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c M (a) for 0 S a S p. Denote by N (a) the subfamily of M (a) consisting of functions of the form (4). It was
f
shown in [28] that if
-
I.
-
I.
na,. < 1,
n=2
n 2 a,. <
and 0 sa s 1, then a neessary and
oo,
n=2
sufficient condition for f to be in N (a) is that
I.
(7)
I.
(n -l)a,.
(n 2
-
n) a,.
n=2 _ _ __ :..:..::-=---- + a .:..:____.::._
(1 -a) n=2
1-
I.
n=2
a,.
1-
I.
n=2
Sl.
na,.
This condition is still necessary when a > 1. Inequality (7) was then used to find the (positive) order of starlikeness and sharp coefficient bounds for f e N (a), a > 0. The 122
order of starlikeness of the general class M (a), a
Open Problem 16. Find the extreme points of N ( al. Functions of the form F 11 (z) = z - A,.z" for which quality holds in (7) uniquely maixmize the n'th coefficient, so that (F,. (z)} are extreme points. We have not been able to determine if there are additional extreme points. The conveixty of the families T* (a) and C (a) are immediate consequences of (6). The more complicated condition (7) does not readily lend itself to showing convexity for N (a). Open Problem 17. Is N (a) a convex family ? Finally, we mention a family for which the extreme points are known for some values. A function fin Tis said to be in T* (a, b), a real and b :2:0, if lzf'(z)/f(z))- a I s b for all z in/!. Note that T* (1, 1- ex)= T* (a). In [3] the extreme points ofT* (a, b) were found when a
some b when f is of the form f (z)
=z- L
a2nz2n .
"=1
Open Problems 18. Find the extreme points ofT* (a, b), a< 1.
References 1. A. Baernstein, Integral means, univalent functions and circular symmertization, Acta. Math. 133 (1974), 139- 169. 2. R. Barnard and J. Lewis, A counter example to the two-thirds conjecture, Proc. Amer. Math. Soc. 41 (1973), 523-529. 3. R.D. Berman and H. Silverman, Coefficient inequalities for a subclass of starlike functions, J. Math. Anal. and Appl. 106 (1985), 197-205. 4. L. Brickman, D.J. Hallenbeck, T.H. MacGregor and D.R, Wilken, Convex hulls and extreme points of families of starlike and convex mappings, Trans. Amer. Math. Soc.185 (1973), 413-428. 5. L. De Branges, A proof of the Bieberbach conjecture, Acta Math. 154 (1985), 137 -152. 6. W. Deeb, Extreme and support points of families of univalent functions with real coefficients, Math. Rep. Toyama Univ. 8 (1985), 103- 111. 7. P. Eenigenburg and D. Nelson, On the order of starlikeness of alpha-convex functions, Mathematics (Cluj) 18 (41) (1976), no.2, 143- 146. 8. A.W. Goodman, Univalent Functions, v.2, Mariner Publishing Co. (1983). 9. R.R. Hall, On a conjecture of Clunie and Sheil-Small, Bull. London Math. Soc. 12 (1980), 25- 28. 10. D.J. Hallenbeck and T.H. MacGregor, Support points of families of analytic functions described by subordination, Trans. Amer. Math. Soc. 278, no. 2 (1983), 523-546. 11. M. Lachance, Remark on functions with all derivatives univalent, Inter. J. Math. 3 (1980),193 -196. 12. Z. Lewandowski, Nouvelles remarques sur les theoremes de Schild relatifs a une classe de functions univalentes, Ann. U.M. Curie-Sklodowska 10 Sec. A. (1956), 81-94.
123
13. K. LiSewner, Untersuchungen Uber schichte konforme Abbildungen des Einheitskreises I, Math. Ann. 89 (1923), 103 -121. 14. C.P. McCarty and D.E. Tepper, A note on the 2/3 conjecture for starlike functions, Proc. Amer. Math. Soc. 34, no. 2 (1972), 417- 421. 15. S.S. Miller, P.T. Mocanu, and M.O. Reade, All a-convex functions are starlike and univalent, Proc. Amer. Math. Soc. 37 (1973), 553-554. 16. S.S. Miller, P.T. Mocanu, and M.O. Reade, Starlike integral operators, Pacific J. Math. 79 (1978), no. 1, 157 -168. 17. P. Monte], "Let;ons sur les functions univalentes ou multivalentes•, Paris, Gautheir- Villars, 1933. 18. M.S. Robertson, On the theory of univalent functions, Ann. Math. 37 (1936), 374-408. 19. A Schild, On a problem in conformal mapping of schlicht functions, Proc. Amer. Math. Soc. 4 (1953), 43-51. 20. A Schild, On a class of functions schlicht in the unit circle, Proc. Amer. Math. Soc. 5 (1954), 115 -120. 21. G. Schober, "Univalent Functions - Selected Topics•, Lecture Notes in Mathematics, Springer- Verlag (1975). 22. S.M. Shah and S.Y. Trimble, Univalent functions with univalent derivatives, Bull. Amer. Math. Soc. 75 (1969), 153 -157. 23. H. Silverman, Univalent functions with negative coefficients, Proc. Amer. Math. Soc. 51 (1975), 109-116. 24. H. Silverman, Extreme points of univalent functions with two fixed points, Trans. Amer. Math. Soc. 219 (1976), 387-395. 25. H. Silverman, Univalent functions with varying arguments, Houston J. Math. 7, no. 2 (1981), 283- 287. 26. H. Silverman, Coefficient bounds for quotients of starlike functions, Rocky Mt. J. Math. 13, no.4 (1983) 627-630. 27. H. Silverman, Univalent functions having univalent derivatives, Rocky Mt. J. Math. 16, no.l (1986) 55 - 61. 28. H. Silverman, Extremal properties for a family of a-convex functions, Houston J. Math. 12, no.2 (1986), 267-273. 29. H. Silverman, Coefficient bounds for inverses of classes of starlike functions, Complex Variable, 12 (1989), 23-31. 30. T.J. Suffridge, On univalent polynomials, J. London Math. Soc. 44 (1969), 496- 504.
124
Some Inequalities for Starlike and Spiral-like Functions
V. Singh Department of Mathematics Punjabi University Patiala 147 002, INDIA.
L Introduction and statement of results c1>
In the unit disc U = (z : I z I < 1} let A be the class of analytic functions (0) = 1. We need the following subclasses of A :
c1>
which satisfy
( i) the class P of functions, f e A which satisfy Ref (z) > 0, z e U;
s·
(ii) the class a of functions,
f
(z)
(1)
=z$ (z), cl> E Re
A which for
(l
< 1 satisfy
(z)) (zf' f (z) > U·
This is the class of functions starlike of order a and for 0 S a < 1 these functions are univalent in U. We shall denote the class S~ by S*. (iii) the class Spa of functions, (2)
f
(z)
=z c1> (z),
R e { eia
c1>
e
A which for I al < 1t/2 satisfy
z t..:..S!l} f (z) > 0.
This class of functions is called the class of a-spiral-like functions and these functions are univalent in U. From the above definitions it is evident that (a)
f
e
s:,
a < 1 if and only iff (z)
(3)
=z$ (z), c1> e
A and
z LJ!l f (z) =(1-a)p(z)+a,
for somep e P;
(b) f e Spa, I al < 1t/2 if and only iff (z)
(4)
= z$ (z), c1>
e A and
(z) ( ) · · eia z f' j(z) = cosap z +Lsma,
for some p e P. 125
Further, it is clear that f e S a• if and only ifthere exists a «!> e S* such that
(z)Jl-a, a<1;
«!>
f(z)=z [ -z-
(5)
and f e Spa if and only if there exists a «!> e S* such that
f
(6)
(z) =
«!>
(z >Je
-ia
z [ -z-
cos a
, I a I < rr/2.
In view of the Herglotz representation formula for the class P the following is well known: Theorem A. For each f
E
S* there exists a unique probability measure JJ on [0, 2tr} such that 2lt
i..:J!l _
(7)
z f (z) -
J
1 + zeit 1 -zeit dJJ (t),
0
and 2lt
(8)
log lJ!l =
z
J
log - 11 it dJJ (t). -ze
0
Further for f e S*, I z I
(9)
=r the following inequalities hold : r
r
(1 + r )2 S If (z) I S (1 - r )2 '
and (10)
1-r < 1+r -
Rezf'(z) f (z)
< -
lzf'(z)l f (z)
< 1+r - 1 -
r ·
The Koebe function (11)
is a distinguished member of S* and shows that the inequalities
(9)
s:
and
(10)
are sharp.
The region of values of zf' (z)/f (z), f e or f e Spa in terms of If (z) I has recently been investigated by serveral authors. In view of the equations (3) and (4) and the following (12)
2r Ip(z)- 1+r21 1 _r2 s 1 _r2 , lzl =r,peP
the region of values of zf'lf for f e (7) and (8) the set (13)
R~ =
s•a
or Spa. is readily determined. Moreover in view of
/(~) {5Lill
:fe S*, lf(~)l fixed }
is a compact convex set. From a result of Leeman [2] it also follows that the extreme points of the set of R~ are of the form
126
(14)
1 + ~eir 1 + t;eifl A1-~eir + (1-A) 1-~eiP ,0~A<1,
wherein (y, ~) e [-1t, 1t] x [-1t, 1t]. However this information is usually not adequate to obtain sharp bounds on certain interesting functionals. We mention some of the earlier results which concern us. Theorem B. (Twomey[6]).For fe S*,ze U, lzl =r z['(z)l l f (z) ~ (1 +r) p (r),
(15)
where 1
(16)
P (r)
= -1-.-r
rlog[( 1 ~r) 2 1/(z)IJ + --------,-1-+_r__ (1 - r2) log -1---r
Theorem C (Singh [5]). Under the same assumptions we have 1 - r2 _ zf'(z) (17) -r-1/(z)l ~ ~ 2p(r)-1,
R.eJW
and[4] (18)
l
zf'(z)l 2 ~ 4(1+r2) (r)-3+r2 f (z) 1- r 2 p 1 - r2'
which improves upon (15) whenever (19)
log[( 1 + r) 2 1/ (z) 1] r
~ ~ (!- r)2 log 1 + r r\_1+r
1-r'
in particular, if 0 < r < 0·29. TheoremD (RuscheweyhandSingh£4]). For fe Spa (20)
Re {eia z['(z)} f (z)
~
2 p (r) -cos a, 2
where
(21)
Theorem E (Lebedev and Gromova [1]). Under the assumptions of Theorem A
(22) (1 - r)3
127
2S '
5 ~ 2.
We note that the inequalities (17), (20 and (22) are sharp but sharp upper bounds on
lzJ~;~)~ ,J e
s:
or Spa are not yet known. In the present paper we prove the following
which leads to an improvement of(15) and (18) in U. Theorem 1. Let f e S*. Then for z e U and I z I = r we have
11 + z['(z)l f (z) S 2p (r).
(23)
More generally, iff e
s;
we have
lzf~;~) + 1- 2p I s
(24)
2p1 (r)
where
.!.::.1! (25)
rlog [
(1 + r)2 (1
r
-
P1 (r) = 1 + r +
Pl
I f (z) I
]
1+r r
(1 - r2) log 1 -
and iff e Spa we obtain
Ie•a. z['f
(z) . + e-
(26)
I
< p (r) -
2
•
where p 2 (r) is given by (21).
I
It may be noted that (23) leads to upper bound in (1 7) if we replace 11 + y~!~) by its real
part; similarly, (26) yields (20) and (24) corresponding results for f e
s;
Theorem 2. Let f e S*. Then for I z I = r
z/'(z)l2 ~ 2 (1 + r2) p2 (r) -1 l f (z)
(27)
where p
(r)
is given by (16). More generally, iff e
12 < ~ (r) + 2P)r2) lz['(z) J
(28)
and (29)
s;
z/'(z)
(1-
then 1 (1-2P>. 0
1
1>P~2•
1
where p1 (r) is given by (25); and iff e Spa then (30)
z/'(z)l2 2 s 4p2 (r)- 4p 2 (r) cos (y +a)+ 1, l f (z)
where p2 (r) is given by (21) and yby the equation (31)
2p 2 (r) cos a cosy = p~ (r) (1 - r2) + cos2 a.
128
[4].
The upper bound on I z/' (z) 1/f (z) I for f e S* as given by (27) is smaller than that given by (15) and (18). Further, the upper bound in (29) is sharp.
IzJ~;~) I
It is also possible to obtain upper bounds for
depending upon
IJ~~)I·
In fact
we have
Theorem 3. Let f e S*. Then (32)
lz['{z)l s 1+~ + rlog lf(z)/f(-z)l. f (z) 1- r (1 - r2) log 1 + r 1 -r
More generally, iff e Spa we get . z['(z)l 1+r2e-2ia l f(z) S 1-r2 +
(33)
J}
(1 -
. [ e-2ia f (z) r Re {e•a log -2ia -f(-ze ) [11+r2e-2ial +2rcosa] 2 r ) log 1_r
Inequalities of this form are of some interest in view of the following characterization of classes S; and Spa.
Lemma A (Ruscheweyh[3]). For arbitrary s, t satisfying lsi if t f (sz) sf (tz)
(34)
-<
:S 1, It I :S 1,/ e
s;
if and only
(1 - tz')l
and f e Spa if and only if t f (sz) sf (tz)
(35)
where F
-<
(1 - tz)2e-ia coo a 1 - sz '
-< f means that F is subordinate to f
in U.
The inequalities mentioned above, both obtained earlier and those obtained in this paper are essentially based on Herglotz formula (7) and the fact that for y ~ 1 y-1 ~ (y) =logy
(36)
is a monotone increasing function. We get stronger inequalities if we use the following
Lemma B [4].For 1 5y 5 A and - An+llogAn+l -An+l + 1 (37) lln - An+2logAn+2 -An+2 +1 ,n=0,1,2, ... the functions (38)
yn + 1 -1 - lln (yn logy
+ 2-
1)
, n = 0, 1, 2, 3, ...
are strictly increasing. The case n = 0 and A = 11 + r leads to the following. -r
129
Theorem 4. Let f e S*. Then for I z I = r < 1 we have
I
[
J[
1+r S(1+3r)log 1 _r-2r+ In (39) if we replace 11 + 1~;~> use (1 7) for upper bound on Re for f e
J
1 + ~~ f (z) (1 + r)2 log 11 +_ rr- 2r - (1 + r) log 11 +_ rr- 2r Re ~ f (z)
(39)
s;
2r2log[(1 + r)2
r
If (z) 1]
(1 -r)
I by its real part we obtain upper bound in (17) and if we
1~;~)
we obtain (23). One could similarly obtain results
and f e Spa but we obtain from giving these results here.
In the end we give an elementary proof of Theorem E depending upon the lower bound in (17).
2. Some Preliminary Results In this section we put together some results which we need.
Lemma 1. For lz I = r < 1 we have
(40)
1 ~1 1 -z
1+r 2r I o g ""i"i""-=zT 1+r < - - - < 1 + -----''-=--::-=--'----11-zl( )I 1+r' 1- r og 1 _ r
and
(41)
1 + r2
1 ~1 1 -z
2r log
I~ : ~ I
<--+------'-:-,;__
- 1 - r2
(1 - r)2 log 1 + r 1-r
The inequality (40) is better than (41) for 1 - 3r- r2- r3 S 0 and (41) is better than (40) for 1 - 3r - r2 - r3 ~ 0.
Proof: We use the fact that for y
~
1
L=_!
(42)
logy
. monotone mcreasmg. . . 1+r and for (41), y = 11+zl1+r 1s For (40) we take y = ""j"i""-=zT 1 _ z 1 _ r · In order to see which inequality is superior we use the following easily proven inequality:
1 (43) rlog--11 - z I
if 1 - 3r- r2 - r3 ~ 0, r log -1 1 + log -1 1 -r +r { +log---~ 2 11 + z I !:_ I (r + 1) I + og [ 2 (11 ++ rr ) 1'f 1 - 3r - r 2 - r a < _ o. 2 og r + 2 1
J
130
Lemma 2. For a real and fued and t real let 1 + zeit e-2ia (44) W = 1 -zeit and 1 - zi sin a eit e-ia 1 -zeit
(45)
Then (1 - r2) I WI (1 - r2) 2 1 s; 11 + r2 e-2ia I - 2r cos a s; [ 11 + r2 e-2ia I - 2r cos a]2
(46)
and (47)
(1 - r2) I WI (1 - r2) (1 - r 2 sin a) 1 s; 11 - ir2 sin ae ia I - r cos a s; [ 11 - ir2 sin ae-ia I r cos a]2 ·
Proof: We shall indicate the proof of (46) only because (47) follows on similar lines. If W is defined by (44) then elementary calculation shows that it satisfies
I
(48)
1 + r2 e-2ia
W-
1-r2
I
2r cos a S:~.
This gives rise to (46). We now proceed to collect some formulae for spiral-like functions. In view of (4) and the Herglotz formula (7) we have for f e Spa 2rt
. zj'(z) feia+zeite-ia e•a f (z) = 1 -zeit dJl (t),
(49)
0
(.
f
2rt
~) Re e•a f (z) = cos a
(50)
1-r2 11 -zeit 12 dJl (t)
0
and by integration of (49)
.
e•a log ~ = 2 cos a z
(51)
f
log -1 1 it dJL (t) . -ze
From (51) one obtains that (52)
. e-2ia f (z) ] f Re [ e•a log -f(-ze-2ia) = 2 cos a log
11 +zeit e-2ia I dJl, 1 _zeit
and (53)
. i sin a e-ia f (z)] f Re [ e•a log f (i sin a ze-ia) = 2 cos a log
11 -zeit1 _zeit sin ae-ia I dJl.
Further, from (49) and (50) we get 2rt
(54)
f
l
eia+zeite-ial2 {zf'(z) e-ia+r2eia} 1-zeit dJL(t)- 2Re f(z) 1-r2 -1,
0
131
and
J
1- iz sin a.eit e--<« 12 {z['(z) 1 + ir2 sin a.ei«} 1 1 -zeit $(t) = Re I (z) 1- r2 .
(55)
3. Proof of Theorem 1 From (7) and (40) we obtain for I e S*
(56)
I
(1 +r) z['(z)l - 2 - 1 +1() :s; z
J
1 +r d 1 11- ''I JLS + ze
2rfl~g (1 -
1+r d 11-ze''l Jl 1 . r) log .....:!:...!:. 1-r
This gives (23) in view of the fact that log {(1 + r 2 )
(57)
If 9 e
s;
llJ!)_I} z 2 Jlog 11 _ zei'l 1+r
=
dp. .
then from (3)for some I e S*,
_1_ [z9'(z) _
(58 )
1- p
9 (z)
!3] = zf'(z) I
(z)
·
Substituting for zf' (z)ll (z) interms of 9 from (58) into (56) and using (5) we get (24). Similarly, using (4), (16) and (51) we obtain (26).
4. Proof of Theorem 2 In view of(9) we notice that 1 1 1 + r S P (r) S 1 - r ·
(59) On account of(12) we have (60) Thus
l
z['(z)_l+r2 1 2r I (z) 1 - r2 :s; 1 - r2 ·
,e
z;~;~) I
S* lies in the circles given by (23) and (60) and hence in their
intersection (59) ensures that this intersection is not empty. The max
Iz;~;~) I
is the
distance from the origin of the two points of intersections of these circles which are symmetric with respect to the real axis. Let P be one of the points of intersection of (23) and (60) and let y be the angle which the line joining P to the centre of (23) makes with the real axis. Then (61) and
Op2
( 1 2r - r2
= 4 p2 (r) + 1 -
4p (r) cosy
)2 = 4 p2 {r) + (1 -2 r2) 'u - !.eJ1 1 - r2 cos y
or (62)
2 p (r) cosy = (1 - r2) p2 {r) + 1.
132
Substituting this value of cos yin (61) we get (63) OJII. = 2 (1 + ,.2) p2 (r) -1. This is (27). To see that (27) is superior to (15) we need to show that 2 (1 + ,.2) p2 (r) -1 s (1 + r'JA p2 (r) or equivalently (1 - r)2 p2 (r) S 1 which is true by (59). To see its superiority over (18) it is enough to show that 4(1+r2> 3+r2 2 (1 + r2) p2 (r) -1 S 1 _ r2 P (r)- 1 _ r2 · But this is equivalent to (19) 0::?: (p(r) (1-r)-1)(p(r) (1+r)-1), which is true in view of (59). To prove (28) and (29) we use (24) alongwith lzf'(z) 1 + r 2 (1-2(3)1 2r(1-P) (64) I (z) 1 - r2 S 1 - p:2 • Thus
z~~;~)
lies in the intersection of the disks (24) and (64). In view of the triangle
inequality it is clear that for 1 > p::?: 1/2 the This gives (29) which is sharp for Re
max lzf~~~) Iis attained along the real axis.
z~~)
as shown in [5], (28) follows on similar
lines as (27). To prove (30) we use (26) alongwith (65)
eia + r Ie•a. zl'(z) l(z)-
2 e-
I
S
2r cos a 1-r2 ,leSpa. 001 = sin a, cl = e-ia eia + r2 + e-
= cos a C1A = 2p2 oc1 = 1
AC~~ = y From ll AC1 C2 we obtain cos a"jA _ 4 2 ~r cos a"jA \1-r2)- P2+\1-r2) ~r
or (66)
8 p 2 (r) cos a cosy 1-r2
2p2 (r) cos a cosy = cos2 a+ p: (1 - r2) ;
and from the ll AC1 0 we get (67)
2 I (z) s lz/'(z)l
4p22 (r) + 1-4 p2 cos (y+ a).
133
5. Proof of Theorem 3 We shall prove (33) because (32) is the special case of (33) for a = 0. With W as in Lemma 2 and (1 -yr2) I WI Y = 11 + r2 e-2ia 1- 2r cos a we have 11 +r2e-2ial +2rcosa 1 ~ Y ~ 11 + r2 e-2ia I - 2r cos a · Monotonic increasing property of 1Y - 1 for y
<68>
1 +zeit e-2ia 1 1 - ze•t
I~
11 + r2 e-2ial - 2r cos a 1- r 2r cos a log {
11 +r2e-2ial +2rcosa 11 +zeite-2ial} 1 _ ze•t 1_r
+ (1 - r2) log {
11 + r2 e-2ial + 2r cos a} 1 - r2
Integrating (68) with respect to probability measure 11 (t) and using (49) and (52) we obtain (33). We mention that on using (53), (55) and (47) and adopting the above procedure we can also prove the following for f e Spa. (69) Re {( 1
(z)} S [1 - ir2 I sin1ae-ia I + r cos a] 2 + _ r2
. 2 . -ia) z[' f (z) + Jr sm a e
r +
11
.2 . -ial R [ ia 1 (1 - r2) i sin ae-ia f (z) - Jr sm ae e e og (1 - r2 sin2 a) f (iz sin ae-ia) (1 _ r 2) log 11 - ir2 sin ae-•a I + r cos a
...} (1 - r2) (1 - r2 sin2 a)
6. Proof of Theorem 4 We use Lemma B for n
1+r . =0, y = 111+r _ zeit 1 and A =1 _ r . In th1s case 1-r
llo =-2-
(1
1+r
+r)log~-
2r
+ rr - 2r (1 +r)2 1og 11-
and in view of (50)
134
J
Now
zf'(z)l I1 + f (z) But (38) for n
=0 andy = 111+r _ zeit 1
1+r 11 - ze" I
<70>
(1 +r) -2- ~
~
J
1 +r 11 -zeit I dJ.L (t).
gives
{( 1 + r )"2 1} 1 + Oo 11 - zei'l +
1+r 2 2r log 11 -zeit I [ 1+r (1 - r) (1 + r)2 log 1 _ r- 2r
J·
Integration of this with respect to the probability measure 1.1 (t) and use of above formulae gives (39).
7. Proof of Theorem E For 0 real we wish to find
I
'!'e i:. r
(z) (,
~z) ) 0
I·
In view of the left hand inequality
of(17)
Hence
Hence if o~ 2 we use and if o~ 2 we use
1~1
1~1
1 ~ (1 + r)2 ·
This gives (22). We might mention that the minimum of the same functional in the class Sa, 0 ~a< 1, does not seem to yield to calculus techniques.
References 1.
N.A. Lebedev and L. Gromova, The range of values of the system (If (z)l, lf'(z)l} in the class of starlike functions (Russian), Vestnik Leningrad Univ. Mat. Meckh, Astronom (1985), Vy p 4, 94-95.
2.
G. B. Leeman, Jr. Constrained extremal problems for families of Stieltjes integral, Archieve fiir Rational Mech. and Analysis, 54 (1973), 350- 357.
3.
St. Ruscheweyh, A subordination theorem for t/J-like functions, J. London Math. Soc. 13 (1976), 275-280.
4.
St. Ruscheweyh and V. Singh, Some inequalities for starlike functions, Manatshefte ftir Math. 97 (1984) 325-330.
V. Singh, Bounds on curvature of level lines under certain classes of univalent and locally univalent mappings, Indian J. Pure Appl. Math. 10 (2) (1979), 129 -144. 6. J.B. Twomey, On starlike functions, Proc. Amer. Math. Soc. 24 (1970), 95- 97.
5.
135
Extreme Points and Support Points of Certain Class of Analytic Functions
V. Srinivas, G.P. Kapoor and O.P. Juneja Department of Mathematics Indian Institute of Technology, Kanpur - 208 016, INDIA.
Introduction Let A (n) be the class of functions of the form f~) = z+
-
L,a.r.z 11
(1)
k =n +1
which are analytic in U = (z : lz I < 1) and (B,r,)n-+
1
be a sequence of nonzero complex
numbers, where n e N = (1, 2, 3, ... ). A function off (z) in A (n) is said to be in the class A [n, B,r,l if and only if it satisfies the conditions
L B,r, a,r,
~ 1,
and arg a,r, = -arg B,r, for ak
~ 0.
k=n+1
For B,r, real with B,r, < 0, the class A Ln, B,r,l coincides with the class A (n, -B,r,) which was introduced by Sekine [3]. The class A (n, -B,r,) gives almost all classes of analytic functions with negative coefficients for particualr values of n and B,r, (see [3]). We define A* (n, -B,r,) =A (n, -B,r,) with Bk ~ -k. For Bk, Ck inC\ (0) with arg Bk = arg C1o and IBk I 2: IC,r, I fork 2: n + 1, the class A In, B,r,l !;;; A [n, Ckl· The present paper mainly consists of two sections. In Section 1 we find the extreme points of the class A [n, Bkl and in Section 2, the support points of the subclass A* [n, Bkl = {f e A [n, Bkl : IBk I 2: kl are obtained. For a particular choice of Bk our results give those of Ow a et. al. [2].
L Extreme Points First we observe that the class A In, Bkl is convex in
Theorem 1. The class A In, Bkl is a convex subfamily of A (n). Proof: Let/;~)= z +
-
L, ak, i zk e A [n, B~ol fori= 1, 2. For 0 < t < 1, k =n +1
136
F (z) = t 11 (z) + (1 - t) f2 (z)
-
L
=z+
-
(tak,1+(1-t)a.~z,21zk =z+
k=n+1
L_A,~zzk.
k=n+1
Now, for A,~z * 0, arg A,~z =- arg B,~z. And
L B,~z A11 :S 1. =n +1 and proof of the theorem is complete. k
Hence F (z) e A [n, B,~zl
In the 'if' part of Theorem 2 of Owa et. al. [2], a condition for f (z) to be analytic has to be imposed. Our following theorem takes care of this. Theorem 2. Let 11 (z) = z and
!1z (z) =
zk z + B,~z ,
(2)
(k
Then a function f e A [n, B,~z] if and only if f (z) can be expressed as
L A.,~z fk (z)
/(z) = A.1 11 (z) +
(3)
k =n +1
i
where 0 s; A.1 , 0 :S A.,lz, A.1 + k
Alz = 1 and lim sup (Akl I B k I ) 1111 s; 1. k-+-
=n +1
Proof: Let a function f (z) be expressed as (3). Then we have, f(z)=A111(z)+
L_A.,~zfl, (z)=z+
-
L_(A.,~z/Bk)zk =z+
-
L_A,~zzk.
The limit superior condition gives that f (z) is analytic in U. Further, we have arg A,~z = - arg B,~z for A,~z # 0, and
L B,~z A,~z
L
= Ak = 1- A1 :S 1. k=n+1
k=n+1
Thus, we have f e A [n, B,~z]. Conversely, let a function f (z) ofform (1) be in A [n, B,~zl. PutA.,Iz =
B,~za,~z
/(z) = z+
L
fork 2: n + 1 and A1 = 1-
-
L_a,.z" k =n +1
k
i
A.lf1Czl+
Alz. Now, we have
=n +1
4(z+~:)=A.111(z)+ f.A.J,~z(z)
k=n+1
k=n+1
which completes the proof of the theorem.
Corollary 1. The extreme points of A [n, B,.J are the functions !I (z) and fk (z) defined in (2) fork~ n + 1.
Corollary 2. (Owa et. al [2]). The extreme points of A [n, B,.J =A (n, -BJJ where B,. is real and B,. < 0, are the functions !I (z) and !k (z) defined in (2) fork ~ n + 1. 137
2. Support Points Let A • [n, B 11l be the subclass of A [n, B 111 such that IB 11l ~ k. Then a function I (z) belonging to A• [n, B11l is univalent in the unit disk U, and A• [n, B11l is a convex subfamily of A (n). A function I (z) in the class A• [n, B11l is said to be a support point A• [n, B1.l if there exists a continuous linear functional J on A (n) such that Re (J (/)) ~ Re (J (g)) for all g e A• [n, B~ol, andRe (J) is nonconstat on A• [n, B~o]. We denote by Supp (A• [n, B,.]) the set of support points of A • [n, B,.], and also the set of extreme points of A • [n, Bk] is denoted by Ext (A• [n, B11ll. Finally, we find Supp {A* [n, Bk]l making use of the following lemma. Lemma (Brickman et. al. [1]). Let (bJJ be a sequence of complex numbers such that
lim suplb,.l 11"<1,
.. 2,
..
k-+oo
and set J (/) = akbk for I (z) e A= ff (z) = l,akzk :1 (z) is analytic in the unit disc U}. k=O k=O Then J is a continuous linear functional on A. Conversely, any continuous linear functional on A is given by such a sequence (bJJ.
In the converse part of Proof of Theorem 3 (Owa et. al. [2]), it was given that Ext (GJ)
t;;
ffk e Ext (A• [n, Bk]) : k ~ n + 1, k "'- Jl where GJ = ffk e Ext (A• [n,Bk]l :Re J (/) = Re J l for Bk < 0 and somej ~ n + 1. But there exists a continuous linear functional J such that 11 e Ext (GJ) and lj I! Ext (GJ) for somej ~ n + 1, Our following proof takes care of this
situation. Theorem 3. Supp {A* [n, B~o1l =
If E A•
[n, Bk]: I (z)
=Z
..
'f. o. k!Bk) zk' 0 s Ak,
+ 1r
=n +1
'f. Ak s 1 and Aj = 0 for some ;1. k=n+1
Proof: Let a functionl(z) = z +
..
'f. (Ak!Bk) zk e A* Ln, B~ol with 0 s AJ.,
2, AJ. S 1 and
k=n+1 k=n+1 Aj = 0 for somej. Define b1 = 1, bj = -BjiBjl and b~r = 0 fork~ n + 1 and k cF-j. Now by
..
setting J (g) = 2, akbk for g (z) = k=O
..
2, akzk e A, the above lemma gives that J is a k=O
continuous linear functional on A. Now J (/) = 1 and J (/j) = 1-
~~-I
< J (/) where
J
0
lj (z) = z + ;. e A• [n, Bkl· Thus Re (J) is nonconstant on A• [n,
..
B~ol.
For the function
J
g(z)=z+
l,a~ozk eA•[n,Bk],wehaveJ(g)=1-1aj1 SJ(/).Thusle Supp{A*[n,B~o]). k =n +1
Conversely, let I e Supp (A* [n, Bkl ). Then there exists a continuous linear functional Jon A (n) such that Re (J) is non constant on A • [n, B~o] and Re (J (/))
G =(he A• [n, B~ol: Re (J (h))=
138
max Re(J) = Re (J (/)). A*[n,B,l
The function f e G and so G is non-empty. In addition, G is closed, convex and locally uniformly bounded. Thus G is compact. Now by the Krein-Mil'man theorem Ext (G) is nonempty. When 0 < t < 1 and tg + (1 - t) h e G with functions g and h in A* [n, B,.J, we have the functions g and h are in G also. So Ext (G) = (l (z) : l (z) is an extreme point of GJ!:; Ext (A* [n, B,.]J. SinceRe (J} is nonconstant on A* [n, B,.J, we have Ext (G)~ Ext (A* [n, B,.]J. Thus there exists aj e N such that /j E Ext (G). Let m be the smallest of such/s.
-
When m > 1, we have Ext (G)!:; !fie Ext (A* [n, B,.]J : i 2: n + 1 and i
m} u
1!1
(z) and
I, (A.,.IB_.) zk, 0 :S Ak, I, ').._. :S 1 and Am = 0 k=n+l k=n+l (z))) < Re J (f). Thus f e R.H.S.
by Theorem 2, the function because Re [J
f (z) = z +
c#
When m = 1, we have Ext (G)!:; lfi e Ext (A* [n, Bkl ): i 2: n + 1). Since Bn's tend to oo as n goes to oo and G is closed, there are only a finite number of fk's in Ext (G). Now by applying Theorem 2 and that/ e G, we get that/ e R.H.S., which is the assertion of our theorem. Corollary (Owa et. al.[2]} For Bk real and Bk :S -k,
Supp {A* [n, Bkll
=Supp
[A* (n,- Bk)l
= If e A* (n,- Bk): f
(z)
=z
-
+ k
0 :S ~.
I, ('A.k/Bk) zk, =n +1
I, A.,. :S 1 and Aj = 0 for some 11. k=n+l
References 1. L. Brickman, T.H. MacGregor, and D.R. Wilken, Convex hulls of some classical families of univalent functions, Trans. Amer. Math. Soc. 156 (1971), 91 - 107. 2. S. Owa, T. Sekine, T. Yaguchi, M. Nunokawa and D. Pashkouleva, Res. notes on certain class of analytic functioins with negative coefficients, lnternat. J. Math & Math. Sci. 12 (1989),199- 204. 3. T. Sekine, On new generalized class of analytic functions with negative coefficients, Report of the Research Institute of Science and Technology, Nihon University, 32 (1987),1- 26.
139
Some Subclasses of k-Valent Meromorphic Functions
J. Stankiewicz and Z. Stankiewicz Department of Mathematics Technical University of Rzesow 35959, Rzeszow P.O. Box 85 POLAND
L Introduction Let U = (z : lz I < 1) and U' = {z : 0 < lz I < 1) = U\ {O)o By M (k) we denote a family of functions which are regular in U' and have at the point z = 0 a pole of order k of the following form F(z) = r«+b1-11z1 -k +ooo +bo+b1z+ooo' ze U' (1.1) and by M (k) we denote a subclass of M (k) of such functions that (1.2) F (z) = zoo« + bo + b1Z + z E u~ Letg, G be two regular or meromorphic functions in Uo We say thatg is subordinate to G and write g -< G if there exists a function w (z) regular in U, Iw (z) I ~ Iz I for z e U and such that g (z) = G(w (z))for z e Uo 0
0
0,
IfG is univalent then g -< G if and only if g (0) = G (0) andg (U) c G (U)o
Let k be a fixed natural number and let A, B be some complex numbers such that IAI ~1. IBI ~1,A+B,. OoByl?(k,A,B)wedenoteaclassoffunctions (1.3) p(z) = 1+GfkZJo+Oie+1z"+ 1 +ooo,ZE U such that (1.4) p (z) -< (1 + Az)/(1 - Bz)o The classes l? (k, A, B) were introduce and investigated in the papers [3, 4, 5] and [1 lo Using the classes l? (k, A, B) we can introduced the following subclasses of meromorphic functions : R (k, A, B) {Fe M (k) : -k-1 zll + 1 F' (z) -< (1 + Az)/(1 - Bz)) {Fe M (k): -k-1 zk + 1 F'(z) e P (1,A, B)); 1R (k, A, B)
{Fe M (k): -k-1 z 11 + 1 F'(z) -
z" + 1 F'(z) e
140
l? (k + 1, A, B));
M
M
•
•
{ zF'(z) (k, A, B) = F e M (k) : - k F (z)
-<
1 +Az} 1-Bz
zF'(z) = {Fe M (k) :- k F (z)
-<
P(l,A,B>};
-{ . zF'(z) (k, A, B) - Fe M (k) . - k F (z)
-<
1 +Az} 1-Bz
-<
P B)}.
z F'(z)
= {Fe M (k) : - k F (z)
(k,A,
For A = B = 1 the classes introduced above are the well known classes of k-valent functions with bounded rotation or starlike functions in the ring u~ The aim of this paper is to investigate some properties of the mentioned classes.
2. The classes R (k, A, B) and lE (k, A, B) Now we investigate some properties of k-valent functions with "bounded rotation". Theorem 2.1: IfF e R (k , A, B) then for euery z e (2.1)
1z
,.+ 1
u:
Iz I = r (r e (0,1)) we haue
, 1+ABr2 1 k IA+Bir F (z) + k 1 - IB 12 r2 < 1 - IB 12 r2
and in particular
(2.2)
k
11+ABr21-IA+Bir (1 - IB I 2 r2) fl + 1
,
s IF (z) I s k
11+ABr21 + IA+Bir (1 - IB I 2 r2) fl + 1
Proof: By the definition of the class R (k, A, B) the function -k-1 zk + 1. F'(z) belongs to the class lP (1, A, B). Thus the estimations (2.1) and (2.2) are the simple consequences of Theorem 2.2 and Theorem 2.3 of the paper [5] respectively. Remark 2.1: IfF e R (k, A, B) then by (2.2) we have IF'(z)l > 0 for z e U'. Thus E U' and therefore F is a locally univalent function in u~
F' (z) "' 0 for z
Remark 2.2: The estimations (2.1) and (2.2) are not sharp. IfF e M (k) then the function z" + 1 F' (z) has its kth coefficient equal to zero. Indeed z" + 1 F'(z) = -k + (1 -k) a1-1e z + ... + (k -1-k)a_1 z"- 1 + a1 z" + 1 + 2a2z" + 2 + ... · The extremal functions in Theroem 2.2 and 2.3 of the paper [5] have all coefficients different from zero and therefore these estimations are not sharp in the class R (k, A, B). Theroem 2.2: IfF e R (k, A, B) then for euery z e (2.3)
I F'(z) +
k
zr+T
u:
Iz I = r (r
E
(0,1)) we haue
"+
1+ABr2 2 1 k IA+BI 1- IBI21"2k+2 s 1- IBI2r2k+2
and in particular
(2.4) k
11 + AB r21r. + 21 - lA + B I r" + 1 11 + ABr21r. + 21 + IA + B I r" + 1 (1 - IB I 2 r2le + 2) fl + 1 s IF' (z) I s k (1 - IB I 2 r2A! + 2) fl + 1
141
The estimations are sharp. The extremal function is -k 1+Azk+ 1 (2.5) Fo (z) = z11 + 1 1 _ Bzk + 1 · And we have the equalities at the points z = rt!- 9, where 9 =- (arg B)l(k + 1) or 9 =- (arg B)l(k + 1)- rrl(k + 1) respectively. Proof: The proof is analogous to the proof of Theorem 2.1. It is enough to replace l? (1, A, B) by P (k + 1, A, B). Now the estimations are sharp because the extremal function (2.5) belongs toR (k, A, B) and gives equality in the estimations (2.3) and (2.4). Theorem 2.3: IfF (z) = z-k + b1- k z 1 - k + ... + b_1:r 1 + bo + b1z + ... belongs to the class R (k, A, B) then for every n = 1- k, 2- k, -1, 1, 2, ... we have lbnl
(2.6)
ltn~BI
IfF (z) = z-k + bo + b1z + ... belongs to the class lR (k, A, B) then
(2.7)
Ibn I
!>nk
IA+BI,
n=1,2, ....
For n = 1, 2, ... the estimations (2.6) and (2. 7) are sharp. The extremal functions are given by the equations 1+Azk+n+1
F'(z)=-kz-k-1 1-Bzk+n+1• n=1,2, ... ·
(2.8)
For n E (-1, -2, ... , - (k- 1)) the estimations (2.6) is sharp only if u (n + k + 1) every u = 1, 2, 3, .... Proof: By the definition of the class R (k, A, B) the function 1
-k-
_J,
:z--+
1
~
k for
1- k 2- k 2 F'(z)= 1+-k-b1-kz+-k-b2-kZ + ... + -2 _j, -1 _j, 2 _j, +kb-2z·-2 + kb1z·+ 1 +kb2z.+2+ ...
belongs to the class lP (1, A, B). Now using the estimation of Theorem 2.4 of the paper [5] we obtain
nbnl lk
!> IA+BI
forn=1-k,2-k, ...
,n~O.
This gives the estimation (2.6). If n ~ 1, then an extremal function (among others) is such that zk + 1 F'(z) -k = 1 +(A+ B)zn +k +1 +(A+ B)z2
which gives the condition (2.8). If n E (-1, -2, ... , - (k - 1 )} then an extremal function exists and has the form zk+ 1 F'(z) 1+Azk+n+ 1 -k = 1 - Bzk + n + 1 . only if u (n + k + 1) ~ k for u = 1, 2, 3, ... · Since lR (k, A, B) c R (k, A, B) and the extremal functions for n = 1, 2, ... belong to lR (k, A, B) then (2. 7) holds and give the sharp estimation. 142
3. The classes M* (k, A, B) and lMI* (k, A, B) On the beginning we give an integral representation for the functions of the classes M* (k, A, B) and M (k, A, B).
Theorem 3.1: A function F written in a form (3.1)
E
F (z) = z-k exp { -k
M (k) belongs to the class M* (k, A. B) if and only if it may be
1
exp
{-J P~
dt;
for some function p e l? (1, A, B). Proof: By the definition of the class M* (k, A, B) there exists a function p e l? (1, A, B) such that z F'(z) -k F (z) =P (z).
(3.2)
From (3.2) we have F' (z) F (z)
( 3 _3 )
+! _ -k
p (z) -1
z -
z
Both sides of (3.3) are regular functions in U and therefore integrating both sides from 0 to z we obtain log [zk F (z)] = -k
j
P (~) - 1 dt;
c
0
or equivalently
i
F(z) =z-k exp {-k
P
1 dt;}.
This gives the formula (3.1). Corollary 3.1. A function F belongs toM* (k, A, B) if and vnly if there exists a function H belonging to the class M* (1, A, B) such that (3.4) F (z) = [H (z)Jk .
Proof: Let Fe M* (k,A, B). Thus by Theorem 3.1 there exists a function p e l? (1, A, B) which satisfies the condition (3.1). If we put H (z) = z-1 exp { -k
(3.5)
Jz P
1
dt;}
then H belongs toM* (1, A, B) and H satisfies the condition (3.4). Reciprocally, if H belongs toM* (1, A, B) then by (3.1), fork = 1, we obtain (3.5) with some function p e l? (1, A, B). Thus a function F given by (3.4) satisfy (3.1) with given k and therefore F belongs toM* (k, A, B).
Theorem 3.2. A function F be written in a form (3.6)
F (z)
E
=z-k exp { -k
M (k) belongs to the class M* (k, A, B) if and only if it may
i
P
<~~- 1 dt;} =[z-1 exp {-
for some function p e l? (k, A, B). 143
i
P
r
Proof: In the same way as in the proof of Theorem 3.1 we obtain (3.2). Since Fe M (k) then z F' (z) ( b1) _J. p(z)= -k F (z) =1- bo + k z-+ ...
(3.7)
belongs to lP' (k, A, B). Thus by integrating (3. 7) we obtan (3.6).
Corollary 3.2: A function Fe M* (k, A, B) if and only if there exists an univalent meromorphic function H(z) =
z +Ck-111'-1 + ...
1
belonging to the class M* (1, A, B) such that F (z) = [H (z)]k. This means then F is a meromorphic, k-valent, starlike function. Theorem 3.3.lf F (z) belongs toM* (k, A, B) and has the form (1.2) then k Ibm I s m + k lA + B I,
(3.8)
m = 0,1, ... ·
The estimation is sharp. The extremal functions are the solutions of the equation zF'(z) 1+A7Jzm+k F (z) = -k 1 - B1)zm + k '
<3 -9 )
where ITJ I = 1. Proof: By the defintion of the class M* (k, A, B) we have zF'(z) -k F (z}
(3.10)
1+Aw(z) 1 - B w (z) '
where (3.11) and (3.12) I w (z) I S Iz Ik < 1 for z e U. From (3.10) we have (3.13) [Bz F' (z) - k A F (z)l w (z) = k F (z) + z F' (z). Now by (3.11) and (3.13) we obtain (3.14)
[-k (A + B)z-k +
I, (nA - kB) b,. z"] [ n=k I, a,.z" J= n=O I, (k + n) b,. z".
n=O
Comparing the coefficients ofboth sides of(3.14) we obtain the equalities: (3.15) -k(A+B)a,.+k=(k+n)b,., n=0,1,2, ... ,k-1. Since I a,. I S lfor all n e N then from (3.15) we obtain
k
lb,.l sk + n IA+BI, n=0,1, ... ,k-1.
(3.16)
This proves our theorem for n S k - 1. For every m C!: k the equality (3.14) may be written in the following form (3.17)
[-k(A+B)+
:~JnA-kB)b,.zn+"']w1(z)= ,.;Jk+n)b,.z" + n=~/t"z"
where
L.
c,. z" n=m+1
m
L. (k
n=m+1
m-k+1 + n) b,. z" - w (z)
L. (nA- kB) b,. z"
n=O
144
and WJ. (z) = w (z)lz!t, I w1 (z) I S 1 for z e U.
Now, we shall use an analogous reasoning as in Clunie's paper [2]. Integrating the squares of modulus the both sides of equality (3.17) along the circle lz I = r, 0 < r < 1, and using the inequality Iw1 (z) I S 1 we obtain (for r -+ 1) the inequality (3.18)
lk2(A+B)21 +
m -A
l:
lnA-kB121b,.l2
~
n=O
m
l:
-
l:
(k+n)21b,.l2 +
n=O
lc,.12.
n=m+l
Ignoring the last sum in (3.18) we obtain after some calculations m
(3.19)
(m+k)21bm12 s k21A+BI2-
-k
l:
[(k+n)2-lnA-kB12]1b,.l2
... o
m
-k
l:
(k
+ n)2 lb,.l2.
n=m-k+l
Inviewofinequalities IAI S1,1BI S1wehave lnA-kBI Sn IAI +k IBI Sn+k. Therefore both sums in (3.19) are non-negative and we can drop them. Then (3.20)
(m
+ k)2 Ibm 12 s k2 lA + B 12
holds form= k, k + 1, ... ·This implies the inequality (3.8) form= k, k + 1, ... · It is easy to see that the function f which satisfies (3.9) belongs to the class M* (k, A, B) and has the form Thus we have k Ibm I = m + k IA+BI and therefore the estimation (3.8) is sharp. Remark: For k = 1 the estimation (3.8) gives the estimation of coefficients for the class of merom orphic starlike univalent functions (see [5], p.90). Remark: The method of the proof of Theorem 3.3 cannot be used in the class M (k, A, B) fork ~ 2. We can only prove that: IfF (z) belongs to the class M (k, A, B) and has the form (1.1) then k lb1 _.1sk+ 1 IA+BI.
The estimation for Ibm I, m > -k + 1, are unknown.
References 1. Cz. Burniak, J. Stankiewicz and Z. Stankiewicz, The estimation of coefficients for
some subclasses of spirallike functions, Demons. Math. 15 (1982), 427 - 441. 2. J. Clunie, On Meromorphic Schlicht functions, J. London Math. Soc. 34 (1959), 215-216. 3. D.B. Shaffer, On Bounds for the Derivative of Analytic Functions, Proc. Amer. Math. Soc. 37 (1973), 517-520. 4. D.B. Shaffer, Distortion Theorems for a Special Class of Analytic Functions, Proc. Amer. Math. Soc. 39 (1973), 281 - 287. 5. J. Stankiewicz and J. Waniurski, Some classes of functions subordinate to linear transformation and their applications, Ann. Univ. Mariae Curie-Sklowdswaska 28 (1974), 85- 94,1976.
145
Bazilevic functions with
logarithmic growth
D.K. Thomas Department of Mathematics and Computer Science University College of Swansea, Swansea SA2, SPP, Wales, UNITED KINGDOM.
Introduction Let f be analytic in the unit disc D = (z : lz I < 1), and normalised such that f (0) =0 and /'(0) =1. If a > 0, then f e B (a) if, and only if, there exists g e S* such that for z=nf9e D (1)
where S* is the class of normalised starlike univalent functions. The class B (a) of Bazilevii: functions was introduced in Ill and was developed initially in [18]. Clearly B (a) contains S* and the class K of close-to-convex functions. In his original paper Bazilevii: [1] showed that B (a) contains only univalent functions. We shall be concerned with the subclass B1 (a) consisting of those functions in B (a) with g (z) "'z. This class, which we shall call the class of Bazilevii: functions with logarithmic growth, was first considered by Singh [16]. Thus we see from (1) that/ e B 1(a) if, and only if, for z e D
Re j'(z) (f (z)/z) ex -l > 0.
(2)
We note that B1(1) =R, where R is the class of normalised analytic functions whose derivative has positive real part. In this paper, we shall list some of the known published results for B1 (a). We then present some new theorems and finally suggest some open problems. Throughout the paper, K (a) will denote a constant depending only upon a, and will not necessarily be the same at each occurrence.
Known Results First we give some coefficient and distorition results.
Theorem A [16]. Let f e B J(oJ and be given by j(z)
=
z+
L.anzn, n=2
then I a2l :!> 2/(1 + a) for a > 0. Also
146
(3)
I
I
aa s
. 2 (3 +a) { (2 + a) (1 + a)2 , if 0 < a
s
1
2
2+"(1• 2
I a I S { (3 + a) + 4 2 3 +a'
if a ?. 1
4 (1 - a) (5 + 3a + a 2 ) . 3(2 + a) (1 + a)3 ' 'f 0 < a
s1
ifa?.l.
All these inequalities are sharp. Theorem B [20]. For f E B J(a,) and given by (3), then Ian I SAn for a= 1 IN, where N is a positive integer and where An is the coef(u:ient of z" in the function Fo E B J(aJ defined by z J'a. 1 0~~)dt. ( FoCz>=ajta.-
Theorem C [20]. Let f (i) Q2
(rFa.s
E
B 1( a,) for a> 0. Then with z
(4)
=rei 9 E D
1/Cz)l s Q1 (r)11a..
(ii)lfO
rm- 1 Q2(r)
1-a. 1-a. --1-r --1+r s lf'(z)l sra.- 1 Q1(r) 1 +r 1- r '
and ifa?.1, 1
rm- 1 Q1 (r)
a. --
a. --
-a. a. --
--1-r s lf'(z)l s 1+r
1
rm- 1 Q2(r)
-a. a. --
--1+r 1-r'
where r
Q1 (r) =a Jpa.- 1
0~ p)dp,
and r Q2 (r) =a Jpa.-1
0: P)dp.
Equality holds in all cases for F • e B1 (a), defined by
F,~,=(· }·-' o~~:)af
(5)
with cp = 0 or 1t. We see from Theorem C (i) that functions in B1(a) have rate of growth
0 (1) [log (11(1 - r))] 11a. as r ~ 1 and are thus of logarithmic growth. It also follows that if f (z) "- w for z e D, then Iw I ?. Q2 (1) lla. and so this disc is always covered in thew- plane. The question of obtaining bounds for the logarithmic derivative off e B1 (a) has recently been considered in [4], [13] and [21]. This problem is of considerable interest, since until recently, no estimates had been obtained even for the relatively simple case B1 (1) =R. The following was proved in [13].
147
Theorem D. For f
E
z
B 1 ( o), with = rei 9, then r
ra.a(1(1-r) /jta.-11-t dt < lzf'(z)l < + r) 1 +t - f (z) -
r
rm(1 +r) a(l - r)
/jta.-111 +t dt · - t
Equality is attained for F" and Fo defined in (5).
New Results Writing (2) as Re (/ (z)a.)'/(za.)' > 0, it follows at once from a Theorem of Libera [10] that Re (/ (z)/z)a. > 0 for z e D. Obradovil:[14] has shown that this result can be imporved to Re (/ (z)/z)a. > 1/(1 + 2a). We now give the sharp result. In fact we give a generalised version. For z e D and n =1, 2, ... , define 1 z In (z) = -fin -1 (t) dt, where
z
Io (z) =(/ (z)/z)a.. We prove
0
Theorem 1 [5]. Let f e B 1 ( o) and z = rei 9 e D. Then for n Yn (1), """'(-1)k+1rk-1
where 0 < Yn (r) = -1 + 2a .£... kn (k _ 1 + a) < 1. k =1
Equality occurs for the function Fo defined by (4). We note that when n =0, r
Re ( jz(z))a.
~
raa. fpa. -1 (1--=. . f!.)d "'"'(-1)k + 1 rk -1 \1 + p p = -1 + 2a .£... (k - 1 + a) k =1
which reduces to -1 + (2/r) log (1 + r) when a = 1.
Proof: From (2) write
f' (z) (f (z)/z)a. - 1 = h (z), where h e P, i.e., h (0) = 1 and Re h (z) > 0 for z = rei9 e D. Thus Re (' ;z)r = aRe
(:a. }a.-1 h(t) dt}
Write t = pei9, so that Re (' ;z>r
= ~a. jpa.-1 Re h
~ ~ Jpa.-1
0:p)dp,
0
since h e P. Hence
Re Io (z)
Re(/
;z>y ~ ~
r
Jpa.- 1 0
148
0: p)dp.
,
Next
r
1
J[
- 1 + 2a
~(-1)k+1pk-1) B.
kn (k _ 1 + a) dp = Yn + 1 (r),
0
where the inequality follows by induction. Now set " " ' (-1) k + 1
+n (r) =a .£..J kn (k -
P" -1
1 + a)
k=1
This series is absolutely convergent for n 2: 0 and 0 < r < 1. Suitably rearranging pairs of terms in fl>n (r) shows tha ~ < +n (r) < 1 and so 0 < Yn (r) < 1. Finally we note that since for n
r+r.(r) = Jflln-1(p)dp, 0
induction shows that fj>' n (r) < 0 and so Yn (r) decreases with r as --+ 1 for fixed n and increases to 1 as n --+ oo for fixed r. We now consider some growth problems. Suppose that f e S, the class of normalised univalent functions. Let M (r) = 1 ~t! r If (z) I and C (r) = f (D (r)), where D (r) = [j (z) : Iz I = r). Also denote by L (r), the length of C (r). A classical result of Littlewood and Paley [12] states that iff e S and is given by (3) if M (r) s (1 - r)-11 for some constant ~. then ~ > 1/2, n Ian I SK(~)nll for n
Theorem E. Let f e B ( o) and be given by (3), then for 0 < aS 1 ( i) n I an I s K (a) M (1 -1/n) for n
(ii) L (r) S K (a) M (r) (log 1/(1- r)) for 0 s r < 1.
An example of Hayman given in [18] shows that the logarithmic factor in (ii) is necessary. Clearly iff is bounded, then it follows that n I an I = 0 (1) as n --+ oo, and L (r) = 0 (1) log [1/(1 - r)] as r--+ 1. Both these estimates are best possible, [2], [8]. For any, f e S, if is obvious that A (r) s 7tM (r)2, where A (r) is the area off (D (r)). Thus replacing M (r) by .JA (r) in Theorem E would be an improvement. In the case of starlike functions this was done in [2] and [19]. However even when a= 1, i.e., when f is close-toconvex, it is still an open question as to whether one can replace M (r) by U(r) in Theorem E. In fact the general problem of relating functional& such as Ian I or L (r) with A (r) appears to be rather difficult. 149
We now present some best possible growth estimates for Ian I and L (r) in terms of A (r) when f e B 1 (a). We first give a relatively simple result in terms of M (r), which will put the subsequent area results into context. Theorem 2 [3). Let f e B1 ( aJ and be given by (3). Then ifO < aS 1, (i) n lanl SK(a)M (1-lln) 1 -a forn?.2, (ii) L (r) S raM (r)1-a [27t + 4log((1 + r)/(1- r))] Proof: (i) From (6) we have z (zf' (z))'
= z (za f
for 0 S r < 1.
(z) 1- a h (z))'
(7)
for f e P. Thus, since 1 2lt n 2an = 2 7trn Jz (zf'(z))'e-in9 d9,
we have from (7) that 2lt
2lt
n 2 1an I s 2
:rn J I za zf (z) 1 - a h (z) I d9
+ (1 -a) jlz1+af(z)-a f'(z)h(z)l d9 2~trn
0
1 2lt + - - J1z1+af(z)1-ah'(z)l d9 2~trn 0
=h
(r) + l2 (r) +Ia (r) say.
Since 0
a 1 h(r)s 27trn-aM(r)-
a Jlh(z)l d9s 27trn-a a ( 27t+4log 11 +_r) r, 0
using an elementary estimate. For I2 (r) we note that (6) gives
Jlz2a f (z)1 -2a h2 (z)
(1- a) 2lt
12 (r) = 27trn
Now If (z) I a
?.
I d9.
Q2 (1) from Theorem C and so 2lt
2 ~-Q~)(l) J I za f
(z) 1 - a h 2 (z) I d9,
S
(1 - a) M (r ) 1 - a 2 7tQ 2 (1)rn-a
Jlh(z)l
S
(1 -a) M (r)1- a 4 27tQ2(1)rn-a 1-r2 '
l2 (r) S
2lt
2
d9,
where we have again used a well-known elementary estimate for h e P. Finally, M (r)1- a I a (r) S 27trn -a- 1
2lt
JI h '(z) ld9, 0
and since, I h' (z) I S
2 Re h (z)
1 _ r2
for 0 s r < 1,
it follows that la(r) S rn
2M(r)1-a a 1 (1-r2)
Choosing r = 1 - lin for n?. 2, the estimates for h (r), !2 (r) and Ia (r) give (i). 150
2lt
(ii) Since L (r) =
f Izf' (z) Id8, (6) gives 0
r)
2lt
f1htz)ld8 SL(r)sraM(r)1-a ( 2lt + 4 log 11 +_ r ,
L (r) S ,.aM (r)1 -a
0
as before. Corollary :Let f e B1 (a) for 0
(i) forn~2, nla,.l SK(a)A (1-1/n)_2_ A()
(ii) for 0 s r < 1, L (r) s
,.a(+) -
1-a 2-
1-a
Oogn)-2- , 3-a
1
2(log 1 _ r ) -
Proof: The proof follows at once on noting that for f e S, !tM (r )2 SA ( ..Jr
) log (11(1 -
(8)
r )),
which is easily proved using Cauchy-Schwarz inequality and the fact that for f e S, A ( ..fr ) < KA (r), where K is a constant [9).
Using results of Littlewood [11), the growth estimates in Theorem 2 and Corollary are easily shown to be best possible for the function Fo defined in (4). When f is bounded, Theorem 2 shows that ifO
=
=
On the other hand, the Corollary shows that iff (D) has finite area, then logarithmic factors remain on the right hand sides of (i) and (ii). We shall show that these logarithmic terms can be removed to produce best possible estimates in the case of functions with finite area. We next prove Theorem 3 [3]. Let f e B 1 ( oJ for 0 < a< 1 and be given by (3). Then
(i) forn?:.2, nla,.l sK(a)A(1-lln)
1-a 2
--(log n)
3-a -2
n
1 -
+K(a)A (1-lln) 2
a 4
(ii) forOSr <1, 1-a
L (r) s K (a) A (r) - 2-
1
1
a
(log [1/(1 -r)]) 2 + K (a) A (r) 2- 4 (log [1/(1 -r)]).
We shall first prove the following lemma. Lemma. Let f e B1 (a) for 0 < a < 1 and g be defined for z e D by
g(z)=z(f(z)/z) 1 -a=z+
i, b,.z".
n=2
Then 1-a
(i) forn?:.2, nlb,.l sK(a)A (1-(lln))_2_ (ii) for 0 S r < 1, A (r,g) SK(a)A (r)1-a, where A (r, g) denotes the area of g (D (r)). 151
3-a 2 -, (logn)-
(9)
Proof. (i) For z = reiB, 1 2K nb,. = 27tl"" fzg'(z) e-inB dB 0
and so 1
nlb,.l s 27tl""
2K
J lzg'(z)l
dB
0
2K
S
(~~)
2K
:r" J (I
J
}~;;!I dB+ 2
lzl +«
lz
0
I
;z>) 1 -a dB,
0
on using the definition of g. Using If (z) I a~ Q2 (1) from Theorem C, we deduce that K(a.) a 1 n I b,.l s r" _a M (r)l- log 1 _ r K(a.) 1-a( 1 )3-a < - - A(r) 2 log-2 -r"-a 1-r ' on using (8). Choosing r = 1 -lin gives (i). (ii) For z
= pei8 , r2K
J J1g'(z)l2pd8dp.
A(r,g)=
0 0
Thus from (9), A
s
2£ Jl u -a.> (t;z)ra f'
= 2 (Jl (r) + J2 (r)) say.
From the Hardy-Stein-Spencer identity [7), we have 7t
d
J1 (r) =2r dr
d
i..
r d-H"A.(r) S A/2 A(r)
r
).12
7t
and so taking i.. = 2 (1 -a.), we obtain J 1 (r) s (1 - a.)a:a A (r)l- a. r
NowJ2 (r) = 2xa.2
J p2a -1 H2(1- a) (p) dp.
Again, using the result of Spencer Ooc. cit.),
0
we have r p
J2 (r) s 4 (1 -a.) lt2a.2 J
J p2a-l
0 0
A (t)l- a t
dtdp
and since A (r) increases with r, we have J2 (r) S K (a.) A(r) 1 - a and the lemma is complete. (The singularity at the origin is avoided by integrating away from zero). 152
Proof of Theorem 3: (i) It follows from (6) and (9) that for z e D, we can write zf' (z) = g (z) h (z),
(10)
where h e P. Seth (z) = (1 + Cll (z))/(1 - Cll (z)) , so that Cll is analytic D, Cll (0) = 0 and ICll (z) I < 1 for z e D. Now write Cll (z) =
:E
Cllnzn. Then from (10),
n=l {zf' (z)
+ g (z)) Cll (z) = zf' (z) - g (z)
and so (11) Equating coefficients of Zn in (11), we have for
n
:2: 2,
nan -bn = 2Ciln-l + (2a2 + b2) Clln-2 + ... + [(n -1)an -1 + bn-llCJlt. Thus the coefficient combination nan - bn on the right-hand side of (11) depends only upon the coefficient combination [2a2 + b2l, ... [(n -1) an -1 + bn _don the left-hand side.
Hence for n
~
2, we can write
i ~ka11 + b11)zll} Cll (z) = 11•2 i: (ka11 - b11>z11 + k=n+l }: c11z 11 ,
{2z + n
(12)
11=2
for some unknown coefficient c11. Squaring the moduli of both sides of (12) and integrating around Iz I = r, we obtain, using the fact that I Cll (z) I < 1, n
:E
I ka11 - b11l 2 r 2k +
11=2
-
:E
Ic11 12 r2ll < 4 +
ll=n +1
n
:E Ikak + b11l 2 .
11=2
Letting r-+ 1 and simplifying, we obtain (with Ia1l = Ib1l = 1) n-1 lnan-bn1 2 s 4 :E k la11b11l . II =1
The Cauchy-Schwarz inequality now gives n-1 1 n-1 1 1nan-bn12 l:,k la1112)2 ( l:,k lb1112)2 11·1 k·l 4 n-1 ! n-1 ! S 2n ( l:,k lalll2r2k) 2 ( l:,k lb_.l2r2k) 2
S4(
k=l
r
k=l
!.
4
!.
sA ( r ) 2 A(r g) 2 1tr2n •
for 0 s r < 1. Choosing r = 1 -lin for n :2: 2 and using the Lemma, we obtain (i). (ii) From (6) and (9) we have 2lt
L (r) =
2lt
f lzf'(z)ld8 = f 0 21tr
lg(z)h (z)l dB,
0 21tr
s f flg'(z)h(z)ldpdB+f flg(z)h'(z)ldpd8, 0 0
0 0
h (r) + I2 (r) say. 153
(z=pe'">
Now
where we have again used an elementary estimate for h e P and the definition of A (r, g). Next with z = pei 8• I2(r)
s
(JrJ 2>t
1(rJ j
2>t
lg(z) 2 h'(z)l d8dp) 2 1
J1
lh'(z)l 2 d8dr 2
(13)
1
= (J1 (r)) 2 (J2 (r)) 2 say. Since Re h (z) >for z e D, we may write
111 L2>t
h (z) = 1 +
hnzn = -2
n =1
1
lt
+ze-
dJJ (t),
(14)
2>t
where 1.1 (t) increases with t on [0, 2x] and
JdJJ (t) =2x. Hence 0
J 2>t
1
h, (z) = it
e-it
(1 - ze-
0
and so
f
2>t
1 I h, (z) I Sit
dJJ (t)
-:-11:-=_c:.z..::e-<-T.t""l.,..2 .
0
Thus from (13), (15)
Since g is normalised, we may write g(z) n Sn (t) Z 1 - ze -it = n L =1
(16)
,
and so Parseval's Theorem, 2>t
2>t
JJ
2>t
2 11 ~~:~t~ d6dJJ(t) = 2x n~f2n
j
1Sn(t)1 2 dJJ(t).
We now show that for n?: 1, n
2>t
J 1Sn
s 2x
0
154
l:i i =1
lbpJI.
(17)
From (16),
r Sn (t) zn
n=1
=
(z + :i: bn zR ) ( n=O :i: e-int zn) and so for n
C!: 1
n=2
00
r b,. e-i
Sn(t)=
(18)
n =1
Also writing a1 = 1 and ho = 1, it follows from (10)that nan= rb,.hn-lr
(19)
n =1
1 2lt
Fork C!: 1 we have from (14) h1r = - Ie-ilrt dJ,t (t) and thus from (19) Ito
1
r b,. It n =1 00
nan =-
2lt
Je-i (n - lr)t dJ,t (t)
(20)
0
Now I Sn (t) 12 = Sn (t) Sn (t) , and so (18) gives j
n
1Sn(t)12 = 2Re L
n
Lbi b;e-i
j= 1
/r = 1
lbvl2.
V:1
2lt
Thus integrating and using the fact that I dJ,t (t) = 2x, we obtain 0
n
2lt
I I Sn (t) I 2 dJ,t (t) =2Re L
Lbi b; Ie-i
j=1 lr=1
0
n
2lt
j
0
n
L lbv 12 \1:1
n
=27tRe L,ibj aj -2x j=1
L. lb,12.
vz1
where we have used (20). Thus (17) is proved. We now complete the estimation of J1 (r). From (15) and (17) we have r
f r
s
I
I
A ( .[j; )2 A ( .[j; 1
,g)2
-p
dp.
0
Thus since both A (r) and A (r, g) increase with r we obtain
J 1 (r) S K (A (r) A
!.
(r, g)) 2
1
log 1 _ r ,
where we have again used the fact that iff e S, then A ( ..,Jr ) < KA (r), for some constant K.
155
Finally we estimate J2 (r). As in the estimate of Ia (r) in Theorem 1, we have with z = pei9,
and so r 2rt
J2 (r) s
I I 0
0
2 ~~ ~ 2(z) dSdp,
r
=47t Combining these estimates for J1
J~ 1-p
=47tlog 1 + r · 1-r
0 (r) and J2 (r),
it follows form (13) that !. 1 I 2 (r) S K(A(r)A(r,g)) 4 log 1 _r ·
Applying the Lemma in the estimates for h (r) and I2 (r), we obtain (i). We are now able to state our finite area result, the proof is immediate. Corollary. Let f e B1 (a) for 0
(i) forn~2,nlan1 sK(a), (ii) for OS r < l,L (r)SK(a) log (1/(1-r)). These estimates are best possible in the following sense: (a) there exists a bounded function in B1 (a) such that if a= j;;, where N is a positive integer, then n I an I > 1 for infinitely many values of n. (b) there exists a bounded function in B1 (a) such that for 0 < a s 1, . l1
',!1}
L (r)
1up
log [1/(1 - r)] > 0·
COUNI'ER EXAMPLES We construct examples by modifying those given by Clunie and Keogh in [2] and by Hayman in [8].
L anzn and let (pnl be defined by
(a) Let f (z) = z +
n=2
1+
L,Pn cos nB =
IT<1 +cos 4n8).
n=O Then it was shown in [2] that 0 SPn S 1 andpn = 1 when n = 4m where m ~ 0. - Pn zj'(z) Further :r, - < oo. Now put, f ( )l _ex ex = 1 + ""'PnZn = h1 (z). Then n=2n z z £.J n=l
f
(z)
=z
[1
+
I, .J!!J_zn
(21)
n=f1+a
and Re h1 (z) > 0 for z e D.
156
Thus I e B1 (a) and is bounded since ....J!A_ S & and n+a
i
8!. <
oo.
Euqating coefficients
n ,.. 1 n in (21) and using the fact that a = 1/N , we obtain (n -1) a,. ~ p,. _ 1 and the reuslt follows. (b) Put
I ~{t' !_Z2 za ='I' (z) with 'I' (z) defined as in [8]. Then I e B1 (a). Also, since
r'a,
I
L (r) = rlzj'(z) I dB = 0
Jl
'I' (z)
(I ;z>y- a IdB
0 I -a
21t
~ Q2 (1) I I I I 'I' (z) I dB 0
using Theorem C (i). Hence the result of Hayman Ooc. cit.) shows that with this choice of '1', .
L (r)
h~_.s1up log [(1/1
- r)]
> 0.
SOME OPEN PROBLEMS
We shall not be concerned with generalisations or subclasses. In the main these problems are relatively simple and usually require methods which are slight modifications of those used in B1 (a). We shall therefore restrict ourselves to B1 (a). Problem 1. Find the set of extreme points of the closed convex hull of B 1 (a). In the case a = 1, i.e., for the class R, Hallenbeck [6] has shown that the extreme points are given by -z- 2% log (1-xz), where x E {z: lzl = 1). Problem 2. Show that ifI E B 1 (a) for a > 0 and n :!!: 2, then Ia,. I SA,., where A,. is defined as in (4). This would follow at once from Problem 1. Problem 3.Letl E B 1 (a) for 0
1-a
n Ia,. I SK (a) A (1- (1/n)) 2-a, and L (r) SK(a)A (r) 2 -a log [1/(1-r))] Note also that we have already established this conjecture in the case A. = J.L = (1 the Corollary to Theorem 2. Probelm 4. Let I
E
B 1 (a) and let
')h
be defined by
log~= 'f.r,.z". n=1 K(a) Show that for n ~ 1, ly,.l S n log (n + 1 ).
Note tha~ this is an open question even in the case a= 1.
157
-
a)/2 in
Problem 5. For f e B1 (a), what is the correct rate of growth of 2Jt
Jl ~ido f (z) ' 0
and 2Jt
Jlzf~;~T dB? 0
References 1. I.E. Bazilevil! , On a case of integrability in quadratures of the Loewner-Kufarev equation, Mat. Sb. 37 (1955), 471 -476. 2. J.G. Clunie and F.R. Keogh, On starlike and convex schlicht functions, J. London Math. Soc. 35 (1960), 229-223. 3. R.M. El-Ashwah and D.K Thomas, Some coefficient, length, area results for a subclass of Bazilevi't: functions, Bull. Math. de Ia Soc. Sci. Math. de Ia R.S. de Roumanie (to appear). 4. F. Gray and St. Ruscheweyh, Functions whose derivatives takes values in a halfplane, Proc. Amer. Math. Soc. 104 (1988), 215- 218. 5. S.A. Halim and D. K. Thomas, A note on Bazilevi't: functions, submitted for publication. 6. D.J. Hallenbeck, Convex hulls and extreme points of some families of univalent functions, Trans. Amer. Math Soc. 192 (1974), 285- 292. 7. W.K. Hayman, Multivalent functions, Cambridge University Press, 1958. 8. W.K Hayman, On functions with positive real part, J. London Math. Soc. 36 (1961), 35-48. 9. F. Holland and D.K. Thomas, On the order of a starlike functions, Trans. Amer. Math. Soc. 158 (1971), 189-201. 10. R.J. Libera, Some classes of regular univalent functions, Proc. Amer. Math. Soc. 16 (1965), 755- 758. 11. J. E. Littlewood, Lectures on the theory of function, Oxford University Press, 1944. 12. J.E. Littlewood and R.E.A.C. Paley, A proof that an odd schlicht function has bounded coefficients, J. London Math. Soc. 7 (1932), 167- 169. 13. R.R. London and D.K Thomas, The derivative of Bazilevi't: functions, Proc. Amer. Math. Soc. 104 (1988), 235 238. 14. M. Obradovi(; Some results on Bazilevi't: functions, Math. Sb.37 (1985), 92- 96. 15. Ch. Pommerenke, Univalent functions, Vandenhoeck and Ruprecht, 1975. 16. R. Singh, On Bazilevi't: functions, Proc. Amer. Math. Soc. 38 (1973), 261 - 217. 17. D.C. Spencer, A function-theoritic identity, Amer. J. Math. 65 (1943), 147- 160. 18. D.K Thomas, On Bazilevi't: functions, Trans. Amer. Math. Soc. 132 (1968), 353 361. 20. D. K. Thomas, On a starlike of Bazilevi"t: functions, lnternat. J. Math. and Math. Sci. 8 (1985), 779- 783. 21. D.K. Thomas, On functions whose derivative has positive real part, Proc. Amer. Math. Soc. 98 (1986), 68- 70. 158
Certain Subclasses of Analytic Functions
B.A. Uralegaddi Department of Mathematics Karnatak University Dharwad 580 003, INDIA.
Introduction Let An denote the class of functions of the form I (z) =z +an +1 zn + 1 +an+ 2 zn +2 + ... that are regular in the unit disk E = (z: lz I < 1). A function I (z) in An is said to be starlike if Re (z/' (z}/1 (z)) > 0 for z e E and is said to be convex if Re (1 + zl" (z}/f' (z)) > 0 for z e E. For any integer k, let
Ii' l(z) =
Z+
-
L
mk m =n+1
= z (Dk Observe that
no I
(z)
=I (z), D' I
(z)
-1
I
am zm
(z))'
=zf' (z) and D-1 I
(z)
The above
0
=1, 2, ... was introduced by Salagean [4).
differential operator Dk, fork
In this paper we shall show that if Re (D k I (z }/z) > 0 for z e E then Re (Dk + 11 (z)fDk I (z)) > 0 for I z I < ((n2 + 1)112- n]11n. Some of the earlier results in [2] and [3) are obtained as particular cases.
Main Results Theorem 1. Suppose that I
Dk + 1 I
(z) (z)
forze E,thenRe Dkl(z)
Proof: We have
(z)!z)
=z + an + 1 zn + 1 + ... is regular and satisfies
Re Dk I (z}/z > 0
> Ofor lzl <((n2+1)112_n]lln.
= p (z), where p (z) = 1 + Cn zn + Cn + 1 zn + 1 + ... · 159
Logarithmic differentiation of the above equation yields z (Dk /(z))'
=
Dli f (z)
1
~ +z P (z) ;
that is, Dk + 1 f (z) Dli f (z) = 1 and
p' (z) +Z p (z)
I
Dk+1f(z) p'(z)l 2n lzln Re Dkf(z) :<:1- zp(z) :<:1-1-lzl2n>0 if 1z 1 < ((n2 + 1)112 _ n)11n. Putting k = 0 in Theorem 1, we obtain.
=
Corollary 1. Suppose f (z) z +an+ 1 zn + 1 + ... is regular and satisfzes Re (f (z)/z} > 0 for z e E then Re (zf' (z)lf (z)) > 0 (f (z) is starlike and univalent) in I z I < ((n2 + 1)112- n)11n, {3].
Putting n = 1, in Corollary 1, the following result is obtained. Corollary 2. Suppose that f (z) =z + a2 z 2 + ... is regular in E. If Re (f (z}/z) > 0 in E, then Re zf' (z)lf (z) > 0 in lz I < -./2-1, [2].
Fork = 1, Theorem 1 yields the following result. Corollary 3. Iff (z) = z +an + 1 zn + 1 +an + 2 zn + 2 + ... is regular and satisfzes Ref' (z) > 0 in E then Re (1 + zf" (z}/f' (z)) > 0 in I z I < ((n2 + 1)112- n)lln, [3].
Putting n = 1, in the above Corollary, we obtain. Corollary 4. Suppose that f (z) =z + a2 z2 + ... is regular in E. If Ref' (z) > 0 in E, then Re (1 + zf" (z}/f' (z)) > 0 in I z I < -./2-1, [2]. z
Theorem 2.Letf(z) =z +an +1 zn+ 1 +an +2zn + 2 + ... be regular inE andF(z) = f f Then Re (D~ F (z}/z} > 0 for z e E if and only if Re (Dk- 1 f (z)lz) > 0 for z e E.
0
;t)
dt.
Proof: We have zF (z) = f (z) and Dk - 1 (zF' (z)) = Dk - 1 f (z). That is Dk F (z) = Dk - 1f (z) and the result follows from the hypothesis.
Theorem 3. Let f (z) = z + an + 1 zn + 1 + ... be regular in E and for k ~ 1 any integer, satisfy Die + 1 f (z) ~~. 11(/c + 1) 1 Re Dk f (z) > 0 for z e E. Then Re (Lr f (z}/z} > 2 for z e E. Proof: Define a regular function w (z) on E as follows: (
Dk f (z))11(k+1) =
1
.
1 + w (z)
z
Clearly w (0) = 0 and w (z) "- -1 in E. To prove the Theorem it suffices to show that I w (z) I < 1 for each z e E. Logarithmic differentiation of (•) yields Dk + 1 f (z) Dli f (z)
= 1- (k + 1) 160
zw' (z) . 1 + w (z)
Suppose that Iw (z) I t: 1 for z E E. Then by Jack's Lemma [1] there exists a point zoe E such that zo w' (zo) = m w (zo), with I w (zo) I = 1 and m 2: 1. It can be easily seen that
R Dk + 1 f (zo) e Dk f (z 0 )
2 - (k + l)m 2
~
0
which contradicts the hypothesis. Fork= 1 and n = 1, the above Theorem yields the known result [2]. Corollary 5. Iff (z) = z + a2 z2 + aa z 3 + ... is convex then Re ..J!' (z) >
~
for z e E.
References 1.
I.S. Jack, Functions starlike and convex of order a, J. London Math. Soc. 3 (1971), 469-474.
2. T.H. MacGregar, Functions whose derivative has a positive real part, Trans. Amer. Math. Soc.104 (1962), 532-537. 3.
, The radius of univalence of certain analytic functions, Amer. Math. Soc. 14 (1963), 514- 520.
4.
G.S. Salagean, Subclasses of univalent functions, Lecture notes in Math. 1013, Springer Verlag, Heidelberg, 1983, 292-310.
5.
E. Strohhiicker, Beitrage zur theorie der schlichten Funktionen, Math. Z. 37 (1933), 356-380.
161
Proc.
Open Problems J.M. ANDERSON, U.K. Let 0 < a < 1 and
I e S*
(a)
then
conformal extension there for k = sin
I is not only univalent in ll but has a k-quasi
It:
[Krzyz]. Krzyz explicitly exhibits an extension.
Can another such extension be given by the Ahlfor's method? In particular does there exist g (r), r ~ 1 with g (1) = 1 such that the function F ~) =I~). z E ll
=I
0)g (
Iz I ),
z E C\ll
is a k-quasi conformal extension for k = sin 1t2a ?
R.W. BARNARD, U.S.A. (1) Letl~)=z+ I.anzn e S*(l/2).TheniNCz)=z+ "L,akN+lzkN+l"'Oforany n=2
k=l
{z: 0 < lz I < 1) and for every N. Does there exists a starlike function of order less than 1/2 for which the above result hold? (2) Let I~) = z + . . . be a convex function in I z I < 1. 2rt
Show that
;It JII' I
(lei B) d8
0
:$;
-m...,i:-n--:-'1}==-=-ce"ill)..-:-1 9
T. BULBOACA, Romania. Let A = (f analytic in I z I < 1
:ICO) = 0, /'CO) = 1). Then I e
A, and
ReI' ~) > 0 implies
z
F
J ;o
~) = I
dt e S* [R. Singh and S. Singh].
0
It is true that if
zg'(z)
I
geAandRe g(z) ~ Im
z (zg'(z))'l g(z)
,lzl <1
zf'(z) then Re g (z) > 0 impliesF e S*. For g (z) = z we obtain the result of Singh and Singh. (i) If ReI' (z) > a find the best Q (I)) for which FE S* (Q (I))); or for a given a find the best 'V (a) for which Re /'~) > 'V (a) implies Fe S* (a) in lz I < 1.
(ii) Find a condition not so strong as(*) for which Re
z;~;~)
lz I < 1. Is condition(*) necessary for the above implication to be true?
162
> 0 implies Fe S*, in
G.P. KAPOOR, India. Letf (z) =
I. an zn
be analytic in lz I < R. Define g (z)
n=O
whereg1 (z) =
I. dn n=1
zn - 1 andg2 (z) =
-
I.
= I. dn an zn - 1 = (g1 * g 2 )(z) n=1
an zn - 1 , ldnl being any increasing sequence
n=1
of positive numbers. If dn = n we have for I z I < R, g (z) = I i m f (z + ~)- f (z) . h-.0
Analogously for the general sequence ldnl can we obtain g (z) by a limit process involving the function f (z) and the sequence ldnl?
V. KARUNAKARAN, India. We have a natural map from the class of analytic functions with slow growth (i.e., functions
I. an zn
with I an I
~
C In IP for some constants C and p) into the class of perio-
n =0
die distributions (namely the continuous linear functional on D smooth functions on T given by f ~ A.1 where A.1 (Q) = I i m
(T)
ff
the Frechet space of (rx) Q (x) du (x) for
r -.1- T
QeD (T)). Translate the geometric and analytic properties of the class of analytic functions into those of the corresponding periodic distributions. For instance what property of A.1 will be equivalent to the univalence off ?
S.S. MILLER, USA. Let p (z) = 1 + pz + P2 z2 + ... be analytic in the unit disk U. Let h (z) be convex in U with h (0) =0. If p satisfies the differential subordination,
-<
(zp' (z)) 2
h
(z)
( *)
-< q for all p
then find a dominant for p. i.e., find a function q such that p
satisfying (*).
S.OWA, Japan. Let Tn be the class of functions of the form
f
(z)
=z
-
I. ak zk,
-
(ak 2> 0), n e N
k =n +1
which are analytic in the open unit disc U. A function f (z) e Tn is said to be starlike of complex order b (b # 0; b is complex) iff (z)lz # 0 (z e U) and
Re {1
+
~ (zf~;~) -1 )} > 0,
(z
e
U).
We denote by T• (b) the subclass of Tn consisting of all starlike functions of complex n order b. A function f (z) e Tn is said to be convex of complex order b (b # 0, b is complex) if j'(z) # 0 (z e U) and 1 Re {1 + b
(zf" f' (z))lf > 0, (z)
163
(z e U).
Also we denote by Kn (b) the subclass of Tn consisting of functions which are convex of complex order b. Forb = 1 -a, (0 ~a < 1), r•n (1 -a) and Kn (1 -a) become the classes of starlike and convex functions of order a, respectively. For r•n (b) and Kn (b) can we say (i)
(ii)
(b)~
f
(z) e T 0
f
(z) e Kn (b) ~
n
-
I, (k + I b I -1)
ak :il I b I,
k=n+1
I, (k + I b I -1)
ak :il I b I
?
k =n +1
Forb= 1- a (0 :;a a< 1), we know that (i) and (ii) are true.
M. OBRADOVIC, Yugoslavia. Let f (z) = z + a2 z 2 + ... be an analytic function in the unit disc U = {z: Iz I < 1} and let vn + 1 f (z) 2n + 3 z Re Dn f (z) < 2n + 2 , Z E U, n = 1, 2, ... ,where vn f (z) = (1 _ z)n + 1 * f (z) where * stands
f
for Hadamard product of (1 _;)n + 1 with
(z). Then
f is starlike in U. What about
a 2n + 3 h h Re Dn + 1 f (z) a . I" . I . U? const ant s,.,.. > 2n + 2 sue t at vn f (z) < ,.,.., z e U 1mp 1es umva ence m .
R. PARVATHAM, India. (1) Let P,.(p) denote the class of all functions p p
J
Re
p 0 (z)-p
1_p
dB
s klt
0
for k ~ 2, 0 S p < 1 and z = rei9 e E. Let V k (p) be the class of all functions f P (z) that are holomorphic in E such that /p(O)=O=ff,(0)-1 and zfM (z) 1 e Pk (p),
+#J p
and u,. (p) be the class of all functions fp (z) that are holomorphic in E such thatfp (0) = 0, f;, (0) = 1 and zf' (z)
#J
eP,.(p).
Here V2 (p) is the well-known class of convex functions of order p and U2 (p) is the class of starlike functions of order p. Determine p so that f e v,. (0) implies f e U,.(p). Of course, when k = 2 the answer is known to be p = 1/2. (2) Let P (a) be the class of all functions f holomorphic in E with f (0) = 0 = f' (0) -1,
f (z)f'(z) ,.OinE and satisfying z
164
9:z
J {(
Re a 1 + z/"(z)) j'(z) + (1-a) ~} f (z) d8 >
-lt
e,_
where 0 :S 9]_ < 82 :S 9]_ + 2n, Z =rei9, 0 :S r < 1 and IX~ 0. For a given f e P (a), it is true that the function F (z) defined in E by
F~) = { c +z!/a
i
tc -1 jllot (t)
dt}
ex
is in P (a), for a
S. PONNUSAMY, India. (1)1f k(z)=(l ~z) 2 andpisanalyticin lzl<1,thenitistruethat
Re
{z: ~~~)-a
k (z) p (z)} > 0 implies
Rep~)> 0
in lzl< 1, (a> 0).
Is the above inequality true for any other analytic univalent function in the unit disc? (2) If a< -2 and f be analytic in I z I < 1, then Re [j'(z) + az j"(z)) < ~ implies Re j'(z) > 0
for~
n and a region for real a so that f' ~) + IXZ r ~> E n implies f is starlike in
:S -o/2 (pre print).
Determine conditions on
(3) (a) Iff~)= z +
-
2. a,. z"
I z I < 1.
is analytic in the unit disc ll = (z : I z I < 1) and satisfies
k=2
l
I
zf"(z) /'(z) <1,forze ll,
what can we say about its sections fn~)=z+
n
2.
a,.zk, zell?
k=2
(b) Verify whether the relation
Ref'~)>~
implies lfn'(z)l
~
is true or not? It is known that iff (z) is convex in ll, then f n (z) is convex in I z I < 114 and 114 is the largest radius with this property.
Prem SINGH, India. 1 N-1 Letj(z) = 2.anzn,a1=1, I,n lanl :S1,andjN(z)=N- 2. e-kj(ekz) where n=1 n=2 k=O
e = exp (2lti/N). It is known that Rei!'(~; > 0 for "'N, satisfies nY;_ 2n I an I :S 1.
165
If
I n
Ian I > 1, there does not exist a single function f (z) = z +
n=2
I
an zn for which
n=2
zf'(z) Re/N(z) > O,foranyN~l.
J. S'fANKIEWICZ, Poland. (1) LetP=fp(z)=1+~z+ ... , regularin lzl <1andRep(z)>Ofor lzl
z'lc:~~o)
where z 0 is fixed and f
ranges over this class. (2) Let D = (z : Re z > 0) and S* {D)= (f :f regular in D,
lim D
andRe
zJ~;~) > 0 for z e
)Z--+
f(z)- z =a < oo, 0
E
f(D)
00
D). It is known that f (D) is starlike with respect to the exterior
point w = 0. Find the sharp estimation of If (z) I for z e D.
D.K. THOMAS, U.K.
(1) Let Hq (n) =
be the Hankel determinant of coefficients off (z) = z +
I n
an zn .
=1
Show that IH1 (n) I = Ian an+ 2 - a~+ 1 I S 1 for f e S (or even for f e S*). It is not known whether this is true for n = 2; i.e., show
IH1 (2)1 = la 2 a 4 -a~ Is 1 for fe S (or fe S*). (2) Let :E denote the class of meromorphic functions in D = (z: lz I > 1). Iff, g e :E. define Fby F(z)
I
It is known that k
k IAk 12 < =1
= f(z)1 -ag(z)a
oo
(0SaS1)
(D.K. Thomas, Proc. Amer. Math. Soc.). Is it true that
-
I k
k IAk 12 < 1 ?
=1
166
(3) Denote by K the class of close-to-convex analytic functions defined in the unit disc. Let f e K and be given by
f
(z) = z +
:E an zn
.
n=2
Let z = reiB and c (r) the image of (z: I z I = r) by f. Let L (r) be the length of c (r). It is known that, n I an I S constant M (r) and L (r) S constant M (r) log (11(1-r)}. Show that (i) n I an I S constant .tA (r); (ii)
L (r) S constant .tA (r) log -1 -1 r .
(4) Why not define an Integral operator as follows? Let Io (z) =f'(z), Then
In (z) In+ 1 (z)
h
1 z
1 z
f Io (t) dt ... , In (z) =-z 0fin- I(t) dt. z0
(z) =-
is an integral operator anologous to the Ruscheweyh differential
Io (z) operator. Note that h (z)
zf' (z) (z) ·
=f
Does this operator have a natural place in univalent functions?
167
