#### Abstract

We introduce and investigate two new subclasses and of analytic and bi-univalent functions in the open unit disk For functions belonging to these classes, 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 , Al-Oboudi  introduced the following operator: If is given by (2), then from (5) and (6) we see that with . When , we get Sălăgean’s differential operator .

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  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  (see also [5, 6]). In fact, the aforecited work of Srivastava et al.  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 , Çağlar et al. , Porwal and Darus , and others (see, for example, ).

The object of the present paper is to introduce two new subclasses of the function class and find estimates on the coefficients and for functions in these new subclasses of the function class .

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

Lemma 1 (see ). If , then for each , where is the family of all functions analytic in for which for .

#### 2. Coefficient Bounds for the Function Class

Definition 2. A function given 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 3. In Definition 2, if we choose(i), then we have the class introduced by Çalar et al. ;(ii) and , then we have the class introduced by Frasin and Aouf ;(iii), and , then we have the class introduced by Srivastava et al. ;(iv), and , then we have the class of strongly bi-starlike functions of order , introduced by Brannan and Taha [5, 6];(v) and , then we have the class introduced by Porwal and Darus .

Theorem 4. Let the function given by the Taylor-Maclaurin series expansion (2) be in the function class Then,

Proof. First of all, it follows from conditions (11) that respectively, where in . Now, upon equating the coefficients in (21), we get From (23) and (25), we obtain Also, from (24), (26), and (28), we find that Therefore, we obtain Applying Lemma 1 for the aforementioned equality, we get the desired estimate on the coefficient as asserted in (19).
Next, in order to find the bound on the coefficient , we subtract (26) from (24). We thus get It follows from (27), (28), and (31) that Applying Lemma 1 for the previous equality, we get the desired estimate on the coefficient as asserted in (20).

If we take in Theorem 4, then we have the following corollary.

Corollary 5 (see ). Let the function given by the Taylor-Maclaurin series expansion (2) be in the class . Then,

If we take and in Theorem 4, then we have the following corollary.

Corollary 6 (see ). Let the function given by the Taylor-Maclaurin series expansion (2) be in the class . Then

If we take and in Theorem 4, then we have the following corollary.

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

If we take , and in Theorem 4, then we have the following corollary.

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

If we take and in Theorem 4, then we have the following corollary.

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

#### 3. Coefficient Bounds for the Function Class

Definition 10. A function given by (2) is said to be in the class if the following conditions are satisfied: where the function is defined by (12) and is the Al-Oboudi differential operator.

Remark 11. In Definition 10, if we choose(i), then we have the class introduced by Çalar et al. ;(ii) and , then we have the class introduced by Frasin and Aouf ;(iii), and , then we have the class introduced by Srivastava et al. ;(iv), and , then we have the class of bi-starlike functions of order , introduced by Brannan and Taha [5, 6];(v) and , then we have the class introduced by Porwal and Darus .

Theorem 12. Let the function given by the Taylor-Maclaurin series expansion (2) be in the function class Then,

Proof. First of all, it follows from conditions (38) that respectively, where in . Now, upon equating the coefficients in (47), we get From (49) and (51), we obtain Also, from (50) and (52), we have Therefore, from equalities (54) and (55) we find that respectively, and applying Lemma 1, we get the desired estimate on the coefficient as asserted in (45).
Next, in order to find the bound on the coefficient , we subtract (52) from (50). We thus get which, upon substitution of the value of from (54), yields On the other hand, by using the (55) into (57), it follows that Applying Lemma 1 for (58) and (59), we get the desired estimate on the coefficient as asserted in (46).

If we take in Theorem 12, then we have the following corollary.

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

If we take and in Theorem 12, then we have the following corollary.

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

Remark 15. Corollary 14 provides an improvement of the following estimates obtained by Frasin and Aouf .

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

If we take , and in Theorem 12, then we have the following corollary.

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

Remark 18. Corollary 17 provides an improvement of the following estimates obtained by Srivastava et al. .

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

If we take , and in Theorem 12, then we have the following corollary.

Corollary 20. Let the function given by the Taylor-Maclaurin series expansion (2) be in the class . Then,

Remark 21. Corollary 20 provides an improvement of the following estimates obtained by Brannan and Taha [5, 6] (see also [10, Corollary 3.2]).

Corollary 22 (see [5, 6]). Let the function given by the Taylor-Maclaurin series expansion (2) be in the class . Then,

If we take and in Theorem 12, then we have the following corollary.

Corollary 23. Let the function given by the Taylor-Maclaurin series expansion (2) be in the function class . Then,

Remark 24. Corollary 23 provides an improvement of the following estimates obtained by Porwal and Darus .

Corollary 25 (see ). Let the function given by the Taylor-Maclaurin series expansion (2) be in the function class . Then,