Coefficient Estimate Problem for a New Subclass of Biunivalent Functions

We introduce a unified subclass of the function class of biunivalent functions defined in the open unit disc. Furthermore, we find estimates on the coefficients and for functions in this subclass. In addition, many relevant connections with known or new results are pointed out. 1. Introduction Let denote the class of functions of the form which are analytic in the open unit disc . Further, by , we will denote the class of all functions in which are univalent in . Some of the important and well-investigated subclasses of the univalent function class include, for example, the class of starlike functions of order in and the class of convex functions of order in . It is well known that every function has an inverse , defined by where A function is said to be biunivalent in if both and are univalent in . Let denote the class of biunivalent functions in given by (1). In 1967, Lewin [1] investigated the biunivalent function class and showed that ; on the other hand Brannan and Clunie [2] (see also [3–5]) and Netanyahu [6] made an attempt to introduce various subclasses of biunivalent function class and obtained nonsharp coefficient estimates on the first two coefficients and of (1). But the coefficient problem for each of the following Taylor-Maclaurin coefficients for ; is still an open problem. In this line, following Brannan and Taha [4], recently, many researchers have introduced and investigated several interesting subclasses of biunivalent function class and they have found nonsharp estimates on the first two Taylor-Maclaurin coefficients and ; for details, one can refer to the works of [7–13]. Now, we define of function satisfying the following conditions: for some , where is the extension of to . Similarly, we say that a function belongs to the class if satisfies the following inequalities: for some , where is the extension of to . The classes and were introduced by Prema and Keerthi [14]; furthermore, for these classes, they have found the following estimates on the first two Taylor-Maclaurin coefficients in (1). Theorem 1. If , , and , then Theorem 2. If , , and , then Motivated by the works of Xu et al. [12, 13], we introduce the following generalized subclass of the analytic function class . Definition 3. Let , and let the functions be so constrained that We say that if the following conditions are satisfied: where and the function is the extension of to . We note that by specializing , , and , we get the following interesting subclasses: (1) ; see [12],(2) ,？？ ( ; ) and ,？？ ( ; ); see [14],(3) ( ) and ( ); see [11]. The objective of the present