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The Scientific World Journal
Volume 2013 (2013), Article ID 171039, 6 pages
http://dx.doi.org/10.1155/2013/171039
Research Article

Coefficient Estimates for Initial Taylor-Maclaurin Coefficients for a Subclass of Analytic and Bi-Univalent Functions Defined by Al-Oboudi Differential Operator

Civil Aviation College, Kocaeli University, Arslanbey Campus, 41285 İzmit-Kocaeli, Turkey

Received 5 August 2013; Accepted 7 October 2013

Academic Editors: H. Bulut and J. Park

Copyright © 2013 Serap Bulut. 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

We introduce and investigate an interesting subclass of analytic and bi-univalent functions in the open unit disk . For functions belonging to the class , we obtain estimates on the first two Taylor-Maclaurin coefficients and .

1. Introduction

Let be the set of real numbers, the set of complex numbers, and the set of positive integers.

Let denote the class of all functions of the form which are analytic in the open unit disk We also denote by the class of all functions in the normalized analytic function class which are univalent in .

For two functions and , analytic in , we say that the function is subordinate to in and write if there exists a Schwarz function , which is analytic in with such that Indeed, it is known that Furthermore, if the function is univalent in , then we have the following equivalence:

For , Al-Oboudi [1] introduced the following operator: If is given by (2), then from (10) and (11) we see that with . When , we get Sǎlǎgean’s differential operator , [2].

Since univalent functions are one-to-one, they are invertible and the inverse functions need not be defined on the entire unit disk . In fact, the Koebe one-quarter theorem [3] ensures that the image of under every univalent function contains a disk of radius . Thus every function has an inverse , which is defined by In fact, the inverse function is given by

A function is said to be bi-univalent in if both and are univalent in . Let denote the class of bi-univalent functions in given by (2). For a brief history and interesting examples of functions in the class , see [4] (see also [5, 6]). In fact, the aforecited work of Srivastava et al. [4] essentially revived the investigation of various subclasses of the bi-univalent function class in recent years; it was followed by such works as those by Frasin and Aouf [7], Porwal and Darus [8], and others (see, e.g., [917]).

Motivated by the abovementioned works, we define the following subclass of function class .

Definition 1. Let be a convex univalent function such that A function , defined by (2), is said to be in the class if the following conditions are satisfied: where , the function is given by and is the Al-Oboudi differential operator.

Remark 2. If we set in Definition 1, then the class reduces to the class denoted by which is the subclass of the functions satisfying where , the function is defined by (17), and is the Al-Oboudi differential operator.

Remark 3. If we set in Definition 1, then the class reduces to the class denoted by which is the subclass of the functions satisfying where , the function is defined by (17), and is the Al-Oboudi differential operator.

Remark 4. If we set in Definition 1, then the class reduces to the class denoted by which is the subclass of the functions satisfying where , the function is defined by (17), and is the Sǎlǎgean differential operator.

Remark 5. If we set in Definition 1, then the class reduces to the class denoted by which is the subclass of the functions satisfying where , and the function is defined by (17).

Remark 6. If we set in Definition 1, then the class reduces to the class denoted by which is the subclass of the functions satisfying where and the function is defined by (17).

We note that

Firstly, in order to derive our main results, we need the following lemma.

Lemma 7 (see [18]). Let the function given by be convex in . Suppose also that the function given by is holomorphic in . If , then

The object of the present paper is to find estimates on the Taylor-Maclaurin coefficients and for functions in this new subclass of the function class .

2. A Set of General Coefficient Estimates

In this section, we state and prove our general results involving the bi-univalent function class given by Definition 1.

Theorem 8. Let the function given by the Taylor-Maclaurin series expansion (2) be in the function class with Then

Proof. It follows from (16) that where and have the following Taylor-Maclaurin series expansions: respectively. Now, upon equating the coefficients in (36) and (37), we get
From (40) and (42), we obtain Also, from (41) and (43), we find that Since , according to Lemma 7, we immediately have Applying (47) and Lemma 7 for the coefficients , , , and , from the equalities (45) and (46), we obtain respectively. So we get the desired estimate on the coefficient as asserted in (34).
Next, in order to find the bound on the coefficient , we subtract (43) from (41). We thus get Upon substituting the value of from (45) into (50), it follows that So we get On the other hand, upon substituting the value of from (46) into (50), it follows that And we get Comparing the inequalities in (52) and (54) completes the proof of Theorem 8.

3. Corollaries and Consequences

By setting in Theorem 8, we have the following corollary.

Corollary 9. Let the function given by the Taylor-Maclaurin series expansion (2) be in the function class Then

By setting in Theorem 8, we have the following corollary.

Corollary 10. Let the function given by the Taylor-Maclaurin series expansion (2) be in the function class Then

By setting in Theorem 8, we have the following corollary.

Corollary 11. Let the function given by the Taylor-Maclaurin series expansion (2) be in the function class Then

Remark 12. When , Corollary 11 is an improvement of the following estimates obtained by Porwal and Darus [8].

Corollary 13 (see [8]). Let the function given by the Taylor-Maclaurin series expansion (2) be in the function class Then

By setting in Theorem 8, we have the following corollary.

Corollary 14. Let the function given by the Taylor-Maclaurin series expansion (2) be in the function class Then

Remark 15. When , Corollary 14 is an improvement of the following estimates obtained by Frasin and Aouf [7].

Corollary 16 (see [7]). Let the function given by the Taylor-Maclaurin series expansion (2) be in the function class Then

By setting in Theorem 8, we have the following corollary.

Corollary 17. Let the function given by the Taylor-Maclaurin series expansion (2) be in the function class Then

Remark 18. When , Corollary 17 is an improvement of the following estimates obtained by Srivastava et al. [4].

Corollary 19 (see [4]). Let the function given by the Taylor-Maclaurin series expansion (2) be in the function class Then

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