A. Hencet--oo lim Ft(z) = zi while t--.-00 In case of k < 0 we obtain similarly:

lim Ft(z) = z1

t-.-00 and

lim Ft(z) = z2.

t- *-00

Therefore we need to recognize k. Since u(t, z) = Ft (z) satisfies the equation

au(t, z) + f(u(t, z)) = 0, at

with the initial data: U(0' Z) = Z'

we obtain by solving this equation:

Ft(z) - z1 = Ft(z) - z2

Z - Z1

e-a(=1-z2) t

z-

Z2

Comparing this formula with (4.2.7) we obtain that k must be -a(zl - z2). By using (4.2.5) we calculate: k

=

1 2 -a(zl - z2) _ -21a1 (y4a2_b2+jb+

_

44a1a212b2

=-

1

41x12 - b2 - ib

41 a12 - b2 < 0.

So k is negative in our case, and we have the second situation. Now we are ready to formulate our result.

Proposition 4.2.3 Let f (z) = az2 + ib - a such that b E JR and 21al > Ibl. Then f generates a flow of fractional linear transformations: Ft(z) = e'°°

z - at

1-atz

which converges to

z1=-

2a

ast -+oo, b2 - ib

and to

z2=

2a

41x12-b2+i6

ast-i-00.

Chapter IV

108

In addition,

1 - ekt

1

at=---'-EBA, Z2

z2 - ektz1

where k = -

41aI2 --b 2

and Wt - arg

z2(1 - atz2) zi(1 - atz1) = arg - at z2 - at z1

Exercise 2. Let f c aut(o). Show that the fractional linear transformation G, defined by the formula:

z-c G(z) _

-

1 - zc'

where c = f '(0) has the same fixed points as the null points of f .

Exercise 3. Show that if f E aut(o) generates a one-parameter group of parabolic automorphisms of 0 then f has the form:

f(z) where r E O

'r)

and y E C, Rery = 0. Find explicitly the group S = {Ft}tER

generated by f and show that

lim Ft(z) = t-.-oo lim Ft(z) = T.

t-.oo

Exercise 4. Show that if f E aut(o) has a null point r in 0, i.e., f generates a one-parameter group S = {Ft}tER of elliptic automorphisms, then Re f'(r) = 0. Show that the group S = {Ft}tER can be presented in the form: Ft(z) = Mt (eat Mr (z))

,

where

MT(z) = (z - r)(?rz - 1)-1 and

Rea = 0.

ASYMPTOTIC BEHAVIOR

4.3

109

Embedding of discrete time group into a continuous flow

A classical problem of analysis is: given a self-mapping F(z) (F E Hol(A)). Find a sernigroup {Ft(z)}t>o C Hol(A) with Fi(z) = F(z). In this case we say that F satisfies the embedding property on A. In general, however, this problem cannot be solved globally even if F E Hol(A) is a fractional linear transformation (see Example 4.3.1). Example 1 Consider the fractional linear transformation defined as follows:

F(z) =

az

1 - bz

with a = exp(i3.1) and b = 2. Since dal+Ibl = 1 we have that F is a self-mapping of the units disk A. Therefore its iterations F(") map A into itself. Moreover, F(") converge to zero uniformly on compact subsets of A as n goes to infinity. At the same time the semigroup atz

1-1

bQ

(1 - at)z

is a continuous extension of the discrete time semigroup {F(")} because of the equality F = Fl. However, for some t > 0 (which is not integer) Ft does not map A into itself. If, for example, we choose z = 0.99+0.1i c A then by calculation we obtain that F2 (z)I > 1 (see Fig. 4.1). Nevertheless, by using some geometric methods we will give in Section 5.9 a complete characterization of those fractional linear transformations of the unit disk into inself that satisfy the above embedding property. In this section we show how to solve this problem, when F E Aut(o).

Suppose first that F is a hyperbolic automorphism of A, i.e., F has exactly two distinct fixed points zl and z2 on the boundary A. As we saw above (see section 1.1.3), in this case if F = r,P m, , then

F(z) = L-'(.F(z)), where

A=

1 - dz2

1-dzl

and L is defined by L(z) =

E1[8

+

(4.3.1)

\{0,1}

z - zl

z-z2

(4.3.2)

Chapter IV

110

\\\\\\ \\\\\ \\\\ \ 11

11,

1\

11 zo

(zo).

Figure 4.1: Fractional iterations of the self-mapping F(z).

In addition, if z1 is the sink point of F, then 0 < A < 1. Denote

b= kzl+z2

k

k=logA, a=

andc=

i Zz - z2

Z2 - zl

zl z2 z2

z1

k.

Since Ix11 = 1x21 = 1, and z1 # z2 it is easy to verify that a = -c, and b E R. Indeed, a

k

z2 - z1

z2 - z1

z2 - z1

c

z1 - z2

z1 - z2k

(z2 - Z1)zlz2

2112212 - 221x112

In addition, Re ib

=

k Re zi + zz k

Iz1 - z2I

lzl

Re(z1 + x2)(21

z2)

x212

2(Re1z112+2241

- 2221 - 12212)=0,

i.e., b E R. Therefore the mapping f : f(z) = az2 + ibz + c

is a complete vector field. It follows by the Vietta formulas that z1 and z2 are the roots of the equation f (z) = 0. Hence as above (see section 4.2) the group of automorphisms generated by f has a form:

Ft(z) = L-1(ektL(z)), where L is defined by (4.4.2).

ASYMPTOTIC BEHAVIOR

111

For t = 1 we have:

Fi(z) = L-1(el0

L(z)) = F(z)

because of (4.4.1). This gives us the desirable embedding. As a matter of fact, we have proved somewhat more:

Proposition 4.3.1 Let z1 54 z2 are two arbitrary points on the boundary aA of the unit disk A. Then for each k E R, k 54 0, there is a one-parameter group S = {Ft}, t E R, such that the following assertions hold: (a) Ft(zi) = zi, i = 1,2. (b)

dFt(z) k b = k z1 + z2 It=o= -(az2 + ibz - a) where a = dt Z2 - z1 i z2 - z1

(c) If k < 0 then z1 is a sink point of {Ft}t>o; if k > 0 then z2 is a sink point of {Ft}t>0. (d) In particular, if z1 and z2 are fixed points of a hyperbolic automorphism

F(z) = e'e

z + cr 1+az' aEO,

cpEIlB,

then for k = In

1 - az2

1-dz1

we have the equality:

F1(z) = F(z),

z E A.

Now we turn to a parabolic automorphism F E Aut(o). In this case F has a unique fixed point z0 E ao which is a sink point of F. Moreover, as we already know F satisfies the equation:

e"'-1+

z0

F(z)-zo

z0

z-z0

eiw +1

(4.3.3)

for some cp E R, cp # 7r + 2nir (see section 1.3.1). It is clear that in such a situation we must solve the following Cauchy problem:

au(t, z)

2

+a( Z - zo) - 0,

at

(4.3.4)

u(0,z) = z E 0, with suitable a E C in order to find a one-parameter group {F&)}, Ft = u(t, ), such that: F, (z) = F(z). (4.3.5) Solving (4.3.4) we have that Ft satisfies the equation: 1

Ft(z) - zo

=at+

1

z - 20

Chapter IV

112

Comparing the latter equality with (4.3.3) and (4.3.5) we obtain:

eiw-1 (4.3.6)

eiw + 1

zo

Now we only need to check that the mapping f defined by:

f(z) = a(z - zo)2

(4.3.7)

is a complete vector field. In fact, substitute (4.3.6) in (4.3.7) we obtain:

f(z) = a z2+ibz+c, where 2 c=az0=

and

_2 i

eiw

ezw-1 eiw + 1

-1

zo=-a 4 Im e=w

leiw+12

eaw+1

R.

Thus we have proved the following assertion:

Proposition 4.3.2 Let zo and r

-1 be two unimodular points. Then the map-

ping f defined by (4.3.7) with a=

r-1

r+1 zo

is a generator of the one-parameter group S = {Ft}tcR E Aut(o), such that each Ft is a parabolic automorphism of A, and the point zo E 3i is a sink point of S. If, in particular, F E Aut(o), is a parabolic automorphism of A: F(z) = e'w

z + ce

1+z i

such that F(zo) = zo, then for r = e'w we have: Fi(z) = F(z).

Finally, we consider the simplest situation where F E Aut(o) is an elliptic automorphism of A, i.e., F has a unique fixed point zo E A. In this case F can be presented as:

F = mZO 0 rw 0 m_ZO,

where

mZ.(z) = is a Mobius transformation of A,

(4.3.8)

z+zo 1+zoz

mZO (0) = zo, and rw is a rotation around zero,

i.e.,

cp#27rk,

Setting in this case etwtz

k=0,1,2,.

ASYMPTOTIC BEHAVIOR

113

and

Ft(z) = -m.. o r,p, o m_ZO,

t E R,

(4.3.9)

we have that S = {Ft}, t E R is a one-parameter group of elliptic automorphisms

such that F(zo) = z0 for all t E R and F(z) = F(z), zEO.

of

Exercise 1. Prove that group (4.3.9) is generated by the mapping f : z

f(z)

=

Izol2(z - zo)(1 - !O-Z).

(4.3.10)

1 -

Exercise 2. Prove directly that the mapping f defined by (4.3.10) is a complete vector field.

Finally, we formulate the result:

Proposition 4.3.3 Let zo E A and cp E R, co = 27rk, k = 0, 1, 2 .... Then the mapping f defined by (4.3.10) generates the one-parameter group S = {Ft}, t E R of elliptic automorphisms defined by formula (4.3.9). In particular, if F E Aut(A) is an elliptic automorphism, defined by (4.3.8), then Fl (z) = F(z), z E A.

4.4

Rates of convergence of a flow with an interior

stationary point Let f E Hol(A, C) be a semi-complete vector field on A, and let us assume that Null f n A # 0. Actually, it means that if f is not zero identically, then f has exactly one null point in A (otherwise the semigroup S = {Ft}t>o generated by f should have more than one stationary point in A, and a contradiction results). A standard question of dynamical systems analysis is: given a vector field f, describe the asymptotic behavior of the flow defined by the evolution equation du/dt + f (u) = 0, u(0) = z, in a neighborhood of its singular point r (f (r) = 0). In this section we intend to answer this question for a semi-complete vector field, as much as to find global rates of convergence of the generated flow to its interior stationary point. For our purpose we need the following definition.

Definition 4.4.1 ([40]) A function f E 9Hol(O) is said to be a strongly semicomplete vector field if it has a unique null point r in A which is a uniformly attractive stationary point for the flow S = {Ft}t>o generated by f, that is, the net {Ft}t>o converges to rr uniformly on each compact subset of z

as t , no.

Chapter IV

114

We begin with establishing a simple assertion which is a (nonlinear) analog of the Lyapunov stability theorem.

Proposition 4.4.1 Let f E Hol(A, C) be a semi-complete vector field with f (T) 0 for some T E A. Then: (i) Re f'(T) > 0, (ii) Re f'(-r) > 0 if and only if f is strongly semi-complete.

Proof. Let S = {Ft}t>o be the flow generated by f. Since Ft(T) = T for all t > 0, it follows by the chain rule, that (Ft)'(T) = At satisfies the semigroup property At+s = At A,i Ao = 1, hence At = ekt, with k = f'(T) E C. In addition, it follows by Corollary 1.1.1 that le- ktI < 1.

But this inequality holds if and only if Re k = Re f'(T) > 0 and assertion (i) follows. Furthermore, Re k = Re f'(T) > 0 if and only if for all t > 0 Ie- ktI _ I(Ft)'(T)I < 1,

and then assertion (ii) is a consequence of Proposition 4.1.1. and Corollary 3.1.1.

Exercise 1. Show that if Re f'(T) = 0 then the semigroup S = {Ft}t>o is actually a group of elliptic automorphisms.

Thus, if f E 9 Hol(A) is not a complete vector field of the type f(z) = (z - T)(1 - Tz)a

with T E A and Rea = 0, then the semigroup S = {Ft}t>o generated by f does not consist of elliptic automorphisms and converges uniformly on each compact subset

of A to a point in 0. If this point belongs to A, then it is a unique uniformly attractive stationary point of the semigroup, and the question of finding a rate of convergence arises.

Remark 4.4.1 We already known that for a holomorphic function f on A the property of being semi-complete is equivalent to the property of being monotone

with respect to the Poincare metric p on the unit disk. Also, it can easily be seen that if f is strongly p-monotone then it is strongly semi-complete. For this case one can obtain a rate of convergence in terms of the hyperbolic metric on A (see Section 4.5). The converse, however, does not hold in general: a strongly semi-complete vector field is not necessarily strongly p-monotone. Therefore in this section we will establish some rates of convergence in terms of the Euclidean and pseudo-hyperbolic distance on A. First we consider the case when f (0) = 0, i.e., zero is a stationary point of the semigroup S = {Ft}t>o.

ASYMPTOTIC BEHAVIOR

115

Proposition 4.4.2 (see [58] and [109]) Let f E 9Hol(O) be a strongly semicomplete vector field with f (0) ='0 and A = Re f'(0) > 0, and let S = {Ft}t>o be the semigroup generated by f. Then there exists c E [0, 1] such that for all z E A and t > 0 the following estimates hold: (a) IFt(z)1 <- IzI eXp

(-A1 - cIzIt;

1 + cIzI ) JFt(z)j < exp(-At) (1 IzI (b) (1 - cIF't(z)l)2 - c1zD2

Proof. The procedure that is to follow is based on general flow invariance conditions established in sections 3.4 and 3.5. We start with the consideration of an auxiliary Cauchy problem: 8u(t , z)

at

1 - cu(t , z)

+ p u(t' z) 1 + u(t, z)

--0, (4.4.1)

1 u(0, z) = z,

z c A.

Noting that the function a(z) = p z satisfies the condition 1+z Rea(z)z = Iz12pRe 1 + z > 0

for all z E A, we conclude that a c Hol(O) and the Cauchy problem above has a unique solution u(t, z) E A for all z E A and t > 0. In addition, u(t, 0) = 0 for all t > 0, because of a(0) = 0. Then by the Schwarz Lemma we obtain that Ju(t, z)j < IzI,

z E A.

(4.4.2)

Furthermore, since the function a(s) = ps 1 + s' defined on the interval [0, 1] is real we have by the uniqueness property that the differential equation: dt + a(v) = 0

(4.4.3)

v(0) = s

(4.4.4)

with the initial data: has a unique solution v(t, s) for all t > 0 and s E [0, 1) which coincides with u(t, s) restricted on the real interval [0, 1). Consequently, by (4.4.2) we obtain: v(t,s) < s

for all t > 0 and s E [0, 1). Also, it follows from (4.4.3) that: dv v

_

1 - v dt.

-p1 +v

(4.4.5)

Chapter IV

116

Hence, by (4.4.4):

v=sexp(J -pl+vdt Finally, inequality (4.4.5) implies the estimate:

pt)

v(s,t) < sexp

(_1+s

(4.4.6)

Recall that by Proposition 3.4.5 f E G Hol(0) with f (0) = 0 satisfies the inequality:

1-121 Ref (z)z > Re f'(0)IzI21 + IzI = a(IzI) Izl,

Z E A.

Then by Lemma 3.4.1 and (4.4.6) we have: IFt(z)I <- u(t, Izl) <- Izl exp

(_Ref'(o) 1

Izl

t

This proves (a). To prove (b) we return to (4.4.3) and rewrite it in the form: 1 1+v -dv = -pdt

1-v v

1 + v (we substitute a(v) = vp 1 - v into this equation). Integrating the latter equality

in [0, t] we obtain:

_= e-pt

v (1

v(s,t))2

s (1

s)2

S E [ 0 , 1).

,

Again, setting s = Izl and using the relation IF(t, Izl)I <- v(s, t),

we obtain (b).

Note that estimate (a) with c = 1 is due to K. Gurganus, whilst estimate (b) was established by T. Poreda. Now we are prepared to consider a more general case, when f E G Hol(0), with f (T) = 0 for some r E A, which is not necessarily zero. Proposition 4.4.3 Let A equipped with the pseudo-hyperbolic distance d (z, w) =

w-z 1 -wz

,

and let S f = {Ft}t>o be the semigroup generated by f c 9 Hol(0). Then for some r c A the following conditions are equivalent:

ASYMPTOTIC BEHAVIOR

117

(i) f (T) = 0 with Re f' (T) > p > 0; (ii) for some c E [0,1]:

d (Ft(z), T) < d(z, T) exp (_P::::t)

forallt>0andzEA. In addition, if f is bounded and strongly p-monotone then the number c in (ii) can be chosen as strictly less than 1.

Proof. Define the Mobius transformation MT : A H A by:

MT(z)=

T-z 1 - Tz

,

zEO.

(4.4.7)

It is clear that MT(0) = T,

MT(T) = 0,

(4.4.8)

and it follows that MT is an involution, i.e.,

MT=MT1.

(4.4.9)

Consider the family {Gt}t>o defined as follows:

Gt=MToFt oMT.

(4.4.10)

It is obvious that {G,} is also a semigroup of holomorphic self-mappings of A, and for all t > 0 we obtain Gt(0) = 0, t > 0, (4.4.11) since Ft (T) = T, for all t > 0. Let g be the generator of the semigroup {Gt}t>o , i.e., -dGt(z)

By direct calculations we obtain: dMzT

(Gt(z)) Wt

dt (MT(F't(MT(z)))) = ddT (Ft(MT(z))) dd t (M,-W)

=

d

(F't(MT(z))) (-f (Ft(MT(z))),

z E A.

Hence by (4.4.12)

g(z) =

dd M,

W,(z)) . f (MT (z)),

zEA.

(4.4.13)

Note that the latter formula expresses the direct connection between generators

f and g. Also, (4.4.8) and (4.4.13) imply:

g(0) = 0,

(4.4.14)

Chapter IV

118

which is equivalent to (4.4.11). In turn, differentiating (4.4.13) we calculate z

9'(z) = dMT (MT(z)) . dd T (z) - f(MT(z))

+ ddzT (MT(z)) . d (MT(z)) .

ddz (z).

(4.4.15)

Substituting here z = 0 and using that f (T) = 0 we obtain:

M, 9'(0) = dd (T) d (T)

da M_

Since

dMT

dz

d dz

T-z

171

( 1 - Tz

2

(4.4.16)

(0)

-1

(1 - TZ)2

relation (4.4.16) becomes:

g'(0) = t121- 1 fl(T) . (ITI2 - 1) = fl(-r).

(4.4.17)

So by Proposition 4.4.2 (a) we obtain:

IGt(z)I < IzI exp I - Reg'(0)

1

+

izi

t

and using (4.4.10) and (4.4.17) we obtain from the latter inequality:

I(MToFt oM.,)(z)I < I z I exp(-Re f'(T)

1+zi t).

(4.4.18)

MT(w)\in

For arbitrary w, setting z =

(4.4.18) we obtain:

(M,. o Ft)(w)I < IMT(w)I exp

(_Ref'(T).

1

MT(w)I

t

.

(4.4.19)

1 + MT(w)I

Rewriting (4.4.19) in the form:

T - Ft(w) 1 - Ft(w)T

T-w 1 -fw

exp

- Re f'(T)

t

,

(4.4.20)

we obtain (ii). Conversely, assuming that (ii) holds we first observe that f (T) should be zero. Of course, this is an immediate consequence of (ii) if p > 0, hence Re f'(T) > 0. Indeed, in this case lien Ft (z) = T, t-oo so, T is the stationary point of Sf = {Ft}t>o, whence, a null point of f. But even if Re f'(0) = 0, condition (ii) means that each pseudo-hyperbolic ball Q,(-r) (see Section 1.3) is Ft-invariant for all t > 0, and the same conclusion follows.

ASYMPTOTIC BEHAVIOR

119

To arrive at the second condition of (i) we just need to differentiate (ii) at t = 0+ to obtain: P

f (z) (1 - 17-12) Re (z - 7-)(1 - z7-)

1_ 1+

1-iz -Z 7-

z

1-7-Z

Letting z -+ T we obtain immediately that Re f'(7-) > p as desired. Finely, note that our last assertion follows from the strong Harnack inequality (see Section 3.5 and Proposition 3.6.2).

Remark 4.4.2 Since IFt(w)l < 1 one may obtain from (4.4.20) (setting c = 1) the following estimate lFt(w) - Tl 1+l7l

exp

-Re f'(7-)

l1-fwl-IT-wl l1-fwl+l7--wlt

(4.4.21)

This formula gives an estimated rate of convergence of the semigroup {Ft}t>o to its stationary point T in the usual Euclidean metric in C. As a matter of fact, this estimate can be slightly improved by using the following considerations. Return to formula (4.4.19). Denoting Ft(w) = J and MT(J) = u we have M,(u) = J and MT(0) = T. Now, it follows by Corollary 1.1.3 of the Schwartz Lemma (see section 1.1.1), that: IMT(u) - MT(0)l :5 lul 11

IMT(0)Ilul =

lul 11

lTl lul

or

IFt(w) -7-l


I

ll - ;r-WI

(1 - lTl2) exp (- Re f'(7-) I1-TwI-IT-wI 1-rwI+I7--wI

.

tl /

(4.4.22)

1 - lTl exp - Ref'(T) I1-TwI-IT-wI . t

It is clear that (4.4.22) implies (4.4.21).

Remark 4.4.3 Note that the above proposition may be considered as an analogy of the Schwarz-Pick inequality. Moreover, for the continuous case this inequality has a more precise form because of the exponential factor

1 - d(z,T) exp -Pl+d(z,T)t

.

However, the property of this factor depending also on z E A is sometimes inconvenient for applications. In some situations it would be preferable to find a metric on A in which such a factor becomes uniform. In addition, as in the discrete time case, condition (ii) in our proposition cannot be useful in the study

Chapter IV

120

of the boundary asymptotic behavior of flows with no stationary point inside 0. Therefore we will treat the last question separately in Section 4.6 by using a continuous extension of the Julia-Wolff-Caratheodory Theorem. However, it turns out that one can define a non-Euclidean distance on 0 (which, in fact, is not a metric on 0, but induces the original topology of A) such that the rates of convergence for interior and boundary points have some unified form. This makes it possible to study the dynamic behavior of evolution equations when their stationary points approach the boundary. Moreover, we will show in Section 4.6 that in terms of such a distance, the question about a uniform rate of convergence of the exponential type may be solvable too. Finally, we note that a deficiency of the rates of convergence given by formulas (4.4.20) and (4.4.21) is the presence there the value of the derivative f' at the null point r of a generator f. In fact, if f has a complicated form even the existence of its null point is unclear a priori. Therefore it would be nice to establish a rate of convergence by using another condition which will guarantee also the existence of an interior null point of a semi-complete vector field. The next section is devoted to this matter.

Remark 4.4.4 From the geometrical function theory point of view there is a need to point out a lower bound estimate of the flow behavior. Exactly as in Proposition 4.4.2, by using the left side inequality of (3.5.18) and Lemma 3.5.1 one again obtains: if f (0) = 0 and Re f'(0) = A > 0, then for some c E [0, 1] 1 + CZ (a) IFt(z)l ? lzl exp(-A 1 - cIzI t);

(b)

> exp(-At) (1 - ctFt(I )I)2 (1 + cIzI)2 Moreover, if f is bounded strongly p-monotone then c can be chosen strictly less then 1.

4.5 A rate of convergence in terms of the Poincare metric In this section we will give several sufficient conditions for f E Hol(0, C) to be strongly p-monotone, hence strongly semi-complete on the open unit disk 0 and obtain rates of convergence for the sernigroups generated by such functions in terms of the Poincare hyperbolic metric p on A.

ASYMPTOTIC BEHAVIOR

121

Proposition 4.5.1 ([40]) Let f E Hol(A,C) satisfy the condition: Ref (z)z > a(jzj)jzj,

zEA

(4.5.1)

for some real continuous function a(l) on the interval [0, 1], such that:

a(1) = a > 0.

(4.5.2)

Then f is strongly p-monotone, hence a strongly semi-complete vector field on

A. Thus f has a unique null point T E A which is uniformly attractive for the semigroup S = {Ft}, t > 0, generated by f Moreover, for each pair z, w E A .

p(FF(z), Ft(w)) < exp {In particular, p(FF(z), T) < exp {-

Proof. Consider the mapping g3 : A g.,(z) = z + sf(z) - w,

at

(4.5.3)

2t}p(z, w).

}p(z, w).

(4.5.4)

C defined as follows: where w E A, s > 0.

Let Or be the disk centered at zero with radius r zE3Or={zEC: jzj =r} we have by (4.5.1):

:

0
1.

Regs(z)z = Iz12 + s Re f (z)z - Rewz > r2 + sa(r)r - r1w l = r (r + sa(r) - Iwl) .

For all

(4.5.5)

Since a (l) > 0 it follows that for s > 0 small enough the equation:

W9(r) = r + sa(r) = 1

(4.5.6)

has a solution rs E (0, 1).

In fact, cp,(0) = sa(0) < 1 for 0 < s < 1/a(0) and cps(1) = 1 + sa(1) > 1. It follows by (4.5.5) that for such fixed n and z E 9A, Reg,,

The latter inequality implies that the equation: g9(z) = z + sf(z) - w = 0

has a unique solution z = J3(w) :_ (I + s f)-1 (w) E A, for each w E 0. In other words, the resolvent mapping JS maps A into Ary . This means that J3 has an interior fixed point T E A which is also a unique null point of f in A. Furthermore, we consider the function h(z) = h3(z) defined as follows: h. (z) = J., (z) + sa(T'S) (Js(z) 2

- J,,(w))

Chapter IV

122

where w is an element of A, and r, is the solution of (4.5.6) We obtain: I h, (z) I < r, + see (rs) = 1,

z E A.

Since h,(w) = J,(w) E A it follows by the maximum principle that h, maps A into itself. Therefore by Corollary 1.1.1 Ih',(w)I

1

1-Ih., (W) 12 - 1-1w12

for allzEA. But hs(z) = I 1 + s a(2n) ) [Js(z)], . Hence

<

I [J9 Ml

1

1

1-IJ3(w)12

1

+s°T2

1-Iw12.

s

In other words, the mapping J, is a strict contraction with respect to the infinitesimal Poincare metric dp,, = 1

-Idwl IwI2 , w E A, defined in Section 2.2.

Integrating the above inequality we obtain the same conclusion for the hyperbolic metric p: P (J., (z), J'.M) <

p(z, w).

(4.5.7)

1 + s °(T' 2

Since a(r) is bounded on the interval [0, 1] it follows by (4.5.6) that r, -4 1 as

s --p 0 and a(r,) - a > 0 because of its continuity. Therefore there exist some positive b and c such that for all s E (0, b) we have a(r,) > E > 0. Then for such s we obtain the inequality P (J. (z), JS(w))

1

1 sE

p(z' w)'

which means that f is strongly p-monotone. Now setting s = t/n in (4.5.7) and using the exponential formula: Ft (z) = lim J(') (z) n-...

we obtain by induction:

p(Ft(z),Ft(w))

<-

lini

1

(1+ n. a2rn at

exp{The proposition is proved.

2

}p(z,w).

n P(z, W)

ASYMPTOTIC BEHAVIOR

123

Example 1. Let f E Hol(O, (C) be defined by

f(z)=a-az 2 +bz 1-cz 1 + cz' where a, b E C, Re b > 0, and 0 < c < 1. If we introduce the function

a(s) = -jaI(1 - s2) +

(Reb)s1 - cs

1+cs'

then we obtain

Reg(z)z > a(Izl)lzl and

a(1)=RebI

c>0.

Hence f (z) is a strongly semi-complete vector field on A.

Remark 4.5.1 Note that if f E Hol(A, C) is known to be a semi-complete vector field on A, then condition (4.5.2) can be replaced by a slightly more general condition, namely, a(l) > 0 for some 1 E (0, 11,

which still ensures f to be strongly semi-complete. The above arguments can be employed in the disk Al. This note leads us to the following simple sufficient condition yielding the existence of an interior null point of a semi-complete vector field and its attractiveness.

Corollary 4.5.1 (cf., EM-RS-SD-2000c) Let f E 9 Hol(A) be such that: Re f'(0) > 41f(0)1.

(4.5.8)

Then f has a unique null point T E A and the semigroup S1 = {Ft}t>o converges to T as t goes to infinity. Proof. Consider the function:

a(r) = -a(1 - r2) + br 1

+ r'

where a = If (0) 1, b = Re f'(0). It follows by Proposition 3.5.3 that

Re f(z)z > a(IzI)IzI,

z E A.

Then we have under condition (4.5.8), that a(O) = -a < 0, a'(1) < 0. Hence, there is l E (0, 1), such that a(l) > 0.

(4.5.9)

a(1) = 0, and

Remark 4.5.2 Note that conditions (4.5.8) and (4.5.9) imply that for some b > 0 each disk A, with r E (1- 5, r] is invariant for the semigroup {Ft} generated by f. However these conditions are not sufficient to ensure the validity of the strong p-monotonicity of f.

Chapter IV

124

Remark 4.5.3 Observe also that the above Proposition 4.5.1, Lemma 3.6.1, and Proposition 3.6.2 imply that the property of f E Hol(0, C) being strongly p-monotone is equivalent to the strong flow invariance condition

Ref(z)z>q>0 for all z in the annulus 11-5< z j < 11 for some 6 > 0. In terms of the Poincare metric this property is equivalent to the global uniform exponential convergence (see formula (4.5.3)) of the semigroup {Ft}t>o generated by f on all of A, whilst the property of f to be strongly semi-complete is equivalent to local uniform exponential convergence on each compact subset of 0 (see Proposition 4.4.3).

4.6

Continuous version of the Julia-Wolff-Caratheodory Theorem

Let now f c Hol(A) be a null point free mapping. We already know that in this case the semigroup Sf = {Ft}t>o generated by f is convergent (see Section 4.3). Moreover, there is a point r E 80 (a sink point of Sf), such that for each z E 0, lim Ft(z) = r. However, the study of rates of convergence in this case is more too complicated than in the case when f has a null point in A. Indeed, as we saw above, in the latter case the asymptotic behavior of the semigroup generated by f is completely determined by the value of f'(r). Namely, {Ft}t>o is convergent if and only if Ref'(-r) > 0. Therefore it is natural that the rates of convergence obtained in Section 4.4 are connected with this value. If f E 4 Hol(O) is null point free, i.e., S f = {Ft}t>o has a sink point r E 80, one cannot use the same approach, since f'(r) is not defined in general. In addition,

the above approach of using a Mobius transformation is also unfeasible in this situation. Therefore we need another method to study the asymptotic behavior of the semigroup generated by a null point free holomorphic function. A complete characterization of convergence of a semigroup to its sink point can be done by using the so called nontangential derivative of its generator in the spirit of the Julia-Wolff-Caratheodory Theorem (Proposition 1.4.2). Here we establish a continuous version of this result (Proposition 4.6.2) which will provide another

proof of the classical one. Note in passing that if F E Hol(A) is a holomorphic self-mapping of A, then f = I - F is semi-complete. We first prove some auxiliary assertions.

ASYMPTOTIC BEHAVIOR

125

Lemma 4.6.1 Let f E Hol(A, C) and let e E 80. If 0 := Z lim f (z) exists z-e z - e (finitely) then: (i) L lim f (z) = 0; z-e (ii) the angular limit L lim f'(z) also exists and equals to 3.

z-e

Proof. (i) is trivial. To prove (ii) let us assume that e = 1 and present f in the form f(z) = ,Q(z - 1) + h(z).

(4.6.1)

Then we have

Z lim h(z) = 0

z-l

and Z lim h(z) = 0.

(4.6.2)

L lim h'(z) = 0.

(4.6.3)

z-.1 z - 1

We wish to show that z-.1

To this end take two sectors S and S in A both with vertex at e = 1, so that S C S. For z E S we denote by r(z) the circle with its center at z, such that r(z) is tangent to the boundary of S. By 9 we denote the angle between segments [0, 1] and [z, 1] and let 29s and 29s

be the angles of the sectors S and S, respectively, at vertex e = 1 (see Fig. 4.2). Then sin(9S - Os) < sin(9S - 0) = 1 (z)I, (4.6.4) where r(z) is the radius of the circle r(z). Now observe that when z converges to 1 in the sector S, all points w E ar(z) converge to 1 in the sector S. Therefore using (4.6.2) we obtain that for each E > 0 there is a point z close to 1 such that h(w)I

Iw-11

< E,

w c r(z).

It then follows by the Cauchy formula and (4.6.4) that

h'(z)

=

1

h(z)

27ri

(w - z)2

dw

<E

r(z)

<

E

/

1w-11

27r J Iw-z12 r(z)

max lw-1I

27r wEr(z)

Idwl

,f

lw - z12

=

max Iw-11

E

r(z) wEr(z)

r(z)

r(z) r(z) + I1- zI) = E 1+

I '(-z) I

<

1+

dw

1

sin(9S - Os) )

Chapter IV

126

Figure 4.2: The circle r(z) and the sectors S and S. The latter inequality concludes the proof of the Lemma.

The value 3 := L lim f (z) is called the angular derivative of f at e C ao. z-.e z - e We say that a function f E Ho1(A, C) has the radial derivative at e E ao , if the limits

a := lim f (re) and

lim

f re

a

r-.1- (r - 1)e exist. We will denote it by T f'(e). It is clear that if f has the angular derivative at a point e E OA then it has a radial derivative at this point and

T f'(e) = Zf'(e). We will show that for a null point free semi-complete vector field a somewhat converse statement is also true at the sink point of the flow generated by f. First we establish the following assertion.

Lemma 4.6.2 Let f be a semi-complete vector field on A. Suppose that for a

ASYMPTOTIC BEHAVIOR

127

point e E aA there exists the radial limit derivative f (re)

lim

r-.1- (r -

1)e,

such that

Re3>0.

(4.6.5)

Then f has no null point in A. Moreover, the point e must be the sink point of the semigroup generated by f. Proof. Without loss of generality let as assume that e = 1, and write f (z) by the Berkson-Porta representation

f(z) = (z - r)(1 - zf)p(z),

(4.6.6)

where Re p(z) > 0 everywhere, and r is a point of i. We wish to show that r in (4.6.6) is equal to 1. First we assume that

ReQ>0

(4.6.7)

have by (4.6.6) and (4.6.7) and conversely suppose that r\1. Then we[urn Re,@ = Re

(Urn f (r)

1- r-

I

p(r) = Re r-l- (r - r)(1 - rT) rllJ rr

1 - r12 Re

lr

lim Rep(r)

i- r-1

> 0. J

On the other hand for all r E (0, 1) Re p(r) =

r-1 r-1 Re p(r) < 0, 1

which is a contradiction. To complete our proof for the general case we consider the mapping fE . A F-+ C defined by (4.6.8) fE(z) = f(z) + E(z2 - 1)

with E > 0. Since E(z2 - 1) is a complete vector field, the vector field fE is semicomplete and

Re [j ff(1)]=Re0+2E>0.

(4.6.9)

Therefore by the previous step, for each E > 0 the mapping fE has no null points

in A. Now, if r in (4.6.6) belongs to A, that is, f has an interior null point, then by the Rouchet theorem it follows that one can choose a small enough E such that fE has an interior null point rE close to T. Once again a contradiction. Moreover, (4.6.9) and the step proved above show that for each E > 0 the point 1 must be a sink point of fE. Thus there are functions pE E Hol(A, C) with Rep, > 0, z E A, such that M z) = -(1 - z)2pE(z).

Chapter IV

128

Comparing the latter formula with (4.6.8) we obtain

pE(z) - (z - r)(1 - zt)p(z) + Ez + 1

1-z

-(1-z)2

Since Rep, > 0 for all E > 0 and p,(z) converges to po(z)

(z - r)(1 - z )p(z) -(1 - Z)2

po(z) =

uniformly on each compact subset of A as E goes to 0, we obtain that

Repo>0. But -(1 - z)2po(z) = (z - T)(1 - zT)p(z) = f(z),

hence contradicting the uniqueness of the Berkson-Porta representation. Thus po(z) = p(z) and r = 1.

Proposition 4.6.1 If for a point e E 8A the radial limits

Tlimf'(z)=0 z-+e exist with Re /3 > 0, then

and

Tlimf(z)=0 z-,e

0
and e = r is a sink point of the flow generated by f.

Proof. Indeed, since If'(re)I < M < oo we obtain

<M(1-r).

If (re) l =

Hence lim inf f (re) exists. Let e = 1 and u(z) = Re f (z). Then we obtain r-.l- (r - 1)e if

u(r) _ -

Jr

u'(t)dt = u'(i7)(r - 1),

where 77 E [r, 1). Hence lim

u(r)

r-.l- (r - 1)

= Re/3 > 0.

To establish the converse assertion to Lemma 4.6.2 we will use the RieszHerglotz integral representation of functions of the class P = {p E Hol(O, C) Rep(z) > 0}(see Section 0.2): f (4.6.10) p(z) = 1 + z dµr(() + i Imp(O),

J

ao

1z

ASYMPTOTIC BEHAVIOR

129

where µp : aO F-+ R (the measure characteristic function for p E P) is a positive increasing finite function µp on the unit circle 80, such that

J ao

dpp(C) = Rep(0).

Lemma 4.6.3 Let f c Hol(O, C) be a semi-complete vector field on A with no null point in A. Then there is a point T E 80, such that f has an angular derivative at T. Moreover, if -r is a sink point of the flow generated by f then Z f'(7-) exists and it is a positive real number which is equal to 2µp(7-), where f(z)

p(z) =

(z -,r)(1 - zf)'

Proof. Of course, it is enough to prove only the second assertion of the Lemma.

Let us again present f by the Berkson-Porta formula

f(z) _ (z - 7-)(1 - zT)p(z), where Rep(z) > 0,

(z

z E A. Then by (4.6.10) one can write

(z)) = J (1 - zT) 1 + z( ao

(1 - zf)i Imr f (0).

(4.6.11)

Let zn E A be a sequence of points which converges to T nontangentially. Then if we write zn in the form zn = T(xn + iYn),

we can find 0 < K < oo, such that ynl :5 K(1 - xn)

(4.6.12)

(see Fig. 4.3). Further, consider the sequence of functions gn : aO --* C defined by

(1 - Tzn)(1 + zn() To estimate gn on the unit circle we calculate (1 - TZn)(1 + zn() = wn(1 - zn() or

zn( (1 - TZn + gn) = gn - (1 - TZn).

This implies that IZnI2 11 - TZn +g,I2 = Ign - (1 - TZn)I2

(4.6.13)

Chapter IV

130

Figure 4.3: The nontangential convergence to r.

or

9nI2(1 - IznI2) - 2 Re[gn(1 - Tzn)] (1 + IznI2) = 11 - TznI2(Iz,tI2 - 1). Consequently, 2

gn-

11

(1 - Tzn)(1 + IznI2)

- Tz,,12(1 + IznI2)2 - - Tznl2 11

I-Izn12

(1 - Iznl2)2 11 - TznI2

4

znl2

(1 - Iznl2)2.

Finally,

n -

(1 - Tzn)(1 + IZnj2)

=21z,1I11Tz2I

1 - IznI2

it

1-Iznl

and we obtain 1-Tznl

lwn(()I <4 1 - IznI2

-4

(1-xn)2+yn (1 - xn)2yn

Taking (4.6.12) into account we obtain z

4

1+xnIyn 1-x Since for n large enough

1 +xn - IynIK >

1

2'

1+K2

1+xn-IynIK.

ASYMPTOTIC BEHAVIOR

131

we have for such n 1 + K2.

Ign(()I < 8

In addition, it follows by (4.6.13) that

<#T.

gn(u)- 0,

Then by the Lebesgue Theorem (see, for example, [125]) and (4.6.11) we obtain lim

f (zn)

zn - T

-

- Tz)(1 + z() dµ() - i Im r f (0) lim (1 - znf) zn- T 1 - zn

if OA

2µp(7-),

and the Lemma is proved.

Our next goal is to establish relations between angular derivatives of semicomplete vector fields and the asymptotic behavior of their generated semigroups.

Lemma 4.6.4 Let F E Hol(O, A) be a holomorphic self-mapping of A with no null point in A, and let T E 0A be its sink point. If zn converges to r nontangentially, then so does the sequence F(zn).

Proof. It follows by the Julia-Wolff-Caratheodory Theorem (Proposition 1.4.2) that

0
cpr(F(z)) < c p.,(z),

4.6.14)

where 11 - zfI2 1 - Iz12

OTz)

Iz -TI' 1

- IZI2

If z E I'(7-, k) = {z E A : Iz - TI < k(1 - IzI)} for some k > 1 (see Definition 1.2.1) then (4.6.14) implies IF(z) - TI < aIz - TI

1 - F(z)I -

1 - IzI

Iz - TI

1 +IF(z)I

IF(z) -TI

1 + IZI

(4.6.15)

In turn, The Julia-Caratheodory Theorem states that L lim

Z,, -T

F(z) - T= z-T

a>0.

Therefore the third factor in (4.6.15) is bounded. Thus we can find kl > k, such that F(z) E F(T, kl) and we are done. Finally we are able to formulate our main assertion of this section which is a dynamical analog of the Julia-Wolff-Caratheodory Theorem.

Chapter IV

132

Proposition 4.6.2 (see [42]) Let f E G Hol(A, C) be a semi-complete vector field, and let S = {Ft}t>o be the semigroup generated by f. The following are equivalent: (i) f has no null point in A; (ii) f admits the representation f(z) _ -T(z - T)2p(z)

for some T E 80 and Rep(z) > 0 everywhere; (iii) there is a point T E 80, such that

If'(T)=Q exists and Re, 3 > 0;

(iv) there is a point r E 80, such that

Lf'(T) =a exists and Re,3 > 0; (v) there is a point T E 8A, such that the following limits exist

L lim f'(z) =)3 Z-.T

and L lim f (z) = 0 Z-+T

with Re,3 > 0; (vi) there are a point T E 8i and a real positive number y, such that IFt(z) - T12 < e_t1 IZ - T12 1 - Iz12 1- IFt(z)12 Moreover,

(a) the points T E 80 in (ii)-(vi) and the numbers (3 in (iii)-(v) are the same; (b) )3 is, in fact, a nonnegative real number which is the maximum of all -y > 0 which satisfy (vi).

Proof. Equivalence of (i)-(iv) has been proved in Lemmata 4.6.2-4.6.3, while equivalence (iv) and (v) follows from Lemma 4.6.1 and Proposition 4.6.1. Thus it is enough to show that (v)==>(vi) and (vi) implies one of the conditions (i)-(v). Suppose that (v) holds. We already know that f has no null point in A and e = T is the sink point of its generated semigroup. In addition, if Jr E Hol(0), r > 0, is a resolvent of f :

Jr(z) + rf (Jr(z)) = z, z E 0, (4.6.16) then T is also the sink point for Jr, r > 0. It follows by Lemma 4.6.4 that if z converges nontangentially to T, then so does Jr(z) for each r > 0. Hence

L Z-T lim f'(Jr(z)) _ ,3.

(4.6.17)

On the other hand, it follows by Julia's Lemma that for each r > 0 there is a number CYr, 0 < ar < 1, such that L lira J,'.(z) = cYr Z-.T

(4.6.18)

ASYMPTOTIC BEHAVIOR

133

and IJr(z) -TI2

1 - IJr(z)I2

< ar

Iz - TI2

(4.6.19)

1 - IZI2

At the same time, differentiating (4.6.16) we obtain

J,(z) +rf'(Jr(z)) J,(z) = 1, or

Jr(z) =

1

1 + r f'(Jr(z))

Using (4.6.17) and (4.6.18) we obtain 1

ar

(4.6.20)

1 + r/3

Substituting (4.6.20) into (4.6.19) and applying the exponential formula we obtain cpr(Ft(z))

lim

cpr(z) = exp

1

n-co (1+nQ) where

_ cPr(z) = I1 _

w, (z),

(4.6.21)

zl2

(4.6.22) 2

Thus the implication (v)=(vi) is proved. Obviously (vi) .(i), because in this case the semigroup {Ft} has no stationary point in A. Then it remains to prove assertion (b). In other words, we wish to show that if (vi) holds with some ^y > 0 then -y < ,Q. Indeed, consider the real valued function V) (t, z) = wr(Ft(z)),

(4.6.23)

where cpr is defined in (4.6.22). Since by (vi) z/'(t, z) < e-rytz/)(0, z) we have DO

at

It=o+

lim = t-o+

<

tlimo

V)(t, Z) - V)(0, Z)

t ,0(0, z)(t

1) = -'Y')(0, z .

(4.6.24)

On the other hand, differentiating (4.6.23) directly and using OFt(z)

at

It=o+= -f (z)

we obtain Ot

It=o+= -20(0, z) Ref (z)z*,

(4.6.25)

Chapter IV

134

where z

z z

1-Izl2

1-zIr

Comparing (4.6.24) with (4.6.25) we obtain y < 2 Re f (z)z'.

(4.6.26)

Let us again suppose for simplicity that T = 1. Then we may rewrite the right hand side of (4.6.26) in the form:

2Ref(z)z'=2Ref(z)

z

1 - Iz12

-

1

1

z

2Re

f(z)(z-1)

(1 - z)(1 - Jz12).

Setting here z = r and letting r --+ 1- we obtain by (4.6.26):

y < 2 lim Re r-.1-

= lim f (r) r-.1- (r - 1)

f (r)

(r - 1)(r + 1)

= Q.

The proof is completed. 0

This result will play a crucial role in our study of the spirallike and starlike functions with respect to a boundary point (see Chapter 5). At the end of this section we consider some examples and a consequence of the above proposition which is an extension of the Julia-Wolff-Caratheodory Theorem.

Example 1. Consider a semi-complete vector field f E Hol(0, C) defined as follows

zn+ f(z)=(1-z)2.

z-

Computations show that f'(1) = 2/n > 0, while Re f'(zk) < 0, where zk = -1, k = 1, 2, ... , n. Hence if {Ft}t>o is the selnigroup generated by f, then the point T = 1 is an attractive point of this semigroup and the following rate of exponential convergence holds

IFt(z)-112 1 - IFt(z)12 -

<e211z-112

1 - Iz12

for all z E sand t> 0. As we mentioned above, if F is a self-slapping of 0 then the function f (z) _ z - F(z) defines a semi-complete vector field on A. As a matter of fact, this fact holds even if F E Hol(A, C) is not necessarily a self-mapping of A, but satisfies the following one-sided estimate:

11111 F(rz)z<1forallzE8 r-l-

.

(4.6.27)

Thus we have the following version of the Julia-Wolff-Caratheodory Theorem:

ASYMPTOTIC BEHAVIOR

135

Corollary 4.6.1 Let F E Hol(A, A) satisfy (4.6.27) and z E A. Then the following statements are equivalent: (i) F has no fixed point in A; (ii) for some w E aA there exists the angular limit L lim

Z-W

F(z) - w z-w

.= a

with Re ck < 1;

(iii) F admits the representation F(z) = z + (z - w)2wp(z) for some w E ao and p E Hol(A, C) with Re p(z) > 0. Moreover, c in (ii) is actually a real number and the boundary points w in (ii) and (iii) are the same.

4.7

Lower bounds for p-monotone functions

We have seen in previous sections that the asymptotic behavior of semigroups generated by holomorphic functions can be described in terms of their derivatives.

If f E Q Hol(O) and T E Null(f) then r is (globally) attractive if and only f'(T), the derivative of f at T, lies strictly in the right half-plane. Moreover, f E g Hol(A) has no null point in A if and only if for some T E aA the angular derivative Lf'(T) = Q exists (finitely) with Q > 0. In addition, if S = {Ft}t>o is the flow generated by f then IFt(z) - TI2

1-Ft(z)I2

Iz - TI2 <exp(-t/3)1-Iz12,

and the point T E aA is a (globally) attractive sink point of S (even if /3 = 0). However, if L f'(T) = /3 = 0 the latter formula does not help to establish a rate of convergence of the semigroup to its sink point.

Note also that if f E GNp(O) is not holomorphic the characteristics of the derivatives are not relevant. Actually, we will show that for f E 9 Hol(O) the number 03 = L f'(T) is equal to inf 2 Re f (z)z*, (4.7.1) zE0

Chapter IV

136

where

T

Z

1 - Iz12

Z

1-

zT-'

It turns out that even if f E 9Np(A) is not holomorphic, expression (4.7.1) can serve as a characterization of the asymptotic behavior of flows of p-nonexpansive

mappings in 0 both in the cases of an interior stationary point and a boundary sink point. In this section we will mostly follow the material of [39].

For a fixed T E 0, the closure of 0, and an arbitrary z E 0, we define a non-Euclidean `distance' between z to T by the formula:

dr(z) =

11

- zTl2

1-Izl2

(1 - Q

(4.7.2)

(z,T)) ,

where

o(z T) --

(1 - Iz12) (1 - ITI2)

I1- zTI2

e

(see Section 2.3).

Exercise 1. Show that the sets

E(T,s)={zE0: d,(z)<s}, s>0, have the following geometric interpretation: (a) If T E 0, then these sets are exactly the p-balls

E(T, s) = B(T, r) = (z E 0 : p(z, T) < r ) s S + _1_

centered at T E A and of radius r = tanh-1

_17I2

(b) If T E ao, the boundary of 0, then these sets ;r-12

E(-r,

D

d, (z) =

<s

I

1l

s > 0,

are horocycles in 0 which are internally tangent to the unit circle a0 at T.

Now for fixed 7EDandzEBE(r,s)= consider the nonzero vector _

z

T,

l

/

1

z*

fzEA:dr(z)=ss>0,

1-Q(z,T)

1

1-

1

z2`

1-z? T

(4.7.3)

Exercise 2 ([6)). Show that z* is a so called support functional of the smooth

convex set E(-r, s) at the point z c 0, dr(z) = s, i.e., for all w E E(-r, s) the following inequality holds:

Re(wz*) < Re(zz*).

ASYMPTOTIC BEHAVIOR

137

In order to classify the asymptotic behavior of a flow generated by f E cNp(A) for a point rr E 0 we consider two real nonnegative functions on (0, oo): wb(s) :=

inf 2 Re f (z)z*, d,(z)<s

s > 0,

(4.7.4)

s > 0,

(4.7.5)

and

wd(s) :=

inf

d, (z)=s

2 Re f (z)z*,

where z* is defined by (4.7.3). If r E 2K is a stationary (or sink) point for the flow generated by f E GNp(O), then it follows that Ref (z)z* > 0 by p-monotonicity off (see Section 3.4). Hence WO(s) > Wb(S) > 0

and wb(s) is clearly decreasing on (0, oo). Let M(0, oo) denote the class of all positive functions w on (0, oo) such that is Riemann integrable on each closed interval [a, b] C (0, oo) and ds is divergent. w(s)s

(*) 0+

Note that for each w E M(0, oo) the function Q defined by d, (z)

SZ(s) :=

J

(4.7.6)

W(A)A

S

is a strictly decreasing positive function on (0, d, (z)] and maps this interval onto [0, oo). We denote its inverse function by V : [0, oo) --+ (0, d,(z)].

Definition 4.7.1 We will call a function w E M(0, oo) an appropriate lower bound for f c CJNp(A) if

w(s) < wa(s) =

inf

d., (z)=s

2 Re f (z)z*,

s > 0.

Exercise 3. Show that if wb defined by (4.7.4) is not zero then it is an appropriate lower bound for f E QNp(O).

Exercise 4. Let p E Hol(O, C) be such that Re p(z) > 0, z E A and

Zlim(1-z)p(z)>0. Show that if f E G Hol(O) is defined by the Berkson-Porta formula (see Section 3.6)

f(z) = -(1 - z)2p(z),

then for the sink point T = 1 the function wb is constant which is equal to 3.

Chapter IV

138

Proposition 4.7.1 Let f E GNp(0) be continuous and let S = {Ft}t>o be the flow generated by f . Given a point r E 0 and a function w E M (0, co), the following conditions are equivalent: (i) the function w is an appropriate lower bound for f ; (ii) for any differentiable function W on [0, oo) such that V(t) < W(t), V(0) _

W(0) and V'(0) = W'(0),

dr(Ft(z)) < W(t),

z E 0, t > 0,

where V = Q-' and S2 is defined by (4.7.6). In particular, dr(Ft(z)) < V(t); hence r is a globally attractive stationary point for S.

Proof. Consider the function W : R+ x 0 H R+ defined by I' (t, z) = dT(Ft(z))

(4.7.7)

By direct calculations we have a%p

It=o+= -24, (0, z) Ref (z)7.

at

(4.7.8)

We first assume that condition (ii) holds. Since t(0,z) = d, (z) = W(0), we obtain by (4.7.8) and (ii) that 2'(0, z) Re f (z)z*

_

at I t=o+?

dt [I't'(t)]t=o+

-dt [V(t)]t-o+ _

si'(dT (z))

dr(z)w (dr(z))

Varying z E *E(r, s) = {z E 0 : dr(z) = s} we see that the latter inequality immediately implies (i). Conversely, let condition (i) hold. It follows by (4.7.7) and the semigroup

property that for all z E 0 and s, t > 0,

xP (s+t,z) = T (s,Ft(z)). Hence by (4.7.8) and the continuity of f, 41 is differentiable at each t > 0 and we deduce from (i) and (4.7.8) that: a%k (t, z)

at

< -T(t, z)wa (k (t, z))

-

-1P (t, z)w (1k (t, z)) .

Separating variables we obtain d, (z)

f

d,- (F,. (z))

dq

= Il (drFt(z)) > t.

ASYMPTOTIC BEHAVIOR

139

This is equivalent to condition (ii). Our assertion is proved. As we mentioned above, if wy (s) := inf) < s 2 Re f (z)z* is positive then it may

be used as an appropriate lower bound for f E GNP(A) (see Example 1).

Example 1. Let f : A H C be defined by f (z) _ -(1 - z)2

1 + zn Z"

where n is a positive integer. Since f is holomorphic on A it follows by the Berkson-Porta representation (see Section 3.6) that f generates a flow S = {Ft}t>o of holomorphic self-mappings of A and T = 1 is a sink point of S. If we now set z

1

1-Iz12

1-2

z* =

then we obtain

Ref(z)z*=

l--

2

zn Re 1 +

zn

=di(z)Re 11 +- zn >0. 1 - zn

1

In addition, it can be shown (see Exercise 4 and Proposition 4.7.5 below) that 2

w(s) = wy(s) =

inf

di(z)<s

2 Re f(z)z* = constant =

n

In this case di(z)

St(s)

- n 2

dA

,f

= -n In

A

2

s

di (z)

8

and

Thus we have an exponential rate of convergence of the flow S to the boundary point T = 1: 2 l 11 - z 2 Ft(z) - 12

1-IFt(z)I2

<exp(-nt5 1-Izl2.

Note also that although f has n + 1 null points {ak : k = 1, 2, ... , n + 1} on the unit circle, only a1 = 1 is an attractive point of S = {Ft}t>0. The reason is that Re f'(al) > 0, while Ref'(ak) < 0, k = 2, 3, ... , n + 1 (see Figure 4.4 for n = 3). Remark 4.7.1. However, examples show (see Example 2 below) that sometimes wb may be zero identically, while wO itself belongs to the class M(0, oo). Moreover, we will see below that for a semigroup of holomorphic mappings with a boundary sink point T the function wy is always a constant which coincides with the angular

Chapter IV

140

111111\\' 111 1\\

Y

\\ 1

1

\1

!l!

1/1 H \\\\\ fill\\\\1

\\

11/

\\\\\\\\

I

l

JJJJJ / /

/////// 1/I / >

\\1i/ 11

11111!//rz Figure 4.4: The asymptotic behavior of the flow generated by 1

3

f(z)=-(1-z)2 1 + z 3 derivative of f at T. Also observe that the same estimate as in Example 1 can be obtained by using Proposition 4.6.2. Nevertheless, even for holomorphic mappings Proposition 4.7.1 becomes an effective tool when the angular derivative L f'(T) = 0.

Example 2. Let f : A ti C be defined by f(z)_-(1-z)21+CZ

1 - Cz'2

with lcl < 1. Once again, if we define z* as in Example 1 we have Izi2

Ref (z)z*

2

Re i + Cz,

> dl(z) i +

Icl

> 0.

In this case Lf'(1) = wb(s) = 0 for all s E (0,oo) and we cannot use it as an appropriate lower bound. However, we can define w(s) = as, where a = 1 - cl

1 + Icl,

and we find di(z)

1 f dA A2

a 9

1

a (\s

dl(z)

l

Thus we obtain by Proposition 4.7.1 the following rate of nonexponential con-

ASYMPTOTIC BEHAVIOR

141

vergence:

I1-Ft(z)I2 <

I1-zI2

1

1 - IFt(z)I2

1 + atdl(z) 1 - Iz12

The reason is that f E C Hol(A) has no null point inside A and wy (s) defined at the boundary sink point -r = 1 is equal to zero. Next we consider the case when f E GNP(O) is not holomorphic on A. In this situation the convergence of the flow generated by f may be of nonexponential type even f has an interior null point.

Example 3. Let z = x + iy E A. Define f : A -4 C by f (Z) = x7/3 + iy7l3. Since

Re f(z)z = x10/3 + y10/3 > 0,

f is p-monotone and the origin is the unique null point of f. Then, setting 7 = 0, we have 2 IzIlzlz

do(z)

and WO(s)

= 2 do inf

1

Izlz(1 - Izlz)

s

Re f (z)z

x10/3 + y10/3 2 x2+y2 inf

=

$-j (x2 + y2)(1 - x2 - y2)

21/352/3(1 + 5)113.

Since wv(s) E M(0,oo) we can set w(s) = wd(s) and we have do(z)

_

dA

S2(s) _

(A)A s

do(z)

1

21/3

f

dA

\5/3(A + 1) 1/3

s

Inverting this function we obtain the estimate

do(Ft(z)) < V(t) =

do (z)

3

3/2

124/3

1

td0(z)2/3 + (do(z) + 1)2/3]

The latter inequality is equivalent to the estimate IFt(z)I

IzI 1]3/4.

(21z1)4/3 3 t+

- do (z)

Chapter IV

142

Note that one can calculate Ft directly by solving the Cauchy problem and obtain

=

yz

xz

IFt(z)1z (3x4/3t +

1)3/2 +

((

(

y4/3t +

1)3/2

Thus for x = y we obtain I3I

Ft(z)I

(2 z )4/3 3

3/4

t+1J

So the rate of (nonexponential) convergence we have obtained is sharp.

Remark 4.7.2. We saw above (see Sections 4.4 and 4.5) that a similar phenomenon is impossible for holomorphic mappings: Namely, if a flow of holomorphic self-mappings converges locally uniformly to

an interior stationary point then the convergence must be of exponential type. These examples and Proposition 4.7.1 above motivate the following definitions.

Definition 4.7.2 Let S = {Ft}t>o be a flow with a stationary (or sink) point r E L. We will say that the asymptotic behavior of S at -r is of order not less than a > 0 if there is a function w E M(0, o(o) such that lim inf Jll w/s)

sl/«

I

(1 + !w(dT(z))) dT(z)

dT(Ft(z))

(4.7.9)

(4.7.10)

for all z C A and t > 0.

Definition 4.7.3 We will say that the asymptotic behavior of S at r is of wexponential type if there is a decreasing function w E M(0, oo) such that

dT(Ft(z)) < exp (-tw(dr(z))) dr(z)

(4.7.11)

for all zEL andt>0. In particular, if w can be chosen to be a positive constant a then we will say that S has a global uniform rate of exponential convergence:

d,(Ft(z)) < exp(-ta)dr(z).

(4.7.12)

The following assertion is a consequence of Proposition 4.7.1.

Proposition 4.7.2 Let S = {Ft}t>o be a flow generated by f E cNp(0) with a null (or sink) point -r E A. Then the asymptotic behavior of S at r is of order not less than a > 0 if and only if there exists an appropriate lower bound w E M(0, oo)

for f such that (**)

(/) is decreasing on (0, oo).

s

ASYMPTOTIC BEHAVIOR

143

Proof. We first observe that condition (4.7.10) with some w E M(0, oo) satisfying (4.7.9) is equivalent to the same condition with a function w1 E M(0, oo)

which satisfies both (4.7.9) and (**). Indeed, for a given w E M(0, oo) define a function p : (0, oo) i--' (0, oo) by

µ(s) = inf

f w(l) l1/a

It is clear that µ(s) is decreasing. Setting now wi(s) = s1l" . µ(s) we clearly see that wl satisfies (4.7.9) and that wl(s) < w(s). Hence ds

J wl(s)s

0+

is divergent and w1 E M(0, oo). Then the inequality 1

1

+ -Lwl(s)]'

[1 + aw(s)]

proves our claim. Thus we can assume for the rest of the proof that w satisfies (**). It remains to be shown that w is an appropriate lower bound for f. Indeed, defining 0 : (0, d1(z)] [0, oo) by (1.7) and using (**) we have

S2(s)

_

dr(x)

f

d\ _

I

w(A))

9

d, (z)

<

d.(x)

[dT(z)]° w (dT(z))

A dA +1

w(A).

d.

[s-2; (d,-(z)). w (d,-(z))

- 1]

.

Inverting this expression we obtain

V(t) := 0-1(t) <

1

(1 + lw(d,(z)))

dr(z) := W(t).

It is clear that the function W (t) satisfies all the conditions of Proposition 4.7.1. This completes the proof of Proposition 4.7.2.

Corollary 4.7.1 Let S = {Ft}t>o be a flow generated by f E gNp(O) with a null (or sink) point r E A. Then: (i) The asymptotic behavior of S at r is of w-exponential type if and only if inf rwp (l)

:

I E (0, s] } > 0,

s > 0.

(4.7.13)

Chapter IV

144

(ii) The flow S has a global uniform rate of exponential convergence if and only

if wp(s) > a

(4.7.14)

for some a > 0. Indeed, in both cases (i) and (ii) there is one function w E M(0, oo) such that the asymptotic behavior of S at T is of order not less than a for all positive a. In case (i), w can be chosen to be

w(s):=inf(wp(l): 1E(0,s]5>0, s>0, while in case (ii) w can be chosen to be a constant a. The following example shows that for a semigroup of p-nonexpansive (but not holomorphic!) mappings an asymptotic behavior of w-exponential type does not imply, in general, a global uniform rate of exponential convergence.

Example 4. Define a continuous mapping f : 0 -- C by the following formula:

f(x + iy) = x(1 - x)2 + iy(1 - y)2.

Since Re f (z) 2 > 0 for all z = x + iy E 0, it follows that f is a generator of a semigroup S = {Ft}too of p-nonexpansive mappings such that each disk A, = {z E C : IzJ < r < 1} is Ft-invariant. Setting r = 0 and z* = Izl2(1 z- Izl2), we have x2(1 - x)2 + y2(1 - y)2

wd(s) =

P. f(z)i* =

inf

inf

x2+y2=--&.-

do(z)=s

(x2 + y2)(1 - x2 - y2)

It is easy to see that lim we(s) = 1 while we(s) -+ 0 as s example, y = 0 and x =

oo (take, for

-+ 1).

Remark 4.7.3 For holomorphic mappings, however, condition (4.7.14) holds for some a > 0 whenever condition (4.7.13) holds for a decreasing positive function w. In other words, for holomorphic flows any convergence of w-exponential type implies global uniform exponential convergence.

To explain this phenomenon in terms of lower bounds we observe that for a holomorphic generator with an interior null point the function wy(s) =

inf

dr(z)<s

2 Ref (z)z* < w, (s)

(4.7.15)

is bounded from below by a positive number, while for a boundary sink point the function wy is just a constant.

ASYMPTOTIC BEHAVIOR

145

In both cases (interior stationary point or boundary sink point) the asymptotic behavior of a flow generated by f E B Hol(O) is completely determined by the value

wd (0) := lim inf wd (s) which is related to the value of derivative of f at its null point (for the interior case) or the angular derivative (for the boundary case).

Proposition 4.7.3 Let f E G Hol(O) and let be the flow generated by f . If for some point r E 0 there is a decreasing function w : (0, oo) H (0, oo) such that

z E A, t > 0,

dr(Ft(z)) < e-tW(dr(z)) d, (z),

(4.7.16)

then there exists a number p > 0 such that dT(Ft(z)) < e-tµd,-(z),

z E A, t > 0.

(4.7.17)

Moreover,

(i) if r E A, then p can be chosen as p = wb(0)/4, but p cannot be larger than wy(0) (= lim wb(s)); s-o+ (ii) if -r E 8L, then the maximal p for which (4.7.17) holds is exactly wb(0), that is 0 < p < wb(0). The proof of this Proposition is based on the following two lemmata.

Lemma 4.7.1 Let f E G Hol(O) with f (0) = 0, and let wb and wd be defined by (4.1.4) and (4.1.5), respectively. Then: (i) wd(0) = wb(0) = 2 Re f'(0) := 2v; (ii) v/2 < wb(s) < 2v.

Proof. First we show that wd(0) < 2v,

(4.7.18)

where

v = Re f'(0). Since in our case r = 0, we have Ref (z)z* =

Iz2(11

Izl2) Ref (z) 2.

Now fixing ( E 80, we set z = r(, where r E (0, 1). Then we obtain

Ref(z)z**=Re1 172

f(ro

Therefore

wd(s) < 2 Re

1

2 1

7

f (r() ,

where r2 = Iz12 =

Letting s (hence, r) tend to zero we obtain wd(0) < 2 Re f'(0).

s+

Chapter IV

146

On the other hand, it follows by Harnack inequality that for all z E 0 Ref (z)z > vIzI2

1 - IzI 1 + IzI'

This implies that 2 Ref (z)z*

IzI2(12

=

Ref (z)z

IzI2) 2vIz12

1 - IzI

Iz12(1- Iz12)

1 + IzI

>

-

2v (1 +

Izl)2.

Hence

(s)

_

2v

wb

s (1

d, (z)

+

IzI)2

IzI2
2v

ll+

2v (1 + IzI)2

v/2).

(4.7.19)

+s ) 2

Letting s tend to 0+ in (4.7.19) we see that wb(0) > 2v. Comparing the latter inequality with (4.7.18) and (4.7.15) we obtain (i). On the other hand, substituting now in (4.7.19) v = wb(0)/2, we obtain assertion (ii) and we are done.

Suppose now that 9 Hol(O) has a null point different from zero, say, f (T) _

0, TEA, T#0. Note that if the automorphism MT of 0 is given by /L

A

=

T - z

1-zT'

then the following equality holds: dT (11%IT(z)) = (1

- 17-12 ) do (z).

(4.7.20)

Now let us consider the flow {Gt}t>o C Hol(A) defined by

Gt=MToFtoMT

(4.7.21)

and let g E 9 Hol(O) be its generator, i.e., g(z) = -atGt(z)I t=o, = [(MT)'(z)]-1 f (MT(z)).

(4.7.22)

Then Gt(0) = 0 for all t > 0 and g(0) = 0.

Lemma 4.7.2 The functions wd(s) and wb(s) are invariant under the transformations (4.7.21) and (4.7.22).

ASYMPTOTIC BEHAVIOR

147

Proof. We have seen already in (4.7.7) and (4.7.8) that 8t

[dT(Ft(z))] _ =-2dr(z)Re f(z)z+ t_o+ zl

=-2 1-Q z T

Re

[f(z) ( 1- z2 I

T - 1-z;r

IJ.

(4.7.23)

.

(4.7.24)

On the other hand, by (4.7.20) and (4.7.21) we have

at [dr(Ft(z))] t=o+ =

at

[dT(MTGt(w))] t=o+

-

-at [(1

I_r12)do(Gt(w))] t=o+

Since

at

[0cw]

do

(w) 2 Re

l

((w)

w

-

dT(z)

(1 - Irl2)IwI2

2 Re (g(w)

\

1 - Iw12

we obtain from (4.7.24) and (4.7.23) the required equality

1 - (z, r) Re [f (z) (1_ zIz

2

-1

Zt

) ] = Iw12 Re [g(w)

ID

I

1

The Lemma is proved.

Let the flow {Gt(z)}t>o C Hol(O) and its generator g E 9Ho1(A) be defined by (4.7.21) and (4.7.22). By Lemmata 4.7.2 and 4.7.1 we have inf

do(w)=s

Re gw)w* > (

wy(0)

-

4

2

Then by Corollary 4.7.1(ii) we have do(Gt(w)) < do(w) exp (v/2)

for all w E A.

Finally, setting w = MT(z) and using (4.7.20) we conclude that dT(Ft(z)) < dT(z) exp (v/2).

(4.7.25)

This enable us to point out the following rates of convergence.

Proposition 4.7.4 Let {Ft}t>o be a flow generated by f E 9Ho1(A) and let f (T) = 0, r c A. Then the following estimates are equivalent:

Chapter IV

148

(i)

d.r(Ft(z)) < dr(z)exp(-tµ), z E A, t > 0;

(ii)

I MT(Ft(z))I <- I MT(z)I exp

(iii)

IM., (Ft(z))I

IMT(z)I

-µ 1 - IMr(z)I2t 2 -v1-IMT(z)It

exp

1 + IMT(z)I

,

zEA, t>0;

zE0, t>0,

where the numbers µ in (i) and (ii) can be chosen to be one and the same such that 0 < v/2 < µ < 2v and v in (iii) is defined by 21

v = wb(0) = 2wd(0) = Re f'(r).

(4.7.26)

Proof. First we note that inequalities (ii) and (iii) are equivalent to the following ones: (22*)

IGt(w)I <- IwI exp

(iii*)

IGt(w)I <- IwI exp

(-µl

2wI2t),

Vi +-

twit

),

t > 0,

t > 0,

where w = MT(z) E 0 and the flow {Gt}t>o is defined by (4.7.21). First let us suppose that estimate (i) holds. By using (4.7.20) for this flow we have do (Gt(w)) < do (w) exp (-tµ).

Rewriting the latter inequality in the form IGt(w)I2

1 - IGt(w)I2

<

IwI2

1- IwI2

ex p(-

tµ),

we obtain by direct calculations IGt(w)12 < IwI2 IwI2

+ (1 -IwI2) exp (-tµ)

IwI2 exp (-tµ(1- Iw12)).

which coincides with (ii*). Now we will assume that inequality (ii) (and hence (ii*)) holds. Differentiating

both sides of this inequality with respect to t at t = 0+ we obtain

-IwI Reg(w)w < -Iwlµ 1

2wI2

(4.7.27)

This implies that wv(s) > µ. Thus the function w(s) - µ is an appropriate lower bound and the implication (ii) (i) follows by Proposition 4.7.2. Let us suppose now again that inequality (ii) (hence, (ii*) and (4.7.27)) holds

with some number µ > 0. Setting in (4.7.27) w = r(, ( E 80, r E (0, 1) and

ASYMPTOTIC BEHAVIOR

149

letting r to zero (cf., the proof of Lemma 4.7.1) we obtain Re 9'(0) > µ/2 > 0. A direct calculation shows that

g'(0) = [(MT)'(0)[-' f'('r) (MT)V (0) = f'('r) and so v > 0. Therefore, again by Harnack inequality, we have

Reg(w)w > Reg'(0)

1 - IwI > vl'w121 -

1+1wI -

w1

1+IwI.

On the other hand, aln IGt(w)i

-

Gt(

Re [g(Gt(w))Gt(w)]

w

Also it follows by the Schwarz Lemma that jGt(w)j < jwj. Thus we have eln IGt(w) I

at

-

v1 - I'wl

1+IwI.

Integrating this inequality we obtain the following estimate

lnJGt(w)j -Injwj < -v

1+-

1

which implies (iii*).

Finally, if condition (iii*) holds then differentiating it with respect to t at t = 0+ we obtain > 0. Reg(w)w > (1 + wj)2 > 4 Thus wO(0) > 0 and the result follows.

Let us turn now to the case when f E G Hol(A) has no null points in A. As a matter of fact, if S = {Ft}t>0 converges to a boundary sink point r E aL with a rate of convergence of exponential type, dT (Ft (z))

l

< exp (-tw(dr (z) )) dr (z),

where w E M(0, oo) is a decreasing function, then this estimate can be improved as follows:

d7(Ft(z)) < exp(_tw(0))dT(z),

where w(0) := lim w(s). In other words, we claim that if the inequality wp(s) > w(s)

holds for a decreasing w then the stronger inequality wp(s) > w(0)

Chapter IV

150

also holds. In particular, this property holds for the function w = wy. This implies, in turn, that wy is actually constant: wy(s) = wb(0) = Q for all s E (0,oo) and is

equal to the angular derivative of f at the point r E 8A. Moreover, this number /3 gives the best rate of exponential convergence of S = {Ft}t>0. Proposition 4.7.5 Let the function f E Hol(A, C) be a generator of a semigroup S = {Ft}t>o of holomorphic self-mappings of A. Suppose that f has no null-point in 0 and that T E Oi is the boundary sink point for S. Then the following are equivalent:

(i) the asymptotic behavior of S at T is of w-exponential type; (ii) there is a positive number -y such that

dT(Ft(z)) < dT(z) exp (-try), z E 0 and t > 0. Moreover, the maximal -y which satisfies condition (ii) is

/3 = /f'(-r) = 2 info Ref (z)z*. Proof. Let condition (i) hold, i.e., for some decreasing function w E ,M(0, oo),

dT(Ft(z)) < dr(z) exp (-tw(dr(z))) or explicitly,

1 - Ft(z)TI2 <

1-IFt(z)I2

11

- zTl2 exp (-2tw(d,(z)))

1-Iz12

This is equivalent to the inequality 11

- Ft(z)T12 < 1 I1-zfI2

-

IFt(z)I2

1-IzI2

exp (2tw dT(z)))

(4.7.28)

Once again it follows by the Julia-Caratheodory Theorem that for fixed t > 0, b(Ft)

:= liminf 1 - FF(z)I Z_T 1 - IzI 1 - Ft(z)T = L[Ft]'(T) := lien Z-T 1 - zT

(4.7.29)

where that last limit is taken along an nontangential approach region at T. Let us denote w(0) = lim w(dr(ST)).

s-l-

Thus setting z = sT in (4.7.28) and letting s tend to 1- we obtain b2(Ft)) < b(Ft) exp (-2tw(0)), or

d(Ft) < exp (-ty), where we set y = 2w(0).

STARLIKE AND SPIRALLIKE FUNCTIONS

151

Now by using Julia's Lemma we obtain the implication (i) = (ii). The converse implication can be shown by differentiating the inequality in (ii) at t = 0+. Namely, we obtain

Ref(z)z**> 2 >0. So one can set w(s) - y/2 and the asymptotic behavior of S at r is seen to be of exponential type. In addition, it follows by Proposition 4.6.2 that -y !5,3:= L f'(r). Finally, we observe that

lim Re f (sr)(sr)*

s-*1-

lim Ref (sT)

r-1-

s-r

(1-s2

;r

1-s

Ref (sr)T 1 = 0/2, lim S- 1 1 +S 1-

and the assertion is proved.

Chapter 5

Dynamical approach to starlike and spirallike functions This chapter is devoted to showing some relationships between semigroups and the geometry of domains in the complex plane. Mostly we will study those univalent (one-to-one correspondence) functions on the unit disk whose images are starshaped or spiralshaped domains. Several important aspects, however, had to be omitted, e.g. convex and close-to-convex functions (see, for example, [57, 55]), and other different classes of univalent functions. We have selected the forthcoming material according to the guiding principle that the demonstrated methods may be generalized to higher dimensions. For example, the celebrated Koebe One Quarter Theorem states that the image of a univalent function h on 0 normalized by the condition h(0) = 0 and h'(0) = 1 contains a disk of radius 4. This theorem is no longer true at higher dimensions. Nevertheless, the dynamical approach analogues of the Koebe theorem have been recently established and used for subclasses of starlike (or spirallike) functions (see, for example [141, 109, 26, 56, 14]).

Our second objective is to study the dynamics of starshaped (or spiralshaped) domains when the origin is pushed out to the boundary. For example, a domain which is starshaped with respect to a point may fail to be starshaped with respect to another point. We will study inter alia some unified conditions which describe starlike (or spirallike) functions which are independent of the location of their null points.

153

Chapter V

154

5.1

Generators on biholomorphically equivalent domains

Although the studies in the previous chapters were carried out on the unit disk, one can translate them to any simply connected domain (which differs from C) of the plane as the consequence of the Riemann Mapping Theorem. First we recall some notions and definitions in classical function theory.

Definition 5.1.1 Let D be a domain in C. A function h E Hol(D,C) is said to be univalent on D if for each pair of distinct points z1 and z2 in D we have h(zl) # h(z2).

The set of all univalent functions in a domain D C C will be denoted by Univ(D).

For h E Univ(D) one can define the inverse mapping h-1 : SZ H D, where 11 = h(D). The content of the Open Mapping Theorem (see, for example, [122])

is that 12 = h(D) is also a domain (open connected subset) in C. In addition, h-1

E Hol(1, D). In other words, h is one-to-one and h-1 is also holomorphic on h(D). In this case f is also called a (globally) biholomorphic mapping on D. A mapping h c Hol(D, C) is said to be locally biholomorphic on D if for each point z E D there is a neighborhood V C D, of this point such that h E Univ(V). It is well known that h E Hol(D, C) is locally biholomorphic on D if and only if h'(z) # 0 everywhere (see, for example, [122, 128]). Two domains D and SZ in C are called biholomorphically (or conformally) equivalent if there exists h E Univ(D) such that Sl = f (D). The fundamental Riemann Mapping Theorem states that every simply connected domain Sl in C (but not C itself) is biholomorphically equivalent to the open unit disk A in C. Moreover, for each a E SZ there is a unique h E Univ(O) with 12 = f (A) such that h(0) = a and h'(0) > 0.

For the special case when D = A is the open unit disk in C, the subset of Univ(D) normalized by the conditions h(0) = 0 and h'(0) = 1

will be denoted by S(O). This notation conforms to the one used in the classical geometric function theory. In this case we simply write

S (= s(o)) = {h E Univ(o) : h(0) = 0 and h'(0) =1}. In other words, the class S C Hol(O, C) consists of all the mappings h E Univ(O) such that h has the following Taylor series at the origin: 00

h(z) = z + E akzk. k=2

STARLIKE AND SPIRALLIKE FUNCTIONS

155

Thus, referring to some geometrical properties of a simply connected domain SZ containing the origin, if we are permitted to shrink or expand it we can find a function h E S (= S(O)) for which the domain 11 = h(i) is similar to Q. (Of course, we can translate a domain, if necessary, so that the origin would be its interior point). However, in this way we may sometimes loose some features of the dynamical transformation of a domain 0 if its geometrical characteristics are related to a certain given fixed point in C. In particular, it happens if such a point lies on the boundary of Q. The following simple assertion is the key to our further considerations.

Proposition 5.1.1 (Main Lemma) Let D and SZ be two domains in C, such that S2 = h(D) for some biholomorphic (univalent mapping) h. Then there is a linear invertible operator T on the space Hol(Q, C) onto the space Hol(D, C), which takes the set Q Hol(c) C Hol(Q, C) onto the set Q Hol(D) (i.e., Q Hol(D) = T(QHol(i)). Moreover, such an operatorT : QHol(c) <--ti GHol(D) can be given by the formula: [h'(.)]-1 where cp E Q Hol(SZ).

Proof. Let cp E Q Hol(i) and let S. = {0t}t>o be the semigroup of holomorphic self-mappings on a generated by V. Then it is clear that the family {Ft}t>o defined by:

Ft=h-'ogtoh

(5.1.2)

is a semigroup of holomorphic self-mappings of D, which is continuous with respect

to t > 0. By f E Q Hol(D, C) we denote the generator of this semigroup, i.e.,

f(z) = lim

z - Ft(z)

c

-

d tt(z)

It=o .

(5.1.3)

Then substituting formula (5.1.2) into (5.1.3) and using the chain rule we obtain:

f(z)

[(h')'(&(h(z)).

= (h-1)'(h(z))

dOt(h (z))1

. P(h(z))

c=o

= [h'(z)]-1 W(h(z)),

z e D,

i.e., f = Tco E Q Hol(D), where T is defined by (5.1.1). Since h'(z) 0 it is clear that T is a well defined linear action on Hol(Q, C) into Hol(D, C). In addition, since h is biholomorphic on D the operator T is invertible and for each f E Hol(D, C), its inverse T-1 : Hol(D, C) --* Hol(Q, C) can be defined by

T-1(f)(w) =

[(h-1)'(w)] -1 f(h-1(w))

(5.1.4)

Repeating the above considerations we can see, that if f in (5.1.4) belongs to Q Hol(D), then cp = T-1(f) belongs to Q Hol(c), that is T maps Q Hol(c) onto 9 Hol(D).

Chapter V

156

Remark 5.1.1. It follows by formula (5.1.2) that if cp E aut(1), then f = Tcp E aut(D) and conversely. Thus T maps the set aut(f2) of all complete vector fields on SZ onto the set aut(D) of all complete vector fields on D. If, in particular, D = Sl then aut(D) is an invariant subset of Hol(D, C) for operator T defined by (5.1.1). Remark 5.1.2. If cp E G Hol(S2) has a null point b E 0, then so does f = Tcp, and a = h-1(b) is a null point of f. Indeed, f (a) = [h'(a)]-1 p(h(a)) =

[h'(a)]-1

cp(b) = 0.

In addition, note that by direct calculations we obtain:

f'(a) = cp'(b).

(5.1.5)

Therefore a is an attractive fixed point of the semigroup {Ft} = Sf if and only if b is an attractive fixed point of the semigroup {ct} = Sq.

Remark 5.1.3. By using Proposition 5.1.1 and Remark 5.1.2 we are able to easily obtain a description of the class G Hol(0, T) of all semi-complete vector fields

vanished in A, at a point T E A, which is a particular case of the representation that according to E. Berkson and H. Porta (see Section 4.6). Indeed, let us set D = S2 = A and h = MT E aut(A) for some r E A, where M'-(Z) = T - z

(5.1.6)

1-zT

If f E G Hol(A,T) is a semi-complete vector field, such that f (T) = 0 for some T E A, then cp = T-1 f is an element of G Hol(A, 0) such that cp(0) = 0 (note that h-1(T) = 0). Then cp has a representation: W(z) = z pi (z) when Rep1(z) > 0 for all z > 0. Then by Proposition 5.1.1 we obtain:

f(z) = [h'(z)]-1 W(h(z))

[h'(z)]-1 . h(z) p(MT(z)) (1 - z;r)2 T - z p(MT()) z

ITI2-1

Now, if we denote q(z) _

1

-1ITI2 p(MT(z))

1-zT

we obtain the desired representa-

tion: f(z) = (z - T)(1 - zf)q(z),

(5.1.7)

where Re q(z) > 0 everywhere. As a matter of fact, we know that representation (5.1.7) holds also for a null

point free f c G Hol(A). In this case just T E 80. Therefore we can try to use this representation to characterize different biholomorphic mappings on A and the geometric structure of their images.

STARLIKE AND SPIRALLIKE FUNCTIONS

5.2

157

Starlike and spirallike functions

Definition 5.2.1 A set S2 in C is called starshaped (with respect to the origin) if given any w E 1, the point tw belongs to SI for every t with 0 < t < 1. That is, if Si contains w then it also contains the entire line segment joining w to the origin. If D is a domain in C then a function h E Univ(D) is said to be a starlike

function on D if the image SI = h(D) is a starshaped set (with respect to the origin).

In this definition the origin is in S2. If, in particular, the origin belongs to SI

then we will say that h is a starlike function with respect to an interior point. In this case the function h has a null point r in D. The image of the starlike function f (z) =

z (1 - 0.9x)0.7(1 + 0.9iz)°-5(1 - (0.7 + 0.4i)z)0.6(1 + (0.56 + 0.7i)z)0.3

on the unit disk A is illustrated in Figure 5.1.

Figure 5.1: A starshaped domain (0 E S2).

If the origin belongs to the boundary t%I of Si we say that h is starlike with

respect to a boundary point (or fanlike on D). In this case there is a boundary point -r E 8D and a sequence {zn,}°°_1 E D, converging to r such that lim h(z,,,) = 0. n-oo

The domain Si in Figure 5.2 is the image of a fanlike (starlike with respect to a boundary point) on the unit disk A function (1 + z)2

f(z) -

(1 - 0.9z)°.5(1 + 0.9iz)°.7(1 - (0.7 + 0.5i)z)°.6(1 + (0.6 +

0.67i)z)0.3'

Chapter V

158

Figure 5.2: A starshaped domain (0 E 8SZ).

The set of all univalent functions on D which are starlike on D will be denoted by Star(D). If, in addition, there is r c D such that h(r) = 0,

then we will write h E Star(D, r). Of course, in this case such a point T is unique because

Star(D, T) C Star(D) C Univ(D).

If h E Star(D) has no null point in D (i.e., h is starlike with respect to a boundary point) we will write that h c Fan(D). Finally, keeping the classical notations, for the special case when D = A is the unit disk we will just write S* for Star(A, 0) fl S. That is

S' = {h E Star(A) : h(0) = 0 and h'(0) = 1}. The concept of univalent starlike functions was first introduced by Alexander [10] in 1915. In 1921 Nevanlinna [102] conducted a more detailed study of this class. In particular, the following characterization of the class Star(O, 0) is due to him.

Proposition 5.2.1 Let h be a univalent holomorphic mapping on the unit disk A such that h(0) = 0. (5.2.1)

Then h is a starlike function on Oif and only if

(> 0,

Re[zh(z)] z

zEA.

(5.2.2)

STARLIKE AND SPIRALLIKE FUNCTIONS

159

As we will see in the sequel, if h E Hol(A, C) is locally biholomorphic, i.e., h'(z) 0 everywhere, and satisfies (5.2.2) then it is necessarily univalent. Furthermore, condition (5.2.2) leads to the study of other interesting subclasses of S. In particular, in 1936 Robertson [119] had introduced the class S*(A) of starlike functions of order A:

S*(A)={hES*:Re[z

>A>0, zEA}.

(5.2.3)

In 1978 Wald [1501 characterized starlike functions with respect to another center. Using our notions, his result can be reformulated in the following way.

Proposition 5.2.2 Suppose that h E Hol(A, C) is either of the form h(z) = z + E00 2 akzk or of the form h(z) = E°° 1 bkzk with h'(T) = 1 for some T E A. Then the function h(z) - h(T) belongs to Star(/ , r) if and only if Re q(z) > 0, where q(z)

=

h(z) h'(z)(z - T) (1 - z;r)'

zEz \{r},

and q(T) = 1 - ITI2.

We will see below that Proposition 5.2.2 (as well as Proposition 5.2.1) can be easily obtained by using a different approach in more general settings. Different applications of this assertion are presented in Wald's thesis [150] (see also [57]).

Definition 5.2.2 A set S2 in C is said to be spiralshaped (with respect to the origin) if there is a number µ E C with Re to > 0 such that for each w E 1 and t > 0 the point a-t, w also belongs to Q. (see Figure 5.3) If D is a domain in C, then a univalent function h E Hol(D, C) is said to be spirallike on D if the closure of its image 0 = h(D) is a spiralshaped set. Once again, if the origin belongs to S2 then we will say that h is a spirallike

function on D with respect to an interior point. Otherwise (if the origin belongs to the boundary a of Il) we say that h is spirallike with respect to a

boundary point. The set of all biholomorphic mappings on D which are spirallike on D will be denoted by Spiral(D). If, in addition, there is T E D such that h(T) = 0,

then we will write h E Spiral(D, T). Note also that if in Definition 5.2.2 the number p is a real positive number, then S2 is actually starshaped, i.e., Star(D) C Spiral(D). Consequently, Star(D, r) C Spiral(D, T).

If h E Spiral(D) has no null point in D (i.e., h is spirallike with respect to a boundary point) we will write that h E Snail(D). (We will see below that the class Snail(D) is quite narrow: in particular, each h E Univ(A) which has a biholomorphic extension to 0 is, in fact, in Fan(s), i.e., h(A) is starshaped).

160

Chapter V

Figure 5.3: The spiralshaped domain (0 E a1): e-tILw E SZ for each w E Q.

It seems that the first occurrence of the class Spiral(s, 0) = {h c Spiral(s) : h(0) = 0} appeared when condition (5.2.2) was modified analytically by inserting the factor eie: Re Leie zh(z))1 > 0

(5.2.4)

(see Montel [101] and Spacek [137]). Actually, this definition is compatible with our Definition 5.2.2 (see also, Remark 5.2.2 below)

Proposition 5.2.3 Let h E Hol(s, C) have the form h(z) = z + 0" akzk

(5.2.5)

k=22

(i.e., h(0) = 0 and h'(0) = 1). Then h e Spiral(s,0) if and only if condition (5.2.4) holds for some 0 E (7r/2, 7r/2). Of course, Proposition 5.2.1 follows from Proposition 5.2.3 if we set there 0 = 0.

To prove the latter proposition as well as Proposition 5.2.2 we will need another more general assertion which is a direct consequence of the Main Lemma (Proposition 5.1.1). Proposition 5.2.4 Let h E Hol(D, C) be a spirallike (respectively, starlike) function on D. Then there exists a generator f E 9 Hol(D) of a continuous flow on D such that h satisfies the following differential equation:

ph(z) = h'(z) f (z),

z E D,

(5.2.6)

for some p c C (respectively, p E R) with Rep > 0.

Proof. Let S2 = h(D) be a spiralshaped (respectively, starshaped) domain. Then for some p E C (respectively, p E R) with Re a > 0 the mapping

STARLIKE AND SPIRALLIKE FUNCTIONS

161

g = pI, (g(z) = pz), belongs to G Hol(SZ). Indeed, in this case the integral curve defined by the Cauchy problem av(atw)

+ g(v(t, w)) =

av(atw)

+ 'UV (t, w)

0,

v(0, w) = w, w E SZ,

(5.2.7)

has the form {v(t, w) = e-µtw, t > 0, w E 01 and belongs to SZ by Definition 5.2.2 (respectively, Definition 5.2.1). Then Proposition 5.1.1 implies that f E Hol(D, C) defined by: (5.2.8) f (z) = (Tg)(z) = µ [h'(z)[-1 h(z) is a generator of a flow on D, and we are done.

To establish a converse assertion we may omit the requirement on h to be univalent.

Proposition 5.2.5 Suppose that h E Hol(D) satisfies equation (5.2.6), with y E C, (respectively, p E R) Rep > 0, and f E G Hol(D). Then the set SZ = h(D) is spiralshaped (respectively, starshaped). Moreover, if D is bounded and h has a null point r E D with h'(-r) # 0, then h is univalent, hence spirallike (respectively, starlike) with respect to an interior point.

Proof. Fix w E 0 and find z E D such that h(z) = w. Since f E G Hol(D) generates a continuous flow Sf = {Ft(z)}t>0 on D, setting v(t, z) = h (FL(z)), we have that v(t, z) E 92 for all t > 0. That is for a fixed z E D, the family {v(t, z), t > 0} determines a continuous curve in S1, such that v(0, z) = w. In addition, it follows by the chain rule that av(t, z)

at

,

= h (Ft(. ))

aFt(z) _

at

,

= -h (Ft(z)) f (Ft(z))

On account of formula (5.2.6) we obtain that av(t, z) at

- ph(Ft(z)) = -µv(t, z).

(5.2.9)

Integrating the latter relation with the initial data v (0, z) = w

(5.2.10)

we obtain v(t,z)=e_µt,u

ESZ

for allt>0.

Since w E SZ was arbitrary the first assertion of our proposition follows.

Assume now, in addition, that h has a null point r E D and h'(r) 0. This implies that there is a neighborhood U C D of the point r such that h is univalent

on U and V = h(U) C SZ is a neighborhood of the origin. We want to show that, in fact, h is univalent on the whole of D. Indeed, assuming the contrary, suppose that for some w E Sl there are two distinct points z1 and z2 in D such that h(z1) = h(z2) = w.

Chapter V

162

Observe also that, as a result of equation (5.2.6), the conditions h(T) = 0 and 0 imply that f (T) = 0 and f'(r) = µ with Rep > 0. Since D is bounded the family is a normal family in D. Then exactly as in Proposition 4.4.1 one can conclude that r c D is an attractive stationary point of the flow h'(T)

Sf =

generated by f. In particular, we have that Ft(zi) and Ft(z2)

converge to r as t oc. Note that Ft(zi) and Ft(z2) are different for all t > 0. As above, now define v(t, z) = h (Ft(z)) , t > 0, z E D and choose to > 0 such that Ft(zl) and Ft(z2) belong to U for all t > to. Then for such t the curves v(t, zi), i = 1, 2, lie in V = h(U) C Q. But v(0, zi) = v(0, z2) = w and we have that v(t, zi), i = 1, 2, are the same as the solutions of the differential equation (5.2.9) with the same initial data (5.2.10). Consequently Ft(zi) = Ft(z2) for all t > to. That is a contradiction. Sometimes we will say that h E Univ(D, C) is lc-spirallike if it satisfies equation

(5.2.6) with p E C, Rep > 0, and f E G Hol(D).

Remark 5.2.1. Thus h E Spiral(D) is spirallike with respect to an interior point (that is h E Spiral(D,T) for some r c D) if and only if the generator f in (5.2.6) vanishes at T E D. Moreover, if f is defined, then p = f'(T), and T is an attractive stationary point of the flow Sf = generated by f. In fact, it can be shown (see Section 5.7) that for each f E Hol(D), normalized by the conditions f (T) = 0 and Re f'(r) > 0 there is a unique solution h E Spiral(D, T) of the equation (5.2.6) with it = f'(r) normalized by h'(r) = 1. In addition, if p is a purely real number, then h defined by (5.2.6) is a starlike function on D.

Similarly, we can say that h c Snail(D) if and only if f has no null point in D. In this case the flow Sf = generated by f for each z E D converges to a boundary point r E D. Since for the special (but most important) case when D = A is the unit disk we have a complete description of the class G Hol(O), the proved propositions imply several corollaries. In particular, applying the Berkson-Porta parametric representation of the class Q Hol(O) we obtain the following: Corollary 5.2.1 Let h : A i-+ C be a univalent holomorphic function on A. Then h(O) is spiralshaped if and only if the following equation is fulfilled:

ph(z) = h'(z)(z - r)(1 - zfr)p(z),

z E A,

(5.2.11)

where r E 0, it E C with Rep > 0, and p E Hol(O, C) with Re p(z) > 0 for all

zEA. Remark 5.2.2 Thus, if r c A, then h E Spiral(O, r) (i.e., is spirallike with respect to an interior point); if r E 80, then h E Snail(O) (i.e., spirallike with respect to a boundary point). Separating these two cases we conclude: A locally biholomorphic function h on A belongs to Spiral(0, r) if and only if there exist r E A and ,a E C with Rep. > 0 such that Re

h'(z)(z - r) (1 - z;r) > 0, ,uh(z)

z E A.

STARLIKE AND SPIRALLIKE FUNCTIONS

163

Indeed, equation (5.2.11) can be rewritten in the form Re

h'(z) (z - r)(1 - z;r)

ph(z)

-

= Re

1

-

> 0,

z E s.

,

(5.2.11 )

If r E s, then differentiating (5.2.11) at z = r we obtain p(r) =

A

i-

17-12

which means that inequality (5.2.11') is actually strict. If we normalize p by the condition I M I = 1, we obtain that r and p are uniquely determined by h. Of course, the converse assertion is also true. Moreover, if r = 0

then setting 0 = - arg u we obtain a description of the set Spiral(s, 0) which coincides with the classical description (Proposition 5.2.3).

Letting µ = 1 and r = 0 in (5.2.11') we arrive immediately at Nevanlinna's condition (Proposition 5.2.1) and Wald's condition if r E s, r # 0 (Proposition 5.2.2). Similarly, we conclude from (5.2.11):

A univalent function h on s belongs to Snail(s) (respectively, Fan(s)) if and

only if for some r E 8s and y E C (respectively, p E R) with Rep > 0, the following condition holds: Re

h'(z)(z - 7-)2

µh(z)r

< 0.

-

We will see in the next section that the boundary behavior (at the point r) of h'(z) (z - r) the quotient Q(z) = h(z) (the so called Visser-Ostrowski quotient) in the latter inequality characterizes those functions in Snail(s) which, in fact, belong

to Fan(s).

5.3 A generalized Visser-Ostrowski condition and fanlike functions In this section some relations between classes Snail(s) and Fan(s) (of spirallike and starlike functions with respect to a boundary point) will he studied. First we make the following observation. Suppose that h c Snail(s), that is h E Univ(s) satisfies equation (5.2.6): ph(z) = h'(z) f (z),

z E s,

for some p E C with Re it > 0, and f E 9 Hol(s) with no null point in A.

Chapter V

164

We know that in this case there is a unique boundary point T E OA which is the sink point of the semigroup generated by f. Assume, temporarily, that one of the following conditions holds:

(1) Ih/,(z)I <M
(ii) Z-T lim h(z) = 0 and lim h'(z) = a,

a # 0, oo, as A approaches r nontan-

Z-T gentially. The latter condition is known to define h to be conformal at the boundary point

T E 8i (see, for example, [107]). Without loss of generality one can set r = 1. We recall, that in our situation as a result of the Berkson-Porta formula f E G Hol(d) has the form:

f(z) = -q(z)(1 - z)2,

(5.3.1)

with some q E Hol(0, C) such that Re q > 0 everywhere. In addition, we know by the established continuous version of the Julia-WolffCaratheodory Theorem (Proposition 4.6.1) that the angular derivative

z lim f (z)

z-.1z-1

of f exists at 1 and equals to the angular limit L lim f'(z) = /3 > 0.

(5.3.2)

We want to show that in both cases (i) and (ii) the number µ in equation (5.2.6) must be equal to /3, that is, µ is actually a real number. Indeed, assuming first that (i) holds and differentiating equation (5.2.6), we obtain

f'(z) = Zf (z).

(5.3.3)

Since lim f (r) = 0 we obtain µ = /3 by (5.3.2). r-. If now condition (ii) holds, then one can write, by using equalities (5.2.6) and (5.3.1), that µh(z) _ -h'(z)(1 - z)2q(z) with Re q(z) > 0, z E A. Then we have z

1

l h(z) = h'(z) (1 - z)q(z) = h'(z)z (-)

.

(5.3.4)

Since

y z(41 =Llimh'(z)=a#0,oo, Llim

we see again that µ = /3 is real, hence h is, actually, a fanlike function.

Of course, these simple considerations are related to the question of how can one join conditions (i) and (ii). For example, one can replace condition (ii) by the condition that h be isogonal

at r, that is h and arg h' have finite angular limits at T. It is clear that if h is

STARLIKE AND SPIRALLIKE FUNCTIONS

165

conformal at r then it is isogonal at T. In general the converse does not hold. At the same time some geometrical arguments indicate that there is no properly spirallike function with respect to a boundary point which is isogonal at it. Another (much weaker than isogonality) condition which is sufficient to the above mentioned property is the Visser-Ostrowski condition: Z lim

h'(z) (z - r) - 1.

(5.3.5)

z-r h(z) - h(r)

It is known (see [107]) that this condition is equivalent to the following:

L lim (z - r)

(5.3.6)

0.

Obviously condition (i) implies (5.3.6). Thus the Visser-Ostrowski condition is weaker than both (i) and (ii). We will establish now that a generalized VisserOstrowski condition (see condition (*) below) is necessary and sufficient for h E

Snail(s) to be in Fan(s). Let us consider the Visser-Ostrowski quotient:

h'(z)(z - r)

(5.3.7)

Q(z) = h(z) - h(r)

Proposition 5.3.1 ([38]) Let h be a univalent function on s such that h(s) is a spiralshaped set with

r E 8s.

lim h(rr) = 0,

r--.1-

(5.3.8)

Then h(s) is, in fact, starshaped if and only if the following condition holds: (*)

the angular limit L lim Q(z) := v z-r

exists finitely and is a positive real number.

Proof. Assuming again that the generator f in equation (5.2.6) is presented by the Berkson-Porta formula (5.3.1), we could use a similar argument as in (5.3.4) if we show that the point r E 8s in (5.3.7), (5.3.8) and (*) is equal to 1. Indeed, if this is not the case we have by (5.2.6) and (5.3.1) that ph(z)

(1 - Z)2q(-)

h'(z)(z - r)

(z - r)

(5.3.9)

Hence by (*) the limit 77 := L lim Z-?'

[_(i - z)2 q(z)

z - rJ

=µ

v

(5.3.10)

exists (finitely). On the other hand, setting z = rr, r E (0, 1), and letting r approach 1- we obtain

rl = (1 - r)(r - 1);r lim q(rr) = 11 - r12 lim q(rr)

r-.1- r-1

r=1- r-1

Chapter V

166

and

Re77 = 1 - TI2 lim Re T-+1

&T)

< 0.

r - 1

But the latter equality is impossible because of (5.3.10) and Re y > 0. Contradiction. So T = 1. Once again, since µh(z) =

(f

(zl)h'(z)(z - 1)

we obtain

µ = Z lim (f (zi)Q(z) _ ,3z lim Q(z), and our assertion follows.

Thus the class of properly spirallike (i.e., not starlike) functions with respect to a boundary point contains neither conformal nor isogonal mappings at this boundary point.

Remark.5.3.1. We will show below (see Section 5.6) that the number v = L lim Q(z) yields an important geometrical characteristic of h E Fan(0), namely, Z_T

the angle 0 = 7rv is the smallest one such that h(L) lies in the wedge of the angle 0. Thus, in fact, for h E Fan(i) with lim h(r) = 0 we will show that r--.1-

Z lim Q(z) < 2. Z-+T To do this we need an approximation process which is based on Hummel's representation of the class Star(A, ) of starlike functions with respect to interior points. We will give this representation in Section 5.5.

5.4

An invariance property and approximation problems

The following question naturally arises in approximation function theory: given h E Hol(O, C) such that 0 = h(i) is spiralshaped (respectively, starshaped) and a sequence Q,, C SZ spiralshaped (starshaped) domains such that U SZn = SZ, find the sequence hn such thatn-oo lim h,(z) = h(z) for each z E 0, and hn(O) _ Qn. Theoretically, under certain conditions this problem can be solved because of the Riemann Mapping Theorem and Caratheodory's Kernel Convergence Theorem.

STARLIKE AND SPIRALLIKE FUNCTIONS

167

However, in general, to find such a sequence implicitly has not seemed plausible, even the sequence cz is well described.

At the same time one can define, in a sense, a dual approximation problem which is related to some invariance property of spirallike (respectively, starlike) functions.

Namely, if h E Spiral(s) (respectively, Star(s)), find a `nice' sequence of domains D,, in A such that U D,, = A and h continue to be spirallike (respectively, starlike) on each D,,.

In view of Propositions 5.2.4 and 5.2.5 an answer to the above question is provided by the following observation. If h E Spiral(A) (respectively, Star(s)), then for some µ E C (respectively, it E R) with Re µ > 0 the function f (z) = l.ch'(z)[h(z)]-1 is a semi-complete vector field on A. Then these propositions yield that h continues to be spirallike (respectively, starlike) on a domain D C A if and only if this domain is invariant for the semigroup S = {Ft}t>° generated by f. Thus, due to Proposition 5.2.4 we can formulate the following assertion:

Proposition 5.4.1 Let h c Spiral(A) (respectively, h E Star(s)). Then there is a unique point T E A such that for all K > 1 - 17-12 the sets h(D(T, K)), where r(

D(T,K)={zEA:

11 - Zr 2

1-1x12


are spiralshaped (respectively, starshaped).

We recall that for T E A the sets D(-r, K) = {z E A :

z

TI

1-7-z

< r}, K =

(1 - 17-12)(1 - r2)-1 (see Section 1.1, Exercise 6) are pseudo-hyperbolic balls in A while for T E 8A these sets are horocycles internally tangent to 8A at T. In particular, for T = 0 we have that h E Spiral(A, 0) (respectively, Star(A, 0)) is spirallike (respectively, starlike) on each disk A,. = { 1 z 1 < r, 0 < r < 11 concentric with A. (Note that for the functions of the class

S* = {h E Star(s) : h(0) = 0 and h'(0) = 1} this result was obtained independently by Takahashi and Seidel [18] as an extension to Nevanlinna's theorem (see Proposition 5.2.1). A simple proof of this fact with the use of the Schwarz Lemma can be found in [33]).

One may expect that even when T 0 0 and h E Spiral(A, T) (respectively, h E Star(s)) on the unit disk, then for at least r close enough to 1 it continues to be spirallike (starlike) on the disks A,. However, examples show that in general this conjecture has been answered in the negative.

Example 1. Consider the function h(z) = h°(Mz (z)), z E A, where h (z) = °

z (1 - z)2

Chapter V

168

is the so called Koebe function and

1-z M1/2(z) = 2 1z 2

is a Mobius involution transformation of the unit disk taking the origin to 'r = 2 It is easy to see that the Koebe function belongs to Star(0, 0) (even S*) and it follows that h E Star(O, T), because of the relations h(0) = ho(0), h(T) = 0. We claim that there is a sequence {z,, } C 0, Iz,,, - 1 such that Re Lzh((z))J < 0.

Indeed, calculating Re [ h(z) ]

9 Re z(1 - z)(1 + z)(z - 2)(1 - 2 =4 11 + zI21z -1121- 5z12

and setting z = x + iy we obtain (after several technical manipulations) that the numerator N(z) of the right hand side of the latter equation has the form

N(z) = 2 (x - 2 )(x2 + y2 - 1)(x2 + y2 - 2x). Now it can easily he seen that for any sequence {z,,} E 0, Iz,, 1 such that z, c D = {z = x + iy : (x - 2)(x2 + y2 - 2x) > 0} the expression N(z) is negative (see Figure 5.4). Thus the claimed assertion is proved. .'

Figure 5.4: The set D.

STARLIKE AND SPIRALLIKE FUNCTIONS

169

Nevertheless, it has been shown in [66] that if r is close enough to zero, then the answer to the above question (concerning starlike functions) is affirmative. Proposition 5.4.2 Let h E Star(A, rr) with I T I < 2 - vr3-. Then there exists e > 0 such that for each 1 - e < r < 1 the function h belongs to Star(Or). By using the results in Sections 3.6 and 4.5 we can formulate some sufficient conditions for h E Spiral(A, r) (respectively, Star(O, z)) to be spirallike (respectively, starlike) on each Or for r close enough to 1, which do not depend on the location of r E A. We will say that a function h E Hol(A, C) is strongly spirallike if for some

p E C with Rep > 0 the function f (z) := h z)z is strongly p-monotone on A.

Proposition 5.4.3 Let h E Hol(O, C) be a strongly spirallike function. Then there is T E A and E > 0 such that for each 1 - E < r < 1, h E Spiral(Or) and h(T) = 0. In particular, by using a criterion of strong p-monotonicity we have the following:

Corollary 5.4.1 Let a be a real continuous function on [0, 1] such that a(1) > 0 and let h E Hol(A, C) be such that for some p E C with Rep > 0 the following condition holds Re ph(z)z > a(IzI)IzI, hl(z)

z E A.

Then there exists T E A and E > 0 such that for each 1 - c < r < 1 h E Spiral(Ar) and h(T) = 0.

Exercise 1. Let f (z) = az2 - a + bz for some positive a and b. (i) Show that if equation (5.2.6) is solvable for some p e C, Rep > 0, then p is, in fact, a positive real number, that is, the solution h of (5.2.6) is, actually, a starlike function on A. Find p. (ii) Show that there exists c > 0 such that for each r E (1-c, 1], h E Star(Or). (iii) Set a = 2 and b = 3. Show that c in (ii) can be chosen arbitrarily in the interval [0, 2 ).

Exercise 2. Let f (z) = az2 - a + bz and c such that Re b > 0 and Icl < 1.

1 - cz 1 + cz

with some complex numbers a, b

(i) Show that there are p E C and e > 0 such that equation (5.2.6) has a solution h E Spiral(Or,T) for some r E A and each r c (1 - E,1]. (ii) Find relations between a, b and c such that p is real, i.e., h is starlike. (iii) Setting a = 0 and b > 0, show that h defined by (5.2.6) with an appropriate p > 0 is starlike of order a (see Section 5.2) with a = b1 -

Icl

1+lcl

Chapter V

170

Finally, note that if h c Hol(s, C) is known to be in Spiral(s), then sometimes information on values of h and its first and second derivatives at only one point may be useful for solving the dual approximation problem on disks s, concentric with A. Indeed, using Corollary 4.5.1 and Remark 4.5.2 we can easily arrive at the following sufficient condition:

Proposition 5.4.4 Let h c Spiral(s) (respectively, Star(s)), satisfy equation (5.3.6) with some p E C (respectively, p E IR) with Rep > 0 and f E Hol(s, C). Then

O

Reph"(0)h(0)

fh'(o)12


(ii) if 41 h (0) I < [1 - h[h'(0)]

0 ]

then for some E > 0 and each r E (1 - E, 1]

the function h E Spiral(sr.) (respectively, Star(sr.)).

Remark 5.4.1. Of course, the problem above is not relevant for the class Fan(s). Indeed, it is clear that if h E Star(s) has no null points ins then there is no disk O. concentric with s such that h(sr.) is starshaped. In this case another approximation problem arises: given h E Fan(s); find a sequence hn E Star(s, ) of starlike functions with interior null points such that hn converge to h uniformly on compact subsets of s as n oo. It looks as if the following procedure should work in solving this problem. If, for example, h E Fan(s) is isogonal at its boundary null point, say T = 1, then it satisfies the equation ph(z) = h'(z) f (z), z E s, (5.4.1)

for some f E 9 Hol(s): f(z) = -q(z) (1 - z)2,

with Req(z) > 0, z E s, and p = Lf'(1) > 0. Define an approximation sequence fn to f by using the Berkson-Porta representation formula fn(z) = (z - 7n)(1 Tnz)q(z), where {Tn} C s is any sequence which converges to 1 as n tends to oo. If, in addition, we can choose this sequence such that the numbers f, ,(-r,) are real, then we may try to solve the equations

pnhn(z) = h,,(z) fn(z),

z c s,

(5.4.2)

with pn = f'(Tn) in order to define a sequence of univalent functions hn which are starlike with respect to interior points (h(Tn) = 0). However, in this way there is a risk that hn may not converge to h. Indeed, putting it otherwise, we obtain that the numbers pn = f'n (Tn) = (1 - J r, 12)q(7-,,) must converge to

p = L lim f'(z) = L lim f (z) = L lim(1 - z)q(z). z-.1 Z-1 z-.t z - 1 On the other hand, if Tn achieves T = 1 along the real axis we have

lim pn = lim (1 - ITnl2)q(Tn) = 2 lim (1 - T,,)q(Tn) = 2p. n-oo n-.oo n-oo

STARLIKE AND SPIRALLIKE FUNCTIONS

171

That is a contradiction. Also it is easy to construct a counter example of the latter relation.

Example 2. Consider a semi-complete (even complete) vector field f E Hol(O, C) defined as follows: f(z) = z2 - 1.

There are two boundary null points z = 1 and z = -1 of f. Since f'(1) = 2 > 0 it follows by Proposition 4.4.1 (a continuous version of the Julia-WolffCaratheodory Theorem) that z = 1 is a sink point of the semigroup generated by f. Therefore, one can present f by the Berkson-Porta formula:

f(z) _ -(1 - z)21 +z with q(z) = 1 + Z. Obviously, q admits real values if and only if z E A are real. 1

So, we have to choose a sequence {T,,, } E A, T,,, -+ 1, in the above approximation procedure to be real. Further, if we define

fn (z) = (z -T-)(1 -T;,z) we obtain f, (T,) = 0, and f,, (z) fn(Tn) _ (1 - -r,2

+z 1 - z' 1

f (z) in A, while + Tn

= (1 + 7-n)2 - 4 # f'(1).

As a matter of fact, it can be shown that if h,, are solutions of (5.4.2) normalized by the condition h,,(0) = 1, then they converge to a fanlike function h which is a power of h defined by (5.2.1). However, in general, we do not know whether the numbers µn are real (i.e., whether the functions hn are starlike). Thus a procedure of using the Berkson-Porta multiplier (z -Tn)(1 -Tnz) has been shown to be inappropriate for the approximation of starlike functions with respect to a boundary point by starlike functions with respect to interior points. Nevertheless, it turns out that a modification of this multiplier in the spirit of J.A. Hummel makes such a procedure very effective. Hummel's multiplier has been used by A. Lyzzaik to prove a conjecture of M.S. Robertson on a description of starlike functions with respect to a boundary point, the images of which lie in a half plane. The next section is devoted to this approach.

Chapter V

172

Hummel's multiplier and parametric representations of starlike functions

5.5

Hummel's multiplier is a (meromorphic) function of the form HT (z) =

(z - T)(1 - zT) z

zEA

'

where T is a given complex number with Jri < 1. For the case of In < 1, J.A. Hummel showed in [65] and [67] that this function plays a special role in the study of starlike functions. In fact, it turns out that by the multiplication operation this function translates the set Star(O, 0) onto the set

Star(O). Moreover, if r E A then Star(O, 0) is translated onto Star(A, r). More precisely:

Proposition 5.5.1 Let h E Ho1(O, C), h(0) = 0, and g E HOl(O, C) be two functions related by the formula

g(z) = H,-(z) h(z).

(5.5.1)

Then g(O) is starshaped if and only h(O) is starshaped.

In this section we will treat only the case when r E A. In this situation it is more convenient (for some symmetry) to consider the meromorphic function T, on A defined by 41,(Z)

-11

(z - T)(1 - zT),

2 Hr(z) 1ITI2

z E A.

(5.5.2)

Of course, it is sufficient to prove Proposition 5.5.1 replacing HT in (5.5.2) by fir. Moreover, in this case the inverse translation is just the same multiplier composed with a Mobius involution transformation.

Proposition 5.5.2 Let h E Hol(A, C) and g E Hol(i, C) be two functions related by formula:

g(z) = lkr(z) h(z).

(5.5.3)

Then g E Star(O, T) if and only if h E Star(O, 0). Moreover, the inverse transformation can be defined by the formula

h(z) _' (MT (z)) . g(z), where

MT(z) =

T - z

1-zrf

To prove this assertion we need some properties of the function 'T defined by (5.5.2).

STARLIKE AND SPIRALLIKE FUNCTIONS

173

Lemma 5.5.1 For the function %PT the following properties hold: (i) for all z E 8A : IzI = 1, Q,(z) is a positive real number;

(ii) forallzEDA:IzI=1, zWT(z)

Re

= 0;

'kT(z)

(iii) if MT denotes the Mobius transformation z

M., (z)=

1-zr

then

'kr(z) 'I'T(M,.(z)) = 1.

Proof. (i) If we set z = e'w, 0 < cp < 2ir we obtain: (1 - e4r)(1 - ei`pT)

(ei`p - 7-)(1 - ei`PT) (1 - I7-I2)eiw

1 - ITI2

I1-ei' 1-ITI2 >0. 2

(ii) By direct calculations we have:

T;(z) =

IITI2 L(1

1

- z ) (1 - zT)] z2-zT- (1- z)TJ

1-lITI2

=

z-

T

1

ITI2

1 - ITI2 [z2

1

T

1 -ITI2

z ]

(T

T)

\ z2

Again setting z = e4G we obtain: Re

z'T(z) `I'r(z)

e2iw I

F - T)

= Re (eip - r)(1 - ei9 f)

- Re

T - Te2iW

eiw 11 -

e-iwnI2

iw - _iw

1

I1 - e-iPT

(iii) Substitute MT(z) = T - z to 1PT instead of z. We obtain:

1-zn

T-Z

'J'r(Mr(z))

(1 - ZT)

=

i-zf

-T

(1

- -Z 1-zf - T

(T - z - T + zITI2)(1 - zT + z; r-

(1 -z(1- ITI2) (T - z)(1 - zT)

)

T-z

1-ITI2 17-12) (7-

_

- z)(1 - zT) 1

%'T(z)

1-r 12)

Chapter V

174

The lemma is proved.

Following J.A. Hummel [65, 671 we will say that a meromorphic function g on A satisfies the condition (A) if for each E > 0 there is p E (0, 1) such that for all z in the annulus p < Izl < 1 the following condition holds: Re

g'() g(z)

> -E.

(5.5.4)

Lemma 5.5.2 Let g E Star(A,T) for some r E A. Then g satisfies condition (A).

Proof. Since g(0) is starshaped and the origin is an interior point of g(0), the function g satisfies the equation: g(z) = g'(z)(z - T)(1 - zT)p(z)

with some T E A, such that g(T) = 0 and p c Hol(A, C) with Rep > 0 everywhere. Then by Lemma 5.5.1(iii) we obtain:

z g'(z)

(1 - T I2)

__

1

g(z)(z) p(z)

= T(M (z))

(5.5.5)

p(z)

Denoting w = MM(z) we have in turn from (5.5.4): z

g'(z)

g(z)

where q(w) =

(1

- T2

)p(

(

MT w

= TT(w) q(w),

(5.5.6)

)) (recall, that MT is an involution).

Hence Re q(w) > 0 for all w E A. Now for each w E A the right hand side of (5.5.5) can be presented as: T,_(w)

q(w) = (w (

where f (z) =

(w - T)(1 - wT) q(w) (1

- T2I)

wT) q(w)

1(1

_ f (w)

(5.5.7)

ITI2)

defines a semi-complete vector field by the

Berkson-Porta representation. At the same time, it follows by Proposition 3.4.4 that f satisfies the condition:

Ref (w) w > Re f (0)w(1 -

Iw12).

Therefore for each E > 0 one can find pi E (0, 1) such that:

Ref (w) =

Ref (w)w > -E,

(5.5.8)

1

whenever pi < Iwl < 1. Note now, that if w runs in the annulus pi < IwI < 1, then z = MT(w) runs in the set A = A \ B(T, r) where B(T, r) c A is a hyperbolic ball centered at T.

But this ball is also strictly inside A, hence there is p E (0, 1) such that the annulus p < I z I < 1 lies in A (see Figure 5.5).

STARLIKE AND SPIRALLIKE FUNCTIONS

175

A=A\B Figure 5.5: Condition (A).

Thus for such z : p < Izj < 1 we obtain from (5.5.5)-(5.5.7) relation (5.5.3). The lemma is proved.

Proof of Proposition 5.5.2. Let h E Hol(O, C), and g E Hol(A, C) be related by (5.5.2). Assume that g E Star(O, T). It is clear that r must be a unique null point of g. Note, that T, satisfies condition (A) due to Lemma 5.5.1(ii). By differentiating (5.5.2) we obtain the equality: z

z g'(z)

h'(z) h(z)

=

g(z)

z%P'(z)

- T(Z)

which implies that h also satisfies condition (A). At the same time, since h E Hol(O, C) we must have h(O) = 0 and h'(0) _ g(0) 0. Therefore it follows by T the minimum principle for harmonic functions that actually Re

z ((z)

>0

(5.5.9)

for all z E A. Thus h E Star(O, 0). The converse assertion can be easily proved by replacing the roles of h and g. Indeed, let now h E Hol(A, C), h(0) = 0 be starlike, and g E Hol(O, C) be defined by (5.5.2) with some r E A. Note that a function gl E Hol(O, C) defined by: g1(z) = h(MT(z)) is also starlike because:

9i (A) = h(A)Again, using Lemma 5.5.1(iii) we obtain: 91(z)

9(MT(z)) = h(MT(z)) . WT(MT(z)) = T, (Z)

Chapter V

176 or

91(z) = P,-(z) . h1(z),

where we denote h1(z) = g(MT(z)). Since, h1(0) = 0 and g1 E Star(i,T), we have that h1 is also starlike, as proved above. But again g(z) = h1(Mr(z)), and we are done. 11

Remark 5.5.1. Generally speaking we have proved the following: If gl and 92 are two functions related by the formula: 91(A1T(z)) = 92(z)'I'T(z),

(5.5.10)

92(Mr(z)) = 91(z)'PT(z).

(5.5.11)

then also:

In addition g1 E Star(A, 0) if and only if 92 E Star(0, T).

To prove Proposition 5.5.1 for the case when r E 80 we need a more detailed treatment of fanlike functions. We will do this separately in the next section.

5.6 A conjecture of Robertson and geometrical characteristics of fanlike functions The first note related somehow to an application of the class Fan(0) of starlike functions with respect to a boundary point to a boundary point seems to be due n

to E. Egervary [35] in connection with Cesaro sums - J(n - k + 1)zk of the k=1

.... In particular, it turned out that the n E(n - k + 1)zk belong to Fan(g) with hn(1) = 0.

geometric series z + z2 + z3 + ... + zn +

functions hn = 1 -

2

n ( n+

1)

k=1

Strangely, although the classes Star(A, T) (of starlike functions with respect to interior points) have been studied by many mathematicians over a long period of time it seems that very few papers have been published up to 1981 on starlike functions with respect to a boundary point. A breakthrough in this matter is due to M.S. Robertson [121], who suggested the inequality

zEO,

(5.6.1)

STARLIKE AND SPIRALLIKE FUNCTIONS

177

as a characterization criterion for those univalent holomorphic h : A i-* C with h(O) = 1 such that h(A) is starlike with respect to the boundary point h(1) := lim h(r) = 0 and lies in the right half-plane. This characterization was partially proved by Robertson himself under the additional assumption that h admits a holomorphic extension to a neighborhood of the closed unit disk. Furthermore, he established that this class is closely related to the class of close-to-convex functions.

In particular, if h satisfying (5.6.1) is not a constant with h(0) = 1, then g(z) _ log h(z), log h(0) = 0, is close-to-convex with Re

[(1 - z)2 h'(z) I < 0.

(5.6.2)

Note, in passing, that because of Proposition 5.2.2 inequality (5.6.2) is exactly a characterization of functions in the class Fan(A) normalized by the condition

h(1) := lim h(r) = 0.

r-.1Different applications of these results to convex functions were also exhibited in Robertson's work (see also [57]). Observe, that the full proof of Robertson's conjecture was given by A. Lyzzaik [90], whilst a generalization of these results was later established by H. Silverman and E.M. Silvia [134]. In view of Caratheodory's Theorem of kernel convergence, a univalent function which is starlike with respect to a boundary point can be approximated by

functions which are starlike with respect to interior points (see Figure 5.6 below). This approximation process can be considered dynamically as an evolution of the null points of these functions from the interior towards a boundary point. As mentioned above, this evolution is somehow connected to the evolution of semi-complete vector fields corresponding to starlike functions and the asymptotic behavior of one-parameter semigroups. So a natural question is how to trace these dynamics analytically in terms of inequalities (5.2.11'), (5.6.1) and (5.6.2). In this chapter we will mostly follow the works [134] and [37]. We will show that condition (5.6.2) is equivalent to a generalized form of Robertson's condition (5.6.1). In addition, we will relate these conditions to some geometric considerations in the spirit of Silverman and Silvia [134]. We begin with the following observation. Let g E Star(A, 0) and let h E Hol(A, C) be given by h(z) - (1 - z)2g(z) (5.6.3)

z

Consider the functions

h,, (z) = Hr (z) (-g(z)) where

,

(z - Tn)(1 - zTn,) Z

are Hummel's multipliers. Since -g also belongs to Star(A, 0) we have by Proposition 5.5.1 that h,, E Star(A, Tn,). Letting Tn, E A tend to 1 we obtain that h

Chapter V

178

uniformly approximates h on each compact subset of s, so h(s) is expected to be starshaped. In addition it follows by Hurwitz's theorem [68] that h is univalent. Thus one may conjecture that condition (5.6.3) is also a characterization of those univalent holomorphic h : s --+ C such that h(s) is starshaped with h(1) lim h(r) = 0, and hereby should be related to conditions (5.6.1) and (5.6.2). r 1-

Indeed, if (5.6.1) holds then setting (because of (5.6.3)) g(z) _

(1

zz)2h(z)

we obtain Re zg'(z)

g(z)

+l+z = Re rl zh'(z) h(z) 1 - z =

1Re 2z'(z) h+1z

()

+1Re1+z>0

hence g E Star(0, 0). This sketches another proof that an h E Hol(s, C) satisfying (5.6.1) ought to belong to Fan(s).

At the same time, if we keep in mind that (5.6.1) characterizes those h E Fan(s) with h(0) = 1, such that h(s) lies in the right half-plane, we reason that a stronger conjecture, namely that (5.6.1) is equivalent to (5.6.3), should be refuted. So it might be worthwhile to replace (5.6.3) by a more qualified condition as well as to replace (5.6.1) by a generalized inequality, which are both related to the same geometrical location of the image h(s). Following [134] for A E [0, 1) we define the class Ga of nonvanishing holomorphic functions h : s -4 C with h(0) = 1 which satisfy the condition:

Re11 1

zh('()z)

+

1+zl

> 0 for allzEA.

(5.6.4)

First we will study some properties of the classes G\,

Lemma 5.6.1 For each . E [0,1) the set oo

U

rn

z

177

1

yi

z(j

n=1 j=1

n

:EAj=2-2A, 1(jl=1 j=1

is dense in GA in the topology of uniform convergence on compact subsets of A.

Proof. Since the function zh'(z) 1 - .\ h(z)

p(y)

1+z

1-z

belongs to the class of Caratheodory: {p E Hol(s, C), p(O) = 1, Re p(z) > 0, z E s}, it follows from the Riesz-Herglotz representation of this class that: p(z) =

f

ISI=1

1 + z(

1 - z(

dm(C),

z E A,

STARLIKE AND SPIRALLIKE FUNCTIONS

179

where m is a probability measure on the unit circle. Approximating the left hand side of the latter equation by the integral sums: n

Ei+:' 1 - Z6

JAmj, Sj E 8A, E Amy = 1, j=1

i=1

and solving the initial value problem 1

zh' (z) = s(z),

1 -A hn(z)

hn(0) = 1,

where

1+z(J - +zAmj s(z)=1: ,j=1111\\1-4 1-Z)

=2

(j-1

J.=,

we obtain our assertion with Al = 2(1 - A) Amj, j = 1, .

. .

Amy,

, n.

To continue we recall that a holomorphic function g : A i--4 C is said to be

starlike of order A E (0, 1) if g(0) = 0, g'(0) = 1 and Re ( I ((z))) > A for all z E A (see Section 5.2). The set of such functions will be denoted by S*(A). The foregoing simple but important fact is due to Silverman and Silvia [134]. Lemma 5.6.2 Let A E [0, 1) and let g and h be holomorphic functions in A related by the equation g(z) := z(1 - z)2A-2h(z). (5.6.5)

Then g E S*(A) if and only if h E Ga.

Proof. This fact follows immediately by the equation (j)

zg'(z))

+

A )

g(z

and the relations g(0)

1 + z

I

(5 .6.6)

0, g'(0) = h(0).

2mm

Lemma 5.6.3 If 0 < Al < A2 < 1, then Gal C Ga1. Proof. Assuming that h E Gal we obtain by Lemma 5.6.2 that g defined by (5.6.5) with A = A2, belongs to S*(A2). Let us now denote g(z)

9(z) =

(1 - Z)2(A2-A1

We have by (5.6.5) g(z) = g(z):= z(1 -

z)2a2-2h(z) = z(1 - z)2a,-2h(z).

(5.6.7)

Chapter V

180

Thus, again in view of Lemma 5.6.2 we have to show that g E S* (A,). Indeed, on account of (5.6.7) we calculate Re

9(())

= Re (g(())/

A1)Re

- 2()2

> A2 - 2(A2 - A,) Re g (0) = 01

zz

1

z >A,, z-1 -

Y (O) = 1,

and we have completed the proof.

Exercise 1. Show that Re

z

< 1, z E 0. z-12

Exercise 2. Prove that if 0 < Al, A2 < 1, then h E Gal if and only if [h(z)]

'-a E Gal .

A natural question related to the last lemma is: a given h of the class Ga for some A E [0, 1) find the maximal A = A*, such that h also belongs to Ga.. In other words, Ga. should be the minimal class which contains h. It turns out, that this question is closely connected to another one: if h is a starlike function with respect to a boundary point, how does one determine the minimal angle 0 such that h(0) lies in the wedge of this angle. In [37] it was discussed how to resolve the above questions.

Proposition 5.6.1 Let h : A H C be holomorphic and let A E [0, 1). If h is not a constant and h(0) = 1, then the following conditions are equivalent. (i)Re[1 1A zh((j) + 1+zl >0 forallzEA.

hz

1-

(ii) There exists a starlike function g : A H C of order A such that

h(z) - (1 -

z)2-2a g(z)

zE

z

(iii) The function h belongs to Fan(0) with h(1) := lim h(r) = 0 and h(0) lies in a wedge of the angle 27r(1 - A). (iv) The function h is a univalent function on A such that f (z) := h(z)/h'(z), z E A, is a semi-complete vector field with L1i mRe 1

f(z)

1

z-1 - 2-2A'

where the limit in the left hand side of the latter inequality exists finitely. Moreover, the equality in (iv) can be reached if and only if A = A*, where Ga. is the minimal class of Ga which includes h, and if and only if the wedge of the angle 2-7r(1 - A*) is the smallest one which contains h(0).

STARLIKE AND SPIRALLIKE FUNCTIONS

181

Proof. The equivalence of conditions (i) and (ii) is the content of Lemma 5.6.2. Our next steps are slightly simpler than in [37]. First we claim that condition (ii) and Lemma 5.6.3 imply that h E Ga \ {1} must be univalent. Indeed, setting in this lemma A = 0, A2 = A > 0 we obtain that h admits a similar representation as (5.6.3):

h(z) = (1 - z)2g(z)

(5.6.3')

z

with some g E S*. Then, as mentioned above, the approximation process:

h,, (z) -

(Tn - z)(1 - zTn)g(z)

Tn E O, Tn -i 1,

z

implies the desired claim. Next we want to show that h belongs to Fan(O). In view of Proposition 5.2.2 this fact will be proved once we show that` Re ((1

- z)2 h'(z)

I < 0,

(5.6.8)

z E A.

To this end let 0 < r < 1, and define hr : A HC by

1-z 1 - rz )

hr(z) := h(rz)

2(1-A)

z E A.

,

If we use the corresponding function g E S* (A) we can write equivalently that

hr(z) =

((rz)) (1) (1 - z)2(1-A). r

z

This last representation of hr shows that it belongs to Ga. Its definition shows h as r ---* 1- and that hr is continuous on the closed disk 0. Therefore the claimed inequality will follow if we inspect it for hr and for z = ei'P E 80. Indeed, for such z we have

that hr

Re

((l_z)2rf2" hr(z)

- Ref (1 - z)2

zh (z) + 1 - A

(rz

)

(

= Re L(z-2+z) (hr(()) +

+

)1-z)

(1-A)

+ (1 - z)2 z

(1

- A)

1+z) +z)]

1 + zl

-z

< 0,

= 2 (cos cp - 1) Re hr (( )) + (1 - A) i + z J

L

as claimed. The fact established in (5.6.8) and the Berkson-Porta formula mean, actually,

that f : A H C, defined by f (z) = h'(z)

Chapter V

182

is a semi-complete vector field. In addition, the same formula implies (because

of the uniqueness property) that r = 1 must be the sink point of the semigroup S = {Ft = h-1 (e-th(z))}t,o generated by f (see Section 5.2). To proceed with the proof of the implication (ii)=(iii) we show now that lim h(r) = 0. In fact, we intend to prove a more general condition:

r1-

L lim h(z) = 0.

(5.6.9)

Z-1

Fix any element zo in 0 and define zt = Ft(zo), t > 0. We have zt --+ 1 and

h(zt) = e-th(z) , 0, as t -+ oo.

(5.6.10)

Thus what we need to show is that h is bounded in any nontangential approach region

r(1, k) = {z E 0 : 11 - zi < k(1 - Izi), k > 11. Returning to the representation (5.6.3') the latter fact is easily seen with the help of Koebe distortion theorem: _

h(z)j =

Ig(z)I <-

_ Iz z1z I1

< kz

(1 IzIlzI)z

whenever z c I'(1, k). Now by the Lindelof's Principle with (5.6.10) valid (5.6.9) results.

To accomplish (iii) we need to show that h(0) lies in a wedge of the angle 27r(1 - A).

By Lemma 5.6.1 we may assume that h is of the form n h(z)= 11

(1_z)AJ

j=1 n

where 1(jI = 1. (j # 1 and EAj = 2(1 - A). Each function wj(z) := j=1

1 - z

1 - z(

maps the open unit disk 0 onto a half-plane. In other words, Re (e' wj(z)) > 0 for some Oj. n

Denoting E Aj Oj by ,0, we have for each z E A j=1

n

larger3h(z)l

=

is

arget'3 [JwA' _ j=1

(a1ge"3 wj) EAj j=1

n

l < EA' 2) j=1 r(

7r(1-A).

Hence g(0) is contained in a wedge of the angle 27r(1 - A) as claimed.

STARLIKE AND SPIRALLIKE FUNCTIONS

Now following the idea suggested in [90] we will show that

183

Let

ho(z) = h(z)1X. Then ho(0) = 1, ho(1) = 0, ho is univalent and ho(O) is starshaped with respect to ho(1) = 0. Set (

1

Dn=ho(O)USzEC:IzI
,

n=1,2,...,

and for each n let hn : A H Dn be the conformal mapping of A onto Dn such that hn(0) = 1 and arghn(0) = argho(0) (see Figure 5.6).

Figure 5.6: An approximation of a starshaped domain with respect to a boundary point. By Caratheodory's Kernel Theorem we know that

lim hn = ho,

n-.oo

uniformly on each compact subset of A. Since each hn(O) is starshaped there are starlike functions gn with gn(0) = 0 and numbers Tn, ITnI < 1, such that hn(z) =

7 - n )

Tnz),

Z E A,

(see Section 5.5).

Note that 1 = hn(0) = -Tng'(0) and that h;,(0) =

1 gn(0)

+ gn(0)(1 +ITnz) 2 gn(0)

for all n. If the sequence had been unbounded then we would have reached a contradiction, because hn(0) -+ h0' (0) and I 4.

Chapter V

184

Thus {g;,(0)} is bounded and we can extract a convergent subsequence of {g,,,}.It is clear that we can assume that the corresponding subsequence of {r,,} converges to a point T E L. Denoting the limit function of the convergent subsequence of {g,,,} by go, we see that

ho(z) = 9o(z) (z - -r)(1 - Tz), z

z E A.

Letting z approach 1 we conclude that T = 1. Hence 1 (1

z J ho(z) = C_ 90 (Z)

- z)z

and

h(z) = (_ 9o(z)1z J 1

A

(1

-

z)2-2a,

where go : A H C is starlike with go(0) = 0. Since the function

9(z) := z (_ 9o(z)1

1-a

= z(1 - z)2

2h(z)

z

is starlike of order A we obtain (ii), as claimed. (iii)=(iv). Let the smallest wedge in which h(0) lies be of an angle 2ir(1-A*). Then A* > A, 1 zh'(z) 1+z Re

[1-A*

h(z) + 1-z >0, ZEL,

and this inequality no longer holds when A* is replaced with any number A* < A < 1. By the Riesz-Herglotz representation theorem we can write 1

1 - Al

zh'(z)

1+z _

h(z) + 1 - z

1+z( dm ( () 1 -z(

,

where m is a probability measure on the unit circle. After some calculations we obtain h(z) = (1 - z)2(1-A*) exp (_2(1

- A*) J log(1 KI=1

Again we note that (5.6.9) no longer holds when A* is replaced with any number

A* < A < 1. Let 6 denote the Dirac measure at ( = 1 E OA. Decomposing m relative to b, we can write in = (1 - a)v + aS, where 0 < a < 1, and v and S are mutually singular probability measures. It follows that

h(z) = (1 - z)2(1--) exp (_2(1 -

A) J log(1 Is)=1

STARLIKE AND SPIRALLIKE FUNCTIONS

185

where \ = 1 - (1 - \*) (1 - a).

If a > 0 we reach a contradiction because . > \*. Thus a = 0 and m = v. Let g = h/h'. Then f is semi-complete by Proposition 5.2.4. Using (5.6.8) or (5.6.9) we see that

Z-1

=

2(1 - a*) f

f(z)

1-

z E A.

dv((),

ISI=1

Let { z } be any sequence in

h(l, k) = {z E A: 11 - zj < kaa(1 - zl)

,

k > 1} .

which tends to 1 Consider the functions h,, : VA H C, n = 1, 2, ..., defined by .

h,, (C)

1

1-( - zn(

E DO.

Since the function h,,, maps the unit circle aA onto the circle 1( - c"' I = c"' j,

where c,,,=(1-z,)/(1-Jzn12), n= 1,2,...,weobtainthat Ihn(()I < 2Ic,,.1 < 2k.

Using (5.6.10) and applying Lebesgue's Bounded Convergence Theorem we now obtain L lim

f

(z)

= =

2(1 - A*) lim J

n-oc (1=1 2(1 - A*) < 2(1 - A).

h...(()dv(() (5.6.11)

In other words, condition (d) holds. Finally, we show the implication (iv) .(iii). Note that by Proposition 4.6.2, T = 1 is the sink point of the semigroup generated by f and L lim f'(z) is, in fact, z-.1 a real number. Therefore

Lli

f (z) 1 1z-1 - 2(1-A)'

Moreover, f (z) = h(z) = -(1 - z)2p(z), where p : A -4 C is holomorphic with

Rep(z) > 0 for all z c A. Again applying Proposition 5.2.4 and repeating the arguments as in the proof of (5.6.9) we obtain that h is starlike with respect to a boundary point with lim h(r) = 0. Let the smallest wedge in which h(O) lies be r-1 of an angle 27r(1 - A*), where A* E [0, 1). As we saw in the proof of implication (iii) .(iv), it follows that L lim

(zl = 2(1

z

1*)

(5.6.12)

Chapter V

186

Comparing the latter equality with (5.6.11) we see that A < A*. Thus h(s) lies in a wedge of an angle 27r(1 - A), as claimed. This concludes the proof of our assertion.

Remark 5.6.1. Thus given h E Fan(s) with lim h(r) = 0, formula (5.6.12) r-.1 infers that the value of smallest angle 0 such that h(s) lies in its wedge is ir multiplied by the angular limit of the Visser-Ostrowski quotient: 0 = 7rL lim

(z - 1)h'(z) h(z)

Corollary 5.6.1 If h E Fan(s) with lim h(r) = 0 satisfies the Visser-Ostrowski r-.1

condition:

Zlim

Z-1

(z-1)h'(z) =1 h(z)

(in particular, if h is conformal, or, more generally, isogonal at 1), then the smallest wedge which contains h(s) is precisely the right half-plane

11+= {zEC:Rez>0}. Remark 5.6.2 Since Ga C Go, it follows by the above proposition that h E Fan(s) with lim h(r) = 0 if and only if it satisfies the equation

r1-

z)2

h(z) = -(1

z

g(z),

where g E Star(s, 0). This proves Proposition 5.5.1 (Hummel's representation formula) for the case r = 1. Geometrically this fact should be understood as h(s) lies in the wedge of angle 2ir.

5.7

Converse theorems on starlike, spirallike and fanlike functions

In this section we consider inter alia the following converse problem: given f E G Hol(s) find conditions such that the equation h(z) = h'(z)f(z)

(5.7.1)

STARLIKE AND SPIRALLIKE FUNCTIONS

187

has a global solution in A which is univalent (consequently, starlike) in A.

If f (r) = 0 for some T E A and we are looking for the solution of (5.7.1) satisfying the initial conditions:

h(r) = 0 and h'(r) # 0,

(5.7.2)

then the necessary restriction is that f'(T) = 1. More generally, if f'(r) = p 54 0 then instead of equation (5.7.1) we must consider equation:

ph(z) = h'(z) f (z)

(5.7.3)

with initial conditions (5.7.2), which determines a spirallike function h.

Proposition 5.7.1 Let f E 9 Hol(O) be such that: f(0) = 0

(5.7.4)

f'(0) = A.

(5.7.5)

and

Suppose that h E Hol(O) is a solution of (5.7.3). Then:

h(z) = h'(0) lim eµt Ft(z)

t00

(5.7.6)

,

where {Ft}t>o is the semigroup, generated by f.

Proof. Let {Ft}t>o = Sf be a sernigroup generated by f. Then it follows by (5.7.3) that p h ( Ft(z)) = h' ( Ft(z)) f ( F t ( z)) or

_ - h'(Ft ( z ) ) aFt(z) .

at

aGt (z)

at

µ G t(z)

(5 . 7 . 7)

,

where Gt(z) = h(Ft(z)) , t > 0 , z E A and Go(z) = h(z).

(5.7.8)

Solving (5.7.7) with initial data (5.7.8) we obtain:

Gt(z) =

e-µ

or

t

. h(z)

e-µ

h(Ft(z)) = t h(z). Consequently, to prove our assertion, it is sufficient to show that .

tlinn eµt[h(Ft(z))

for allzEA.

-

h'(O)Ft(z)]

=

0

(5.7.9)

(5.7.10)

Chapter V

188

Note, that by Taylor's formula we have that for each w E 0 00 1

h(w) - h'(O)w = n=2

;

n

dzn (0)

wn = h(w).

If M = sup h(w) then the Schwarz Lemma (see Section 1.1) implies jwI<1

1h(w) < M Iw12

L. Setting w = Ft(z) we obtain

for all w

Ih(Ft(z))I < M IFt(z)12.

(5.7.11)

In addition, we know that IFt(z)I

< exp (- Re pt) (1

I-IIzI)2

(1 - IFt(z)I) (see Proposition 4.1.1). Hence (5.7.11) implies M IzI2

(Ft (z))

(1 - IzI)4

exp(-2 Re µt),

since (1 - IFt(z)I)2 < 1. Finally, since Re 1 > 0 we have exp(Relit) I h(Ft(z))

I eµt h(Ft(z))

M Iz12

(1 - IzI) as t -i 00

exp(- Re µt)--+0

and the result follows.

Thus if equation (5.7.3) has a solution h c Hol(0) satisfying the condition

h'(0) = k # 0,

(5.7.12)

then there is no other solution satisfying the same condition. Conversely, if the limit

h(z) := k lim eµtFt(z) t-.oo

exists it is easy to see that h(z) satisfies (5.7.3) with (5.7.12). Indeed, in this case h'(z)f(z)

a atzz) f (z) = k tli = k lim et" too k tlim etµ [iiFt(z) + 00

En n=2 "0

1

n etc` f (Ft(z))

zf (0)(Ft(z)n)

STARLIKE AND SPIRALLIKE FUNCTIONS

189

Again, as above, it can be shown that 1 do f

lim (0)(Ft(z)n) = 0. t0. n! dzn n=2

Hence, we obtain:

h'(z) f (z) = µk slim etµFF(z) = ph(z) 00

and we have completed the proof. Thus to solve the problem (5.7.3)-(5.7.12) we only need to show that the limit in (5.7.6) exists. To this end we define

u(t,z) = et"Ft(z) for all t > 0 and z E A. Then for fixed z E A, eu(t, z) = pLeµt Ft (z) at

- eµtf (Ft(z))

= pu(t, z) - eµt f (e-µtu(t, z)) = e2.t (µe-µt u(t, z) f(e-µtu(t, z)))

-

and u(0, z) = z.

Therefore the function u(t, z) : IEB+ - C satisfies the equation:

au t z

+

at

et"g(e-µtu(t z)) = 0, (5.7.13)

u(0, z) = z, where

cc 1

9(z) = f (Z) n=2

in

J , - (0) z".

(5.7.14)

In turn, (5.7.13) implies: t2

u(tl, z) - u(t2, z)

f etug(e-µtu(t, z)) dt t,

for each pair tl, t2 E R+ and z E A. It follows by (5.7.14) that g'(0) = 0. Hence, for each z E A: I9(z)I <_ Ml Iz12,

where Ml = supZEO Ig(z)I. Therefore Ig(e-µtu(t

z)j

= I9(Ft(z))I <_ MiIFt(z)12

< Mle-2 Re µt

Iz12 (1 -

IzI)4.

Chapter V

190

Using (5.7.6) we obtain for each z E A: t2

Ju(ti,z) -u(t2,z) < MI

e-Re µtdt ,

it I

I

z2

1 - Iz14

-> 0

as tl and t2 tend to oo. Therefore the limit

lim u(t, z) = lim et2Ft(z) t-.00

t-, 00

exists. 0 Thus we have proved the following assertion.

Proposition 5.7.2 Let f E 9 Hol(i) be a semi-complete vector field which satisfies the conditions: f (0) = 0, and f'(0) = µ. Then the equation

µh(z) = h'(z) f (z) has a unique solution h E Hol(A) which satisfies the conditions: h(0) = 0, h'(0) _ k # 0. This solution is a univalent spirallike function and it has the form: h(z) = k tlinn where {Ft} is the semigroup, generated by f In particular, when µ is a real number h(z) is a starlike function. .

Turning to a more general situation, assume now that

f(z) = (z - r)(1 - zT)q(z),

(5.7.15)

with Re q(z) > 0 everywhere. We already know that for each f E 9 Hol(0) with an interior null point T representation (5.7.15) holds. In this case:

f'(T) = (1 -

17_12)q(T)

(5.7.16)

If we define

g(z) = (MT)'(MT(z)) f (MT(z)),

we obtain g(0) = 0 and it = g'(0) = (1 -

I7-I2)q(T)

= f'(-r) and g E 9 Hol(0).

Therefore the equation µh1(z) = h' (z) . g(z)

with the initial conditions h(0) = 0, hi (0) # 0 has a unique solution:

hl(z) = hi(0)

lim t00

eN'tGt(z),

where {Gt(z)}t>o = Sg is the semigroup generated by g. Define now

h(z) = hl(MT(z)),

from which we obtain h(r) = h1(0) = 0 and h'(z) = hi(MT(z)) (MT)'(z).

(5.7.17)

STARLIKE AND SPIRALLIKE FUNCTIONS

In particular, h'(-r) = hi

(0)(It12

- 1 )-i

191

(5.7.18)

In addition,

µh(z) = uhi(MT(z)) = p

h' (MT(z))g(MT(z))

= uh,(MT(z)) - (MT)'(z)f (z) = µh'(z)f (z).

(5.7.19)

Note now that if {Ft}two = Sf is the semigroup generated by f, then Gt(z) = MT(Ft(MT(z))) is the semigroup generated by g (recall, that MT is an involution), and we have two symmetric relations: (i) MT(Gt(z)) = Ft(MT(z)); (ii) MT(Ft(z)) = Gt(MT(z))

Since, h(z) = hl(MT(z)) and hl(z) = h'(0) tlirneµtGt(z), we obtain: h(z)

=

h1'(0) . tli neµtGt(MT(z))

h'(T)(ITI2 - 1) . tli eµtMT(Ft(z))

h'(T)(IT2 - 1) lira

t-.co

eµt r - Ft(z) 1 - TFt(z)

h'(T) lim eµt(Ft(z) -T). t-+co

This formula gives us the solution of (5.7.3) under the conditions h(T) _

0,h'(r)#0.

So, we arrive at the following conclusion:

Proposition 5.7.3 Let f c G Hol(O) have the form (5.7.15)

f(z) = (z - T)(1 - zT)q(z) with Req(z) > 0, T E L and q(0)

0. Then the equation

uh(z) = h'(z) f (z) with µ = (1 - Ir12)q(0) has a unique solution h(z) satisfying the conditions:

h'(T) = k L 0.

h(T) = 0,

This solution is a univalent spirallike function on L which can be defined by the formula: h(z) = k lira e1`t(Ft(z) - T). t--' 00

If, in particular, q(T) is real then h is a starlike function on A. Now we consider the case when f E Hol(A) has no null point in A. It looks like this case is simpler, because equation (5.7.3) has no singularity in O, and one

Chapter V

192

can define a local solution of (5.7.3) for each µ E C under a corresponding initial condition, say h(0) = 1,

(5.7.20)

lim h(rr) = 0, r1-

(5.7.21)

and the boundary condition

where T E 8A is the sink point of the semigroup generated by f (see Sections 5.2 and 5.3). But, on the other hand, it is not clear why such a solution has an extension to all of 0 to be a univalent (spirallike) function (with respect to a boundary point). Moreover, if we wish this solution to satisfy the Visser-Ostrowski condition at the point T E OO we must require for the number µ in (5.7.3) to be real.

Proposition 5.7.4 Let f be a semi-complete vector field in 0 with no null point inside, and let r E OA be the sink point of the semigroup generated by f. If Q = Z lim f'(z) then for a real y > 0 the problem Z-T yh(z) = h'(z) f (z), h(0) = 1

(5.7.22)

has a unique univalent solution in 0 if and only if

µ<20.

(5.7.23)

This solution is a fanlike function (i.e., a starlike function with respect to a boundary point), the image of which lies in the wedge of angle 0 = lrp//3.

Proof. Without loss of generality we can assume that T = 1. If h E Univ(O) satisfies (5.7.22) with some real p > 0, then its image h(O) must be starshaped by Proposition 5.2.4. Furthermore, if we define fl(z) = F f(z) we have

Qi:=Llimfi(z)=Q>0 Z-T µ

and h satisfies the equation h(z) = h'(z)fl(z)

with h(0) = 1. Then it follows by Proposition 5.6.1 that there is .\ E [0, 1) such 1 1 that h belongs to the class Ga and 01 := L lim fi(z) > > . This implies Z-+T (5.7.23). In addition the angular limit of the Visser-Ostrowski quotient is

2-2A2

v=Llim z

h'(zh(z-1)

h(z)

=

1

-p

01

0,

and the smallest wedge which contains h(O) is of the angle Conversely. To solve (5.7.22) we first consider the problem: f3h(z) = h'(z) f (z), h(0) = 1,

(5.7.24)

STARLIKE AND SPIRALLIKE FUNCTIONS

193

where ,3 = L lim f'(z).

z- 1Consider the functions: n = 1,2,...

fn(z) = 1 z + f(z),

(5.7.25)

Firstly, it is known that fn E G Hol(0) for all n = 1, 2, ..., since the class 9 Hol(L) is a real cone. Secondly, for each n = 1, 2.... the equation fn(z) = 0

(5.7.26)

has a unique solution Tn E A such that Tn -4 1-, as n -' oo. Indeed, equation (5.7.26) is equivalent to the following one

z+nf(z)=0, which defines the values of the resolvent Jn at the point zero, i.e., Tn = Jn(0).

If we denote µn = 1 + f'(7-n) we obtain that ftn -i ,Q as n oc and fn(Tn) n Mn. Therefore, by the above Proposition 5.7.3, for each n = 1, 2.... the equation

Anh.(z) = hn(z) . fn(z)

(5.7.27)

has a univalent solution h7,(z) determined by h,,(Tn) # 0. Since rn 0 for all n = 1, 2.... and hn is univalent we have hn(0) 54 0 for all n = 1, 2, .... Therefore, we can define the functions: . hn(z)

hn(z) = hn(0)

which also satisfy equation (5.7.27), with hn(Tn) = hn(0) hn(Tn). In addition, h,,(0) = 1 ra = 1, 2, ... .

(5.7.28)

Now for each r E [ 0 , 1 ) we can find N > 0 such that for all n > N, I Tn I > r, that is f,, do not vanish on the disk Jzj < r. Therefore for such z (i.e., Izi < r) we can write by using (5.7.27) and (5.7.28): hn(z) =exp

,,.If

dz fn (Z)

.

0

Furthermore, since f (0) # 0 there is a neighborhood U of the point z = 0 in which (5.7.24) has a unique solution: h(z) = exp Since µ,,

/1

l02

f (zz) } .

(5.7.29)

Q and f,, (z) --> f (z) for all z E A, we have that hn(z) converges to h(z) in this neighborhood. It then follows by the Vitali theorem that h,,(z)

Chapter V

194

converges to h(z) on all of the disk IzI < r. Since r is arbitrary we obtain that h(z) is well defined on all of A. By the Hurwitz theorem (see, for example, [55]) h is univalent on A. _ Now again by Proposition 5.6.1 we have that h is a fanlike function the image of which lies in the wedge of angle 7r. Therefore the function h defined as h(z)

µ/R

= [h(z)1

is a univalent function on 0 whenever (5.7.23) holds. On account of (5.7.24) it is easy to see that h satisfies (5.7.22).

Remark 5.7.1 In the proof of the above proposition we have used an approximation process for the generator f (see formula 5.7.25). In turn, this process induces an approximation sequence of univalent functions in converging to h with interior null points defined by (5.7.27). However, in general we can claim only that these functions hn are spirallike, but not necessarily starlike. In fact, we do not know whether the numbers Fcn in (5.7.27) are real.

5.8

Growth estimates for spirallike, starlike and fanlike functions

The famous Koebe distortion theorem asserts: if It E S = {h E Univ(O) : h(O) = 0, h'(0) = 1} then

(1 +zllzI

< lh(z)I

-

(1

IzIlzI)2

for all z E A.

Equality holds for the Koebe function 00

ILK (Z) _

(1 - Z)2

-

f,Zn. n=1

Usually the proof of this fact is based on the known bound for the second coefficient in Taylor's expansion of It E S, namely la2l < 2. Note that the Bieberbach conjecture, namely that Ia.,I < n for each h E S, an = 1 d n h (0), n = 1, 2, ... , was proved by L. de Branges [20] in his remarkable work h! dzn

in 1985, whilst for It E S*(0) it has been proved earlier by R. Nevanlinna [103].

STARLIKE AND SPIRALLIKE FUNCTIONS

195

Although analogous of the Koebe distortion theorem fail for biholomorphic mappings in the polidisks (or balls) of dimensions greater than 1, it is still relevant for many special cases, in particular, starlike and spirallike mappings. Again as in previous sections one can use the relationships between these classes and semigroups to obtain corresponding bounds for these classes (see, for example, [109, 26]. Moreover, even in the one-dimensional case, by using some characteristics of semi-complete vector fields one can improve the estimates for some subclasses of Spiral(A). 1. We already know that if h E Hol(A, C) is a spirallike function on A, with

h'(T)=a#0, TEA,

h(T) =0, then h satisfies the equation

ph(z) = h'(z) f (z)

(5.8.1)

for some semi-complete vector field f E Hol(A, C) with f (T) = 0, f'(T) = µ E C.

In addition, if {Ft} = Sf is the semigroup generated by f, then h can be defined by the formula

h(z) = a lim eµt (Ft(z) - T)

(5.8.2)

t-.oc

(see section 5.8).

First let us suppose that T = 0. It follows by Proposition 4.4.2 and Remark 4.4.4 that Ft satisfies the following estimate: e-µt

IFt(z)I

IzI

(1 +cIzl)2 - (1 - cIFt(z)I)2

IzI - e- µt (1 - c1zl)2

with some c c [0, 1]. Moreover, if f is a bounded p-monotone function in A then c can be chosen strictly less then 1. This inequality and (5.8.2) immediately imply the following estimates for h: Ih'(0)I

(1 + cIzI)2 < I h(z)I

h'(0)I

(1 -I clIzI)2

(5.8.3)

For the general case, when T j4 0, T E A one can use the Mobius transformation NIT: T - Z NIT (z) _

1-zT'

defining the spirallike mapping Ti = h o Mr (note that the image h(A) = h(A)).

Since h(0) = h(T) = 0 and h'(0) = h'(T)(ITI2 - 1) we obtain by (5.8.3) and replacing h by h, Ih'(T)I(1-

ITI2)

zI

(1 +clzl) 2

< Ih(MT(z))I h'(T)j(1 - ITI2)

IzI

(1- clzI)2

(5.8.4)

196

Chapter V

or (replacing z by MT(z)) d(T,z) Ih'(T)I(1 - I7-I2)

< Ih(z)I

(1 + cd(T,z))2 -

d(T, z)

< I h'(T)I (1- 1r12)

(5 8 5) .

(1 - cd(7-,z)) 2

.

'

where d(T, z) = I MT(z)I is the pseudo-hyperbolic distance on A. Thus we have proved the following assertion.

Proposition 5.8.1 Let h E Hol(., C) be a spirallike (starlike) function on 0, with h(T) = 0, r E A. Then h satisfies estimates (5.8.5). Moreover, if h is strongly spirallike, then c can be chosen to be strictly less than 1.

Remark 5.8.1. To obtain a growth estimate for h E Spiral(0, r) in a form which does not contain a Mobius transformation we recall that the sets

DT(K)=(zEA:cpr(z)= 11

IzTl2

1- Iz12


K>1-ITI2}

are Ft-invariant for all t > 0, and h belongs to Spiral(DT(K), r) for each K > 1 - ITI2. Therefore it may be convenient to rewrite the right hand side of (5.8.5) in terms of the function cp -. By simple calculations we obtain 1-IT12

(1 - ITI2)(1 + cIMT(z)I)2

(1 - cIMT(z)I)2

(1 - IMT(z)I2)2 (1- ITI2)(1 + cIMr(z)I)2 . I1- zTI4 (1- Iz12)2(1- ITI2)2

(1 +clMT(z)I)2 1- IT12

Now (5.8.5) implies h(z)I

h'(0)I

(1 + c)2 [(z)] 1 - ITI2

(5.8.6)

2. If h is a starlike function on 0, normalized by the conditions h(T) = 0,

h(0) = a E C,

(5.8.7)

then one can find an estimate of growth by using Hummel's representation formula (see Section 5.5): h(z) = HT(z)g(z),

where

H,(z) = (Z - T)(1 - zT) z

zE0

'

STARLIKE AND SPIRALLIKE FUNCTIONS

197

and g E Star(A, 0). Indeed, it follows by the right hand inequality in (5.8.3) that I h(z)I

=

iz -

11

TI

= 0T(z)

- zTI

11 - TI

Iz

I9(z)I

I

(1 TI

1 + IzI Iz - TI

191(0)1

I9'(0)I-

1-Izi11-zTI

Similarly, by using the left hand inequality in (5.8.3) we obtain Ih(z)I

?

11

- z_II . I9'(o)I

Ill

+ IzI

Taking into account that g'(0) = -Ta, we obtain the following estimate:

Proposition 5.8.2 Let h E Hol(O, C) be a starlike function on A normalized by conditions (5.8.7). Then the following estimate of growth holds: Ih(0)I ITI d(T, z) ' cOr(z) 1 +

1z' < h(z)I 1 1

IT I d(-r, z) ' cor(z) 17-1

+IzI

1- IzI

.

(5.8.8)

3. Note that the latter arguments are relevant also when T is a boundary of A. Moreover, in this case (5.8.8) can be written in a more precise form. Indeed, using Proposition 5.6.1(ii) we have that if h is a fanlike function on A, normalized by the conditions: h(1) = 0, h(0) = 1, then for some A E [0, 1) it satisfies the equation h(z) - (1 - z)2-2a 9(z),

where g E S*(A) (starlike of order A), i.e., Re

zg'(z) > A. g(z)

Using the latter inequality it is easy to see that r9i(z)li-a

9(z)

z

= L

z

J

'

Chapter V

198

where gl belongs to S' (cf., Section 5.6). Then again by (5.8.3) we obtain

_

12-2A

(11+ Izi)J

12-2A

-

h(z)l < I(' 1- Izi)

(5.8.9)

In particular, we obtain that h is bounded in each nontangential approach region

atr=1.

Thus on account of Proposition 5.6.1 (see also Remark 5.6.1) and the obvious inequality 1- Izl < I1- zI < 1 + Iz1, we obtain from (5.8.9) the following distortion theorem.

Proposition 5.8.3 (see [134]) Let It be a fanlike function on 0 normalized by the conditions h(0) = 1,

h(1) = 0,

and let the image of It lie in the wedge of angle 7rv, 0 < v < 2. Then the following estimates hold

[1_lzI]v 1 +IzI

[1+IzJ]v h(z)l

1 - IzI

These estimates are sharp for the function

1 11

5.9

z

+z1".

Remarks on Schroeder's equation and the Koenigs embedding property

The so called Schroeder's (functional) equation: h(ip) = Ah

(5.9.1)

has been studied since the late nineteenth century (see, for example, [28] and references there). Here cp is a given holomorphic self-mapping of 0, where A E C is a suitable complex number, for which equation (5.9.1) has a solution h E Hol(0,C). Mostly

this equation has been considered as an eigenvalue problem for a composition operator C,, defined by the formula C,, h = It (V)

(5.9.2)

STARLIKE AND SPIRALLIKE FUNCTIONS

199

on the space Hol(A, C) (or its relevant subspaces, in particular, Hardy spaces HP). In this section we shall discuss some relations of Schroeder's equation with spirallike and starlike mappings on A. We assume that cp has an interior fixed point a E A. Of course, we will exclude the trivial cases when cp is a constant or the identity. In addition, we will assume that cp is not an elliptic automorphism of A. In other words, our condition for cp is that a E A is an attractive fixed point of it, or equivalently I cp'(a)I < 1.

(5.9.3)

We will see below that in this case, in fact, cp'(a) # 0

(5.9.4)

is a necessary condition for the global solvability of (5.9.1). First we note that if Schroeder's equation has a nontrivial solution (i.e., h is not constant), then A is neither 0 nor 1.

Indeed, if A = 0 then h(cp(z)) = 0 for z E A and h - 0 by the uniqueness theorem. If A = 1 then h satisfied the equations

n = 1, 2, ... ,

h(cp(n)) = h, and we have

h(z) = lim h(W

n) (Z))

= h(a)

n-.oo

because of assumption (5.9.2). Thus we have to assume that A is neither 0 nor 1. Note that if A # 1 then h(a) = h(cp(a)) = Ah(a), and we have h(a) = 0

(5.9.5)

as a solution h of Schroeder's equation (5.9.1). Hence h must have the form 00

h(z) = n an(z - a)', k > 1.

(5.9.6)

n=k

For simplicity we put a = 0. Then (5.9.5) and (5.9.1) imply 00

=

h(cp(z)) h(z)

k

= (

z

n=k ancP

n k

(z)

00

)

anzn-k

n=k

Since the left hand side of this equality is a constant we obtain, letting z go to zero, that A = [cp,(0)]k

Since A # 0 we obtain (5.9.3). Thus we have proved the following assertion.

(5.9.7)

Chapter V

200

Proposition 5.9.1 Let cp E Hol(0) have an attractive fixed point a E 0 and suppose that Schroeder's equation (5.9.1) has a nontrivial solution h E Hol(O, C) for some A E C. Then

(i) h(a) = 0; 0 for some positive k; (iii) if h is locally univalent, then A = cp'(a) (ii) A = [cp'(0)]k

0.

In 1884 Koenigs proved a remarkable result on the solvability of Schroeder's equation.

Proposition 5.9.2 (see [78]) Let 0, such that

E Ho1(0) have an attractive fixed point a E

cp'(a) := A

0.

Then there is a function h E Hol(0, C) such that h(cp(z)) = Ah(z)

for allzEL. In addition, if cp is univalent then so is h. The idea in the original proof of Koenigs's theorem is based on the convergence of the sequence 00

('L)

hn(z) _

A

(z) = z +

ak

zk,

n = 1, 2....

k=2

which evidently satisfies the recursion equation Ahn+1(z).

(5.9.8)

Its limit function It is called the Koenigs function and it is normalized by the condition h'(a) = 1.

If such a function can be found one can present ,p(z) = It-' (Ah(z))

(5.9.9)

whenever the right hand side of this equality is well defined. Hence O

(n)(z)

= h-1(A h(z))

(5.9.10)

The latter expression then serves as a definition of fractional iterations of cp when n. is not an integer (cf., section 4.4) and large enough. Of course, if cp can be embedded in a globally defined continuous semigroup of holomorphic mappings it must be univalent, and so should h. Although the discrete semigroup of iterates of cp cannot be embedded, in general, in a continuous semigroup of holomorphic self-mappings of 0, depending on what one requires the answer may be yes in some suitable cases (see for example, G. Srekeres [138], J. Hadamard [59], T. Harris [63], C.C. Cowen [27]). The simplest case (though very useful) is, of course, when cp is a fractional linear transformation and so is h. We consider now, in some sense, the dual problem.

STARLIKE AND SPIRALLIKE FUNCTIONS

201

Definition 5.9.1 We will say that a mapping cp E Hol(A) satisfies the Koenigs embedding property (K.e.p.) if its iterations cp(m) : A -p A can be embedded in a continuous semigroup {Ft}t>o of holomorphic self-mappings of A, i.e., F1 = W.

It turns out that the answer to the question of what are the conditions for a univalent self-mapping cp E Hol(O) to satisfy the (K.e.p.) is related to some geometrical properties of the solution of Schroeder's equation. Proposition 5.9.3 Let cp be a univalent self-mapping of A, and let cp(T) = T for some T E A with 0 < I cp'(T) I < 1. Then cp satisfies the Koenigs embedding property

if and only if its Koenigs function is p-spirallike, with it = -log W'(-r)

-

Proof. Sufficiency. Suppose that equation (5.9.10) has a univalent µ-spirallike solution with µ = - log A. Then for each z E A and t > 0 element e-tµh(z) belongs to h(O). If we define

Ft(z) = h-1(e-t, h(z)), we obtain by (5.9.10) that

F1(z) = h-1(eIn ah(z)) = h-1(Ah(z)) = co(z). Necessity. Let cp satisfy the Koenigs embedding property, i.e., there is a semigroup {Ft}t>o C Hol(O) such that Fi(z) = cp(z)

Denote by f = -

dF

It=o+ the generator of {Ft} and consider the equation

µh(z) = f (z) h'(z),

(5.9.11)

where µ = - log A. Since f'(T) 0 it follows by Proposition 5.7.2 that equation (5.9.11) has a univalent µ-spirallike solution h which satisfies the equality

h(Ft(z)) = e-µth(z). Setting here t = 1 we obtain (5.9.10). If cp has no fixed point in A then it follows by the Julia-Cratheodory Theorem that there is a unique point T E 80 such that L lim V(z) = T and Z_T

0
Chapter V

202

Proposition 5.9.4 Let cp E Hol(A) have no fixed point in A, and let T E OA be its sink point. Then cp satisfies the Koenigs embedding property if and only if for some A E (0, 1) there exists a solution of Schroeder's equation h(cp) = Ah,

(5.9.12)

which is a univalent fanlike function on A such that L lim h(z) = 0. Z-r

Remark 5.9.1 In fact, Proposition 5.7.3 and the Julia-Wolff-Caratheodory Theorem imply that if cp satisfies the Koenigs embedding property then Shroeder's equation (5.8.11) is solvable if and only if a2 < A < 1, where a = Z lim cp'(z) > 0.

Z-r

Another direct consequence of the above propositions is a result originally established by A. Siskakis (see [136]).

Corollary 5.9.1 Let Ft be a continuous semigroup of holomorphic self-mappings

of A, and suppose that there is a point T E z, such that t-.oo lim Ft(z) = T, and Z lim Re f'(z) # 0, where f =

Z-r

-dF dt

Then there are A E C, 0 < IAl < 1, and h c Hol(A,C) such that for all t > 0 h(FF) = Ah.

(5.9.13)

In other words, there is a solution of Schroder's equation which does not depend

ont>0.

In addition, the solution of (5.9.12) can be found by solving the differential equations (5.9.11) with p = - log A. z

Example 1. Let cp(z) =

.

It is easy to verify that cp satisfies

z2 - e2(z2 - 1) Koenigs embedding property with Ft(z) = z2 - ez (z2 - 1), Ft(0) = 0, for all

t>0. Therefore, instead of (5.9.10) one can try to solve (5.9.12) with A = y'(0) = l/e or (5.9.11) with it = 1. _ dFt(z) Since f() z It=o+= z - Z33 equation (5.9.11) becomes dt

h(z) = h'(z) (z - z3). Solving the latter equation we obtain that

h(z) =

az

1-z2

a E C,

is a solution of equations (5.9.10) and (5.9.12). The converse scheme does work if we know in advance the solution of Schroeder's

equation. In particular, it can always be solved for a fractional linear transformation of the unit disk with an interior fixed point. Indeed, let p E Hol(A) be a

STARLIKE AND SPIRALLIKE FUNCTIONS

203

fractional linear mapping of A with a fixed point T E A. Without loss of generality we can consider the case r = 0. Then cp can be written in the form az

0(z)= 1-bz.

(5.9.14)

We will investigate the sufficient and necessary conditions on coefficients a and b such that cp will satisfy the Koenigs embedding property. First we note that the condition cp(z)I < 1 for all Izj < 1 imply jal + JbI < 1.

(5.9.15)

In addition, we have cp'(0) = a. Then it is easy to see that the function z h(z) =

(5.9.16)

1 - kz'

b

Ah(z) with A = a where k = 1 - a, is the Koenigs function for gyp, i.e., and h'(0) = 1. Thus we tend to find a condition which will force h to be univalent p-spirallike function with p = - log a. In other words, we have to check the inequality µh(z) (5.9.17) Re > 0. hl(z) By (5.9.15) inequality (5.9.17) becomes

Repz(l - kz) > 0,

(5.9.18)

since by (5.9.15) kI < 1 the latter condition can be rewritten in the form

cos arg(li) > k.

(5.9.19)

In particular, this condition always holds when p is real. Thus we have proved

Proposition 5.9.5 Let cp(z) =

az with jal + IbI < 1, and 0 < at < 1. Then 1 - bz cp satisfies the Koenigs embedding property if and only if

-

b

cos I arg I log a1)) > 1-a

(5.9.20)

In particular, if a is a real number then cp can be always embedded in a continuous semigroup of holomorphic self-mappings of A. az , as in Example 1 of Section 4.3, with 1 - bz a = s exp(i3.1) and b = 3, then it is easy to see that jal + JbI = 1 whilst condition (5.9.20) does not hold. Thus : A i-+ A cannot be embedded in a continuous semigroup of self-mappings of A (see Figure 4.1).

If we consider the mapping O(z) =

Chapter V

204

Exercise 1. Prove the equivalence of conditions (5.9.17) and (5.9.18).

Exercise 2. Under condition (5.9.19) find the semigroup Ft : A , 0, t > 0, such that Fl = cp, where W has the form (5.9.14).

More detailed discussion of this approach for higher dimensions can be found in [76].

Because of our intention in a modest text to emphasize the dynamical flavor of the subject a big part of the theory which is primarily geometrical or functional analytic in nature has not been included.

We refer the reader to books of A.W. Goodman [57], P. Duren [33], and J.H. Shapiro [131], which could be good guides to complete the knowledge in these topics. Also, to advance to futher study on the boundary behavior of holomorphic mappings we mention an excellent book of C. Pommerenke [107]. Finally, we point out that many applicable subjects as branching processes, optimization theory, functional calculus etc., mentioned in the Preface may motivate an investigation in this direction.

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Author and Subject Index Latuskin, Y., vii, 210 Lyzzaik, A., viii, 171, 177, 211 MacCluer, B., vii, viii, 207, 211 Mazet, P., vii, 211 Mellon, P., vii, 211 Mercer, P., vii, 211 Montel, P., 6, 211 Nevanlinna, R., viii, 94, 158, 194, 212 Pfaltzgraff, J.A., ix, 212 Pick, G., 9 Poincare, H., 7, 43, 44 Porta, H., vii, 63, 93, 95, 101, 206 Potapov, V.P., vii, 212 Reich, S., vii, 205, 207-210, 212 Riemann, B., 7 Riesz, F., 5 Robertson, M.S., viii, ix, 159, 171,

Abate, M., vii, 101, 205 Alexander, J. W., 158, 205 Behan, D.F., 37, 206 Berkson, E., vii, 63, 93, 95, 101, 206 Bernardi, S.D., viii, 206 Brickman, L., ix, 206 Caratheodory, C., 9, 17, 206 Chen, G.N., vii, 206 Cowen, C. C., viii, 200, 207 de Branges, L., ix, 194, 207 Denjoy, A., vi, 9, 32, 35, 207 Earle, C., vii, 207 Egeruary, E., 176, 207 Fan, K., vii, 208 Forelli, F., vii, 208 Galton, F., v, 215 Glicksberg, I., vii, 208 Goebel, K., vii, 208 Gurganus, K.R., ix, 209 Hadamard, J., 200, 209 Hamilton, R., vii, 207 Harris, L.A., 13, 209 Harris, T.E., vi, 200, 209 Herglotz, G., 5 Hummel, J.A., 171, 172, 174, 209

176, 213

Schroeder, E., 198, 213 Schwarz, H.A., 9 Sekowski, T., vii, 208 Sevastyanov, B.A., vi, 213 Shafrir, L, vii, 213 Shields, A. L., 36, 213 Silverman, H., viii, 177, 179, 213 Silvia, E.M., viii, 177, 179, 213 Siskakis, A.G., vii, 214 Srekeres, G., 200, 214 Stachura, A., vii, 208, 210 Steffensen, J.F., vi, 214 Stepin, M., vii, 210

Jafari, F., vii, 209 Julia, G., vii, 9, 17, 209 Koenigs, G., vi, 200, 210 Kubota, Y., vii, 210 Kuczumov, T., vii, 210 Lowner, K., ix, 211 216

AUTHOR AND SUBJECT INDEX Suffridge, T.J., ix, 212, 214 Vesentini, E., vii, 101, 208, 214 Vigue, J.P., vii, 211 Wald, J.K., viii, 159, 215 Watson, H. W., v, 215 Wlodarczyk, K., vii, 215 Wolff, J., vi, vii, 9, 32, 35, 101, 215 Yale, K., vii, 209

217

Cauchy problem, 66, 68, 69, 72, 76, 82, 83, 87, 103, 161 Cauchy-Schwarz formula, 5 class of Caratheodory, 89, 178 commuting family, 36, 101 complete metric space, 7, 43, 44 complete vector field, 104, 156 contraction, 7, 17 strict, 8, 31, 52, 70, 99, 122

admissible curve, 42, 45-47

angular derivative, 23, 32, 35, 135, 140, 145, 150, 164 angular limit, 20, 23, 186 approximating curve, vii, 53, 55, 56 asymptotic behavior, vii, 83, 98, 103, 113, 135, 142-145, 150, 177

Denjoy-Wolff Theorem, vii, 30, 32, 36, 101, 102

embedding property, 109, Chapter V Euclidean distance, 7, 13, 114 exponential formula, 68, 70, 97, 122

automorphism, vi, 9, 15, 19, 26, 30, 35, 56, 83, 146 elliptic, 27, 30, 36, 103, 105, 112, 113

hyperbolic, 27-29, 32, 37, 105, 109, 111

parabolic, 27, 28, 30, 32, 105, 111, 112

Banach Fixed Point Principle, 8, 31, 52, 99 Berkson-Porta representation, 95, 98, 99, 139, 162, 170, 171, 174, 181

Bieberbach conjecture, ix, 194 Bohl-Poincare theorem, 81 boundary behavior, 17, 163 branching process, v, vi

Brouwer's Fixed Point Principle, 8, 31

Caratheodory Kernel Theorem, viii, 6, 166, 177, 183 Cauchy Integral Formula, 4

fixed point, 8, 27, 31, 32, 53, 54, 57, 99, 111

attractive, 32, 35, 156, 199, 200 boundary, 26, 29, 30, 32, 35 common, 36, 101 free mapping, 33, 37, 54 interior, vi, vii, 25, 30-32, 56 of automorphisms, 26 of holomorphic self-mapping, 25, 31

of nonexpansive mapping, 52 flow invariance condition, vii, 82, 98 fractional linear transformation, 10, 29, 40, 107, 109, 200, 202

function p-monotone, 79-82, 98, 141, 195

strongly, 98, 99, 169 close-to-convex, ix, 153, 177 fanlike, 157, 176, 192, 198 harmonic, 5, 85, 86, 97, 175 holomorphic, 4, 9 spirallike, see spirallike function starlike, see starlike function

AUTHOR AND SUBJECT INDEX

218

univalent, ix, 4, 6, 153, 154, 158, 159, 161, 162, 170, 177, 178,

Julia-Wolff-Caratheodory Theorem, 33, 164, 171, 202

180

Koebe distortion theorem, 182, 194, generator, 66, 68, 81, 82, 144, 147,

150, 160, 162

of a one-parameter group, 82, 83, 104

of a o ne-param eter sem igrou p, 70, 76, 83

geodesic segment, 15, 45, 46 geodesics, 44, 47 group of automorphisms, 82, 104 one-parameter, 60 growth estimate, 194, 196

Harnack inequality, 89, 146, 149 strong, 90, 119 horocycle, 18, 19, 32, 37, 54, 97, 136, 167

Hummel's multiplier, 171, 172, 177 Hurwitz convergence theorem, 6 hyperbolic ball, 136, 174 distance, 45 length, 39, 42, 45

metric, vii, 7, 8, 39, 72, 79, 98, 99, 122

Implicit Function Theorem, 72, 82 infinitesimal generator, 66, 70, 76, 83 involution property, 11 isometry, p-isometry, 52, 60 Julia number, 19, 33, 35 Julia's Lemma, vi, 18, 19, 24, 32, 33, 35, 151

Julia-Caratheodory Theorem, 22, 23, 32, 35, 150, 201

195

Koebe function, 168, 194 Koebe One-Quarter Theorem, 153 Koenigs embedding property, 198, 201203

Koenigs function, 200, 201 Lebesgue's Bounded Convergence Theorem, 185 Lindelof's inequality, 12 Lindelof's Principle, 23, 182 lower bound, 80, 135, 144 appropriate , 137 , 140 , 142 , 143 , 148

Mobius transformation, vii, 10, 11, 15, 40, 46, 83, 168, 172, 173, 195, 196

mapping conformal at the boundary point, 164

identity mapping, 11 isogonal at the boundary point, 164

nonexpansive, 7, 8, 39, 43 p-nonexpansive , 52 , 56 , 57 , 6870, 75-77, 79, 81, 82, 136, 144

fixed point free, 54 fixed point of, 52 maximum principle for harmonic functions, 5, 85, 86, 96

maximum modulus principle, 5, 10, 35, 47 Montel Theorem, 6

AUTHOR AND SUBJECT INDEX Nevanlinna's condition, ix, 163, 167 nontangential approach region, 20, 23, 150, 182, 198 nontangential limit, 20, 21, 34 normal family, 6 null point, 98, 104, 135, 144, 156-158, 161, 171

Open Mapping Theorem, 154

Poincare metric, 7, 8, 39, 43-46, 70, 72, 79, 98 infinitesimal, 44, 45, 122 power convergence, 32, 35, 36, 56, 57 pseudo-hyperbolic ball, 17, 25, 118, 167 disk, 32 distance, 14, 17, 39, 114, 196

metric, 7 range condition, 69, 70, 72, 74, 77,

79, 82, 88 rate of convergence, 29, 35, 98, 113, 135, 139, 142, 144, 147, 149 resolvent, 73-78, 81, 82, 96, 99, 193 nonlinear, 67, 69, 72, 79, 82 resolvent identity, 74-78 retraction, 56, 78 Riemann Mapping Theorem, viii, 154, 166

Riesz-Herglotz representation, 5, 93, 178, 184

Schroeder's equation, 198-201 Schwarz Lemma, vi, 9, 12, 27, 87, 89, 91, 149, 167, 188 Schwarz-Pick inequality, 11, 13, 17, 19, 43, 82 Schwarz-Pick Lemma, 11, 14, 17, 19, 25, 26, 31, 32, 57 boundary version, 35

219

semi-complete vector field, 91, 95, 156, 174, 192, 195 semigroup, ix, 9, 60, 68, 97, 98, 135, 150, 155, 187, 190

one-parameter, v, 70 continuous, 60, 63, 66, 109 discrete, 26, 60, 109 sink point, 33, 35-37, 55-57, 78, 97, 105,110-112,135,139,142, 144, 149, 150, 171, 182, 185, 192

spirallike function, viii, ix, 159, 160, 167, 190, 191, 195 strongly, 169, 196 with respect to a boundary point, 159, 162

with respect to an interior point, 159, 161, 162

spiralshaped set, 153, 159, 160, 166, 167

starlike function, viii, ix, 157, 158, 160, 162, 167, 169, 170, 190, 191, 196

of order A, 159, 179, 197 with respect to a boundary point, viii, 157, 158, 171, 176, 177, 180, 185, 192 with respect to an interior point, 157, 159, 161, 170, 171, 176, 177

starshaped set, 153, 157, 159-161, 166, 167, 172, 178 stationary point, 101, 104, 136, 138, 142

Stolz angle, 21

strict contraction, 8, 31, 52, 70, 99, 122

support functional, 136 Taylor representation, 5, 10, 154, 194

AUTHOR AND SUBJECT INDEX

220

uniform convergence, 6, 54, 142 uniform Lipschitz condition, 13 vector field, 2

complete, 83, 85, 93, 96, 104, 110, 112, 156, 171 semi-complete, 83, 91, 95, 97, 98, 113, 156, 171, 174, 177, 180, 182, 190, 192, 195 Visser-Ostrowski condition, 165, 186, 192

Visser-Ostrowski quotient, 163, 165, 186, 192

Vitali theorem, 6, 31 Wald's condition, ix, 163 Weierstrass convergence theorem, 6 Wolff's Lemma, 32, 33, 35, 54

List of Figures The function w = f (z) . . . . . . . . . . . The translation f (z) = z + a, a = 4 + 2i . 0.3 The rotation f (z) = ei"Bz, 9 = -21r/3 .. . 0.4 The contraction f (z) = kz, k = 1/3 . . . . 0.5 The vector field w = f (z) . . . . . . . 0.1

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An orthogonal circle to aA and its image under an automorphism. A Mobius transformation of A .. . . . . . . . . . . . . . . . . . 1.3 A horocycle at the point ( E O . . . . . . . . . . . . . . . . . 1.4 A nontangential approach region at a boundary point. . . . . . . . 1.5 A Stolz angle at a boundary point . . . . . . . . .. . . . . . 1.6 Boundary behavior of a self-mapping of A . . . .. . . . 1.7 Elliptic automorphism.. . . . . . . . . . . . . . . . . . . 1.8 Hyperbolic automorphism .. . . . . . . . . . . . . . . 1.9 Parabolic automorphism . . . . . . . . . . . . . . . . . . . . . 1.10 Uniqueness of a point ( . . . . . . . . . . . . . . . . . . . .

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The points of anharmonic relation

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Boundary condition for f E aut(o) .

3.2 3.3

Boundary flow invariance condition . . . . Values of functions of Caratheodory's class .

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Fractional iterations of the self-mapping F(z) .. The circle F(z) and the sectors S and S .. . . . 4.3 The nontangential convergence to T . . . . 4.4 The asymptotic behavior of the flow . . . . . . . 4.1

4.2

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A starshaped domain (0 E 1) .. . . . A starshaped domain (0 E 80) . . . . 5.3 The spiralshaped domain (0 E an) ..

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LIST OF FIGURES

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5.5 5.6

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168

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183

An approximation of a starshaped domain .