Research Article | Open Access
Initial Coefficient Bounds for a General Class of Biunivalent Functions
We find estimates on the coefficients and for functions in the function class . The results presented in this paper improve or generalize the recent work of Srutha Keerthi and Raja (2013).
1. Introduction and Definitions
Let denote the class of analytic functions in the unit disk that have the form and let be the class of all functions from which are univalent in .
The Koebe one-quarter theorem  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 .
If the functions and are analytic in , then is said to be subordinate to , written as , if there exists a Schwarz function such that .
Let denote the class of biunivalent functions defined in the unit disk . For a brief history and interesting examples in the class , (see ).
Lewin  studied the class of biunivalent functions, obtaining the bound 1.51 for modulus of the second coefficient . Subsequently, Brannan and Clunie  conjectured that for . Netanyahu  showed that if .
Brannan and Taha  introduced certain subclasses of the biunivalent function class similar to the familiar subclasses, and of starlike and convex function of order , respectively (see ). Thus, following Brannan and Taha , a function is the class of strongly biconvex functions of order if each of the following conditions is satisfied: where is the extension of to . The classes and of bistarlike functions of order and biconvex functions of order , corresponding to the function classes and , were also introduced analogously. For each of the function classes and , they found nonsharp estimates on the initial coefficients. Recently, many authors investigated bounds for various subclasses of biunivalent functions ([2, 7, 8]). The coefficient estimate problem for each of the following Taylor-Maclaurin coefficients for ; is presumably still an open problem.
In this paper, by using the method  different from that used by other authors, we obtain bounds for the coefficients and for the subclasses of biunivalent functions considered by Srutha Keerthi and Raja and get more accurate estimates than that given in .
2. Coefficient Estimates
In the following, let be an analytic function with positive real part in , with and . Also, let be starlike with respect to 1 and symmetric with respect to the real axis. Thus, has the Taylor series expansion Suppose that , and are analytic in the unit disk with , , , and suppose that It is well known that Next, (6) and (7) lead to
Definition 1. A function is said to be in the class , , if the following subordinations hold: where .
Theorem 2. Let given by (2) be in the class . Then
Proof. Let , , and . Then there are analytic functions given by (7) such that
where . Since
it follows from (9) and (13) that
From (15) and (17) we obtain
By adding (18) to (16), further computations using (15) to (19) lead to
Equations (19) and (20), together with (8), give
From (15) and (21) we get Next, in order to find the bound on , by subtracting (18) from (16), we obtain Then, in view of (8) and (19), we have Notice that (11), we get
Remark 3. If we let then inequalities (11) and (12) become The bounds on and given by (27) are more accurate than those given in Theorem 2.2 in .
Remark 4. If we let then inequalities (11) and (12) become The bounds on and given by (29) are more accurate than those given in Theorem 3.3 in .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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Copyright © 2014 Şahsene Altınkaya and Sibel Yalçın. 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.