Abstract

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

1. Introduction

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 .

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 [1] 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

Denote by the Hadamard product (or convolution) of the functions and ; that is, if is given by (1) and is given by then

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:

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 (1). For a brief history and interesting examples of functions in the class , see [2] (see also [3]). In fact, the aforecited work of Srivastava et al. [2] 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 Tang et al. [4], El-Ashwah [5], Frasin and Aouf [6], Aouf et al. [7], and others (see, e.g., [2, 8–15]).

Throughout this paper, we assume that is an analytic function with positive real part in the unit disk , satisfying , , and is symmetric with respect to the real axis. Such a function has a series expansion of the form

With this assumption on , we now introduce the following subclass of bi-univalent functions.

Definition 1. Let the function , defined by (1), be in the analytic function class and let . We say that if the following conditions are satisfied: where the function is given by

Remark 2. If we let then the class reduces to the class denoted by which is the subclass of the functions satisfying where the function is defined by which was introduced and studied recently by Tang et al. [4].

Remark 3. If we let then the class reduces to the new class denoted by which is the subclass of the functions satisfying where the function is defined by (15). Also we have the following classes:(i)introduced and studied recently by El-Ashwah [5];(ii)introduced and studied recently by Çağlar et al. [10];(iii)introduced and studied by Frasin and Aouf [6];(iv)of strongly bi-starlike functions of order introduced and studied by Brannan and Taha [3];(v)introduced and studied recently by Aouf et al. [7].

Remark 4. If we let then the class reduces to the new class denoted by which is the subclass of the functions satisfying where the function is defined by (15). Also we have the following classes:(i)introduced and studied recently by El-Ashwah [5];(ii)introduced and studied recently by Çağlar et al. [10];(iii)introduced and studied by Frasin and Aouf [6];(iv)of bi-starlike functions of order introduced and studied by Brannan and Taha [3];(v)introduced and studied recently by Aouf et al. [7].

For more results see also [2, 8, 13, 15–17].

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

Lemma 5 (see [18]). If , then for each , where is the family of all functions analytic in for which for .

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 6. Let the function given by the Taylor-Maclaurin series expansion (1) be in the function class with Then

Proof. Let . Then there are analytic functions , with , satisfying Define the functions and by or, equivalently, Then and are analytic in with . Since , the functions and have a positive real part in , and by Lemma 5   and for each . Also it is clear that Using (40) and (41) together with (12), we get Now, upon equating the coefficients in (42)-(43), we get From (44) and (46), it follows that Now (45), (47), and (48) yield Also from (45), (47), and (49), we get
Applying Lemma 5 for (44), (50), and (51), we get the desired estimate on the coefficient as asserted in (36).
Next, in order to find the bound on the coefficient , by subtracting (47) from (45), we obtain Upon substituting the value of from (44), (50), and (51) into (52), it follows that respectively. Applying Lemma 5 for (53), (54), and (55), we get the desired estimate on the coefficient as asserted in (37).

If we take in Theorem 6, then we have the following corollary.

Corollary 7. Let the function given by the Taylor-Maclaurin series expansion (1) be in the function class Then

Remark 8. Corollary 7 is an improvement of the estimates which were given by Tang et al. [4, Theorem 2.1].

Remark 9. If we set and in Corollary 7, then we have an improvement of the estimates which were given by Ali et al. [16, Theorem 2.1].

If we take in Theorem 6, then we have the following corollary.

Corollary 10. Let the function given by the Taylor-Maclaurin series expansion (1) be in the function class with Then