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Coefficient estimates for close-to-convex functions with argument [beta].

1 Introduction

Let A be the family of functions f analytic in the unit disc D = {z [member of] C : [absolute value of z] < 1}, and [A.sub.1] be the subset of A consisting of functions / which are normalized by f(0) = f'(0) - 1 = 0. A function f [member of] [A.sub.1] is said to be starlike (denoted by f [member of] [S.sup.*]) if f maps D univalently onto a domain starlike with respect to the origin.

Let

[P.sub.[beta]] = {p [member of] A: p(0) = 1, Re [e.sup.i[beta]] p > o} .

Here and hereafter we always suppose -[pi]/2 < [beta] < [pi]/2. It is easy to see that

p [member of] [P.sub.[beta]] [??] [e.sup.i[beta]] p - i sin [beta]/cos [beta] [member of][P.sub.0]. (1)

Herglotz representation formula (see [4]) together with (1) yield the following equivalence

p [member of] [P.sub.[beta]] [??] p(z) = [[integral].sub.[partial derivative]D] 1 + [e.sup.-2i[beta]]xz/1 - xz d[mu](x) (2)

for a Borel probability measure [mu] on the boundary [partial derivative]D of D. This correspondence is 1-1.

Since Vo is the well-known Caratheodory class, we call [P.sub.[beta]] the tilted Caratheodory class by angle [beta]. Some equivalent definitions and basic estimates are known (for a short survey, see [7]).

Definition 1. A function f [member of] [A.sub.1] is said to be close-to-convex (denoted by f [member of] CL) if there exist a starlike function g and a real number [beta] [member of](-[pi]/2, [pi]/2) such that

zf'/g [member of][P.sub.[beta]].

This definition involving a real number [beta] is slightly different from the original one due to Kaplan [5]. An equivalent definition of CL by using Kaplan class and some related sets of univalent functions can be found in [6]. If we specify the real number [beta] in the above definition, the corresponding function is called a close-to-convex function with argument [beta] and we denote the class of all such functions by CL([beta]) (see [1, II, Definition 11.4]). Note that the union of class CL([beta]) over [beta] [member of] (-[pi]/2, [pi]/2) is precisely CL while the intersection is the class of convex functions. These results were given in [2] without proof. Since the former one is obvious, we will only give an outline of the proof of the latter one. Choose a sequence {[[beta].sub.n]} [subset] (-[pi]/2, [pi]/2) such that [[beta].sub.n] [right arrow] [pi]/2 as n [right arrow] [infinity]. The assertion follows from the facts that the class of starlike functions is compact in the sense of locally uniform convergence and any function sequence {[p.sub.n]} where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] converges to the constant function 1 locally uniform as [[beta].sub.n] [right arrow]/2.

In the literature, when studying the close-to-convex functions, some authors focus only on the case [beta] = 0. A. W. Goodman and E. B. Saff [2] were the first to point out explicitly that CL([beta]) and CL are different when [beta] [not equal to] 0 and more deeply the class CL([beta]) has no inclusion relation with respect to [beta]. Therefore it is useful to consider the individual class CL([beta]). The present paper follows their way in this direction and improves their result concerning the class CL([beta]);

Theorem A (Goodman-Saff [2]) Suppose f(z) = z + [[infinity].summation over (n=2)] [a.sub.n][z.sup.n] [member of] CL([beta]) for a

[beta] [member of] (-[pi]/2, [pi]/2). Then

[absolute value of [a.sub.n]] [less than or equal to] 1 + (n - 1) cos [beta].

for n = 2,3, ... . If either n = 2 or [beta] = 0, the inequality is sharp.

In the above mentioned paper, they also stated that the problem of finding the maximum for [absolute value of [a.sub.n]] in the class CL([beta]) was difficult for n [greater than or equal to] 3. With regard to their problem, in the present paper we shall establish the following theorems:

Theorem 1. Suppose f(z) = z + [[infinity].summation over ([a.sub.n][z.sup.n])][member of] CL([beta]) for a [beta] [member of] (-[pi]/2, [pi]/2), then the sharp inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)

holds for n = 2,3, ... . Extremal functions are given by

f'(z) = 1/[(1 - yz).sup.2] 1 + [e.sup.2i[beta]][yu.sub.n]z/1 - [yu.sub.n]z

for y [member of] [partial derivative]D, where [u.sub.n] [member of] [partial derivative]D is a point at which the above maximum is attained.

We mention here that it seems that there are no extremal functions other than the form given above in Theorem 1. Theorem A follows from Theorem 1 immediately by the elementary inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for any u [member of] [partial derivative]D.

The expression in (3) is implicit. When n = 3, we can give a more concrete estimate and also show the extremal functions are unique;

Theorem 2. Suppose f(z) = z + [[infinity].summation over (n=2)][a.sub.n][z.sup.n] [member of] CL([beta]), then the sharp inequality

[absolute value of [a.sub.3]] [less than or equal to] 2 cos [beta]/3 [square root of 5 + 9/4 [cos.sup.2] [beta] + 13/1 - [t.sub.0]] (4)

holds, where [t.sub.0] is the unique root of the equation

[t.sup.3] - (4/3 [cos.sup.2] [beta] + 6) [t.sup.2] + (40/9 [cos.sup.2] [beta] + 9) t + 4 [cos.sup.2] [beta] - 4 = 0 (5)

in 0 [less than or equal to] t < 1. Equality holds in (4) if and only if

f'(z) = 1/[(1 - yz).sup.2] 1 + [e.sup.-2i[beta]][yu.sub.3]z/1 - [yu.sub.3]z

for some y [member of] [partial derivative]D, where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Remark 1. Comparing Theorem A and Theorem 2, it is not difficult to see that

1 + 2 cos [beta] = 2 cos [beta]/3 [square root of 5 + 9/4 [cos.sup.2] [beta] + 13/1 - [t.sub.0]]

if and only if

[t.sub.0] = 9-9 cos [beta]/9 + 4 cos [beta].

Since this [t.sub.0] is a root of (5) in [0,1) only when [beta] = 0, Theorem A is sharp only when [beta] = 0 for n = 3.

Finally we give an example to show how Theorem 2 works.

Example. Let [beta] = [pi]/4. Applying Mathematica, we may get the root of equation (5) which belongs to [0,1) is 0.201 ..., therefore in this case

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is less than 1 + [square root of 2][approximately equal to] 2.414 by Theorem A.

2 Proof of Theorems

In order to prove our theorems, we shall need the following lemma

Lemma 1. (see [3] p. 52) If f [member of] [S.sup.*], then there exists a Borel probability measure v on [partial derivative]D such that

f(z) = [[integral].sub.[partial derivative]D z/[(1 - yz).sup.2] dv (y).

Proof of Theorem 1:

Equivalence (2) and Lemma 1 imply that if f [member of] CL([beta]), then there exist two Borel probability measures [mu] and v on [partial derivative]D such that f' can be represented as

f'(z) = [[integral].sub.[partial derivative]D][[integral].sub.[partial derivative]D] 1/[(1 - yz).sup.2] 1 + [e.sup.- 2i[beta]] xz/1 - xz d[mu](x)dv(y).

Thus in order to estimate the coefficients of f, it is sufficient to estimate those of functions

1/[(1 - yz).sup.2] 1 + [e.sup.-2i[beta]] xz/1 - xz

when [absolute value of] = [absolute value of] y] = 1.

Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

after letting u = x/y, we can easily obtain (3). The extremal functions can be obtained easily by the proof of this theorem.

Proof of Theorem 2: By Theorem 1, we have the sharp inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

h([alpha]) = [[absolute value of 1 + [2e.sup.i[alpha]] + 3/1 + [e.sup.-2i[beta]][e.sup.2i[alpha]]].sup.2]. (6)

Straightforward calculations give

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

and

h'([alpha]) = -4 sin [alpha] - 12 sin 2[beta] + 3[alpha]/2 cos [alpha]/2/cos [beta]

= -(10 sin [alpha] + 6 sin 2[alpha]) - 6 tan [beta] (cos 2[alpha] + cos [alpha]), (8)

h"([alpha]) = -(10 cos [alpha] + 12 cos 2[alpha] + 6 tan [beta] (2 sin 2[alpha] + sin [alpha]). (9)

Since h'([pi]) = 0 and h"([pi]) < 0, h([alpha]) attains a local maximum h([pi]) = (9 - 8 [cos.sup.2] [beta]) I(4 cos2 ft) at n. It follows from h(n) < h(0) that n is not a global maximum point of h(tx). Since h([alpha]) is periodic and continuous, its maximum point exists over (-[pi], [pi]), thus we may suppose that h(a) attains its maximum at some point [[alpha].sub.0] in (-[pi], [pi]), then

h'([[alpha].sub.0]) = 0 (10)

and

h"([[alpha].sub.0]) [less than or equal to] 0. (11)

Combining (8) and (10), we may represent tan [beta] in term of [[alpha].sub.0];

tan [beta] = - 5 sin [[alpha].sub.0] + 3 sin 2[[alpha].sub.0]/3(cos [[alpha].sub.0] + cos 2[[alpha].sub.0]). (12)

Substituting it into (9) shows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)

Since

11 + 11 cos [alpha] + 4 [sin.sup.2] [alpha] cos [alpha] > 0

whenever - [pi] < [alpha] < [pi], hence from (11) and (13), we deduce that

cos [[alpha].sub.0] + cos 2[[alpha].sub.0] > 0

which is fulfilled only when cos [[alpha].sub.0] > 1/2 i.e. [[alpha].sub.0] [member of] (-[pi]/3, [pi]/3).

Let g([[alpha].sub.0]) denote the quantity given in the right hand side of (12). Since g'([alpha]) < 0 over (-[pi]/3, [pi]/3), there exists one and only one [[alpha].sub.0] which satisfies (10) and (11) and h([alpha]) assumes its maximum

5 + 9/4 [cos.sup.2] [beta] + 13/1 - 4 [sin.sup.2] [[alpha].sub.0]/2

at [[alpha].sub.0]

(8) and (10) also imply

cos [[alpha].sub.0]/2 (2 sin [[alpha].sub.0]/2 + 3 sin 3[[alpha].sub.0] + 2[beta]/2/cos [beta]) = 0 (14)

Since [[alpha].sub.0] [not equal to] [pi], after letting [x.sub.0] = sin([[alpha].sub.0]/2), (14) implies that xq is the unique root of the following equation

11x - [12x.sup.3] + 3 tan [beta] [square root of 1 - [x.sup.2]](1 - [4x.sup.2]) = 0.

in (-1/2,1/2). Writing [t.sub.0] = [4x.sup.2.sub.0] and t = [4x.sup.2], we get [t.sub.0] is a root of equation (5) in [0,1).

Let v(t) be the polynomial in the left hand of (5), it is easy to verify that v(0) [less than or equal to] 0, v(1) > 0 and v'(t) > 0 in 0 [less than or equal to] t < 1 which together assure the uniqueness of root [t.sub.0] [member of] [0,1) of equation (5).

Therefore Theorem 2 is complete.

Acknowledgements: The author is grateful to Professor Toshiyuki Sugawa for his constant encouragement and useful discussions during the preparation of this paper. I also would like to acknowledge the referee for corrections.

References

[1] A. W. Goodman, Univalent Functions, 2 vols., Mariner Publishing Co. Inc., 1983.

[2] A. W. Goodman and E. B. Saff, On the definition of a close-to-convex function, Internat. J. Math. Math. Sci. Vol. 1(1978) 125-132.

[3] D. J. Hallenbeck and T. H. MacGregor, Linear Problems and Convexity Techniques in Geometric Function theory, Pitman(1984).

[4] G. Herglotz, Uber Potenzreihen mit positivem, reellen Teil in Einheitskreis, Ber. Verh. Sachs. Akad. Wiss. Leipzig(1911) 501-511.

[5] W. Kaplan, Close-to-convex schlicht functions, Michigan Math. J., 1(1952) 169185.

[6] S. Ruscheweyh, Convolution in Geometric Function Theory, Sem. Math. Sup. 83, University of Montreal, Montreal, Quebec, Canada 1982.

[7] Limei Wang, The tilted Caratheodory class and its applications, In preparation.

Limei Wang *

* This work is a part of the author's Ph.D. thesis, under the supervision of Professor Toshiyuki Sugawa.

Received by the editors January 2010.

Communicated by F. Brackx.

2000 Mathematics Subject Classification : Primary 30C45.

Division of Mathematics

Graduate School of Information Sciences

Tohoku University, Sendai

980-8579 JAPAN

e-mail: rime@ims.is.tohoku.ac.jp
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Author:Wang, Limei
Publication:Bulletin of the Belgian Mathematical Society - Simon Stevin
Article Type:Report
Date:May 1, 2011
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