Coefficient Estimates for New Subclasses of Analytic and Bi-Univalent Functions Defined by Al-Oboudi Differential Operator

Serap Bulut Civil Aviation College, Kocaeli University, Arslanbey Campus, İzmit 41285 Kocaeli, Turkey Correspondence should be addressed to Serap Bulut; bulutserap@yahoo.com Received 7 May 2013; Accepted 4 October 2013 Academic Editor: Stanislav Hencl 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.


Introduction
Let R = (−∞, ∞) be the set of real numbers, C the set of complex numbers, and the set of positive integers.Let A denote the class of all functions of the form which are analytic in the open unit disk We also denote by S the class of all functions in the normalized analytic function class A which are univalent in U.
Since univalent functions are one-to-one, they are invertible and the inverse functions need not be defined on the entire unit disk U.In fact, the Koebe one-quarter theorem [3] ensures that the image of U under every univalent function  ∈ S contains a disk of radius 1/4.Thus, every function  ∈ A has an inverse  −1 , which is defined by In fact, the inverse function  −1 is given by A function  ∈ A is said to be bi-univalent in U if both  and  −1 are univalent in U. Let Σ denote the class of biunivalent functions in U given by (2).For a brief history and interesting examples of functions in the class Σ, 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 Σ in recent years; it was followed by such works as those by Frasin and Aouf [7], C ¸aglar et al. [8], Porwal and Darus [9], and others (see, for example, [10][11][12][13]).
The object of the present paper is to introduce two new subclasses of the function class Σ and find estimates on the coefficients | 2 | and | 3 | for functions in these new subclasses of the function class Σ.
Firstly, in order to derive our main results, we need the following lemma.
Lemma 1 (see [14]).If  ∈ P, then |  | ≤ 2 for each , where P is the family of all functions  analytic in U for which for  ∈ U.

Coefficient
where the function  is given by and    is the Al-Oboudi differential operator.

Theorem 4. Let the function 𝑓(𝑧)
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 P. 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 | 2 | as asserted in (19).
Next, in order to find the bound on the coefficient | 3 |, 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 | 3 | as asserted in (20).
If we take  = 0 in Theorem 4, then we have the following corollary.
Corollary 5 (see [8]).Let the function () given by the Taylor-Maclaurin series expansion (2) be in the class If we take  = 0 and  = 1 in Theorem 4, then we have the following corollary.
Corollary 6 (see [7]).Let the function () given by the Taylor-Maclaurin series expansion (2) be in the class If we take  = 0,  = 1 and  = 1 in Theorem 4, then we have the following corollary.
Corollary 9 (see [9]).Let the function () given by the Taylor-Maclaurin series expansion (2) where the function  is defined by (12) and    is the Al-Oboudi differential operator.
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 P. Now, upon equating the coefficients in (47), we get (1 + 2)  (2 + )  3 From ( 49) and (51), we obtain Also, from (50) and (52), we have respectively, and applying Lemma 1, we get the desired estimate on the coefficient | 2 | as asserted in (45).Next, in order to find the bound on the coefficient | 3 |, we subtract (52) from (50).We thus get which, upon substitution of the value of  2 2 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 | 3 | as asserted in (46).
If we take  = 0 in Theorem 12, then we have the following corollary.
Corollary 13 (see [8]).Let the function () given by the Taylor-Maclaurin series expansion (2) be in the class N (61) Remark 15.Corollary 14 provides an improvement of the following estimates obtained by Frasin and Aouf [7].