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

Serap Bulut

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., [9–17]).

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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