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International Journal of Mathematics and Mathematical Sciences
Volume 2013, Article ID 190560, 4 pages
http://dx.doi.org/10.1155/2013/190560
Research Article

Coefficient Estimates for Certain Classes of Bi-Univalent Functions

1Department of Mathematical Sciences, Kent State University, Burton, OH 44021-9500, USA
2Institute of Mathematical Sciences, Faculty of Science, University of Malaya, 50603 Kuala Lumpur, Malaysia

Received 29 April 2013; Revised 27 July 2013; Accepted 31 July 2013

Academic Editor: Heinrich Begehr

Copyright © 2013 Jay M. Jahangiri and Samaneh G. Hamidi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A function analytic in the open unit disk is said to be bi-univalent in if both the function and its inverse map are univalent there. The bi-univalency condition imposed on the functions analytic in makes the behavior of their coefficients unpredictable. Not much is known about the behavior of the higher order coefficients of classes of bi-univalent functions. We use Faber polynomial expansions of bi-univalent functions to obtain estimates for their general coefficients subject to certain gap series as well as providing bounds for early coefficients of such functions.

1. Introduction

Let denote the class of functions which are analytic in the open unit disk and normalized by

Let denote the class of functions that are univalent in and let be the class of functions that are analytic in and satisfy the condition in . By the Caratheodory lemma (e.g., see [1]) we have .

For and we let denote the family of analytic functions so that

We note that is the class of bounded boundary turning functions and also that if . For , the class and was first defined and investigated by Ding et al. [2].

It is well known that every function has an inverse satisfying for and for , according to Kobe One Quarter Theorem (e.g., see [1]).

A function is said to be bi-univalent in if both and . Finding bounds for the coefficients of classes of bi-univalent functions dates back to 1967 (see Lewin [3]). But the interest on the bounds for the coefficients of classes of bi-univalent functions picked up by the publications of Brannan and Taha [4], Srivastava et al. [5], Frasin and Aouf [6], Ali et al. [7], and Hamidi et al. [8]. The bi-univalency condition imposed on the functions makes the behavior of their coefficients unpredictable. Not much is known about the behavior of the higher order coefficients of classes of bi-univalent functions, as Ali et al. [7] also remarked that finding the bounds for when is an open problem. Here in this paper we let and and use the Faber polynomial coefficient expansions to provide bounds for the general coefficients of such functions with a given gap series. We also obtain estimates for the first two coefficients and of these functions as well as providing an estimate for their coefficient body . The bounds provided in this paper prove to be better than those estimates provided by Srivastava et al. [5] and Frasin and Aouf [6].

2. Main Results

Using the Faber polynomial expansion of functions of the form (1), the coefficients of its inverse map may be expressed as, [9], where such that with is a homogeneous polynomial in the variables [10]. In particular, the first three terms of are

In general, for any , an expansion of is as, [9, page 183], where and by [11] or [12], while , and the sum is taken over all nonnegative integers satisfying

Evidently, , [13].

Theorem 1. For and let and . If ; , then

Proof. For analytic functions of the form (1) we have and for its inverse map, , we have
On the other hand, since and , by definition, there exist two positive real part functions and where and in so that
Comparing the corresponding coefficients of (10) and (12) yields and similarly, from (11) and (13) we obtain
Note that for ; we have and so
Now taking the absolute values of either of the above two equations and applying the Caratheodory lemma, we obtain

Theorem 2. For and let and . Then one has the following

Proof. Replacing by and in (14) and (15), respectively, we deduce
Dividing (19) or (21) by , taking their absolute values, and applying the Caratheodory lemma, we obtain
Adding (20) to (22) implies or
An application of Caratheodory lemma followed by taking the square roots yields
Now the bounds given in Theorem 2 () for follow upon noting that if , then
Dividing (20) by , taking the absolute values of both sides, and applying the Caratheodory lemma yield
Dividing (22) by , taking the absolute values of both sides, and applying the Caratheodory lemma, we obtain

Corollary 3. For let and . Then one has the following

Remark 4. The above two estimates for and show that the bounds given in Theorem 2 are better than those given by Srivastava et al. ([5, page 1191, Theorem 2] and Frasin and Aouf [6, page 1572, Theorem 3.2]).

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