Abstract

We obtain the Fekete-Szegö inequalities for the classes and of biunivalent functions denoted by subordination. The results presented in this paper improve the recent work of Crisan (2013).

1. Introduction and Definitions

Let denote the class of analytic functions in the unit disk that have the form Further, by we will denote the class of all functions in which are univalent in .

The Koebe one-quarter theorem [1] states that the image of under every function from contains a disk of radius . Thus every such univalent function has an inverse which satisfies where

A function is said to be biunivalent in if both and are univalent in . Let denote the class of biunivalent functions defined in the unit disk .

If the functions and are analytic in , then is said to be subordinate to , written as if there exists a Schwarz function , analytic in , with such that

Lewin [2] studied the class of biunivalent functions, obtaining the bound for modulus of the second coefficient . Subsequently, Brannan and Clunie [3] conjectured that for . Netanyahu [4] showed that if . Brannan and Taha [5] introduced certain subclasses of the biunivalent function class similar to the familiar subclasses of univalent functions consisting of strongly starlike, starlike, and convex functions. They introduced bistarlike functions and obtained estimates on the initial coefficients. Bounds for the initial coefficients of several classes of functions were also investigated in [6–12]. The coefficient estimate problem for each of the following Taylor-Maclaurin coefficients for is presumably still an open problem.

Let be an analytic and univalent function with positive real part in with , , and maps the unit disk onto a region starlike with respect to 1 and symmetric with respect to the real axis. Taylor’s series expansion of such function is of the form where all coefficients are real and .

By and we denote the following classes of functions:

The classes and are the extensions of classical sets of starlike and convex functions and in such a form were defined and studied by Ma and Minda [13]. They investigated growth and distortion properties of functions in and as well as Fekete-Szegö inequalities for and . Their proof of Fekete-Szegö inequalities requires the univalence of . Ali et al. [14] have investigated Fekete-Szegö problems for various other classes and their proof does not require the univalence or starlikeness of . So in this paper, we assume that has series expansion , are real, and . A function is bistarlike of Ma-Minda type or biconvex of Ma-Minda type if both and are, respectively, Ma-Minda starlike or convex. These classes are denoted, respectively, by and (see [15]).

In [16], Sakaguchi introduced the class of starlike functions with respect to symmetric points in , consisting of functions that satisfy the condition , . Similarly, in [17], Wang et al. introduced the class of convex functions with respect to symmetric points in , consisting of functions that satisfy the condition , . In the style of Ma and Minda, Ravichandran (see [18]) defined the classes and .

A function is in the class if and in the class if

In this paper, motivated by the earlier work of Zaprawa [19], we obtain the Fekete-Szegö inequalities for the classes and . These inequalities will result in bounds of the third coefficient which are, in some cases, better than these obtained in [7].

In order to derive our main results, we require the following lemma.

Lemma 1 (see [20]). If is an analytic function in with positive real part, then

2. Fekete-Szegö Inequalities for the Function Class

Definition 2 (see [7]). A function is said to be in the class if the following subordination holds: where .

We note that, for , the class reduces to the class introduced by Ravichandran [18].

Theorem 3. Let given by (2) be in the class and . Then

Let and be the analytic extension of to . Then there exist two functions and , analytic in with , , , , such that Next, define the functions and by Clearly, and . From (16) one can derive Combining (8), (15), and (17), From (18), we deduce and From (19) and (21) we obtain Subtracting (20) from (22) and applying (23) we have By adding (20) to (22), we get Combining this with (19) and (21) leads to From (24) and (26) it follows that where Then, in view of (8) and (12), we conclude that Taking or we get the following.

Corollary 4. If then

Corollary 5. If then

Corollary 6. If then inequalities (30) and (31) become

Corollary 7. If then inequalities (30) and (31) become

Remark 8. Corollaries 6 and 7 provide an improvement of the estimate obtained by Crisan [7].

3. Fekete-Szegö Inequalities for the Function Class

Definition 9 (see [7]). A function is said to be if the following subordination holds: where .

We note that, for , the class reduces to the class introduced by Ravichandran [18].

Theorem 10. Let given by (2) be in the class and . Then

Let and be the analytic extension of to . Then there exist two functions and , analytic in with , , , , such that From (38), we deduce and From (39) and (41) we obtain Subtracting (40) from (42) and applying (43) we have By adding (40) to (42), we get Combining this with (39) and (41) leads to From (44) and (46) it follows that where Then, in view of (8) and (12), we conclude that Taking or we get the following.