%0 Journal Article
%T Fekete-Szeg£¿ Type Coefficient Inequalities for Certain Subclass of Analytic Functions and Their Applications Involving the Owa-Srivastava Fractional Operator
%A Serap Bulut
%J International Journal of Analysis
%D 2014
%I Hindawi Publishing Corporation
%R 10.1155/2014/490359
%X A new subclass of analytic functions is introduced. For this class, firstly the Fekete-Szeg£¿ type coefficient inequalities are derived. Various known or new special cases of our results are also pointed out. Secondly some applications of our main results involving the Owa-Srivastava fractional operator are considered. Thus, as one of these applications of our result, we obtain the Fekete-Szeg£¿ type inequality for a class of normalized analytic functions, which is defined here by means of the Hadamard product (or convolution) and the Owa-Srivastava fractional operator. 1. Introduction and Definitions Let denote the class of functions of the form which are analytic in the unit disk Also let denote the subclass of consisting of univalent functions in . Fekete and Szeg£¿ [1] proved a noticeable result that the estimate holds for . The result is sharp in the sense that for each there is a function in the class under consideration for which equality holds. The coefficient functional on represents various geometric quantities as well as in the sense that this behaves well with respect to the rotation; namely, In fact, rather than the simplest case when we have several important ones. For example, represents , where denotes the Schwarzian derivative Moreover, the first two nontrivial coefficients of the th root transform of with the power series (1) are written by so that where Thus it is quite natural to ask about inequalities for corresponding to subclasses of . This is called Fekete-Szeg£¿ problem. Actually, many authors have considered this problem for typical classes of univalent functions (see, e.g., [1¨C12]). For two functions and , analytic in£¿£¿ , we say that the function is subordinate to in , and we write if there exists a Schwarz function , analytic in , with such that In particular, if the function is univalent in , the above subordination is equivalent to Let be an analytic function with which maps the open unit disk onto a star-like region with respect to and is symmetric with respect to the real axis. This paper contains analogues of (3) for the following classes of analytic functions. Definition 1. Let A function is said to be in the class if it satisfies the following subordination condition: where is defined to be the same as above for . Remark 2. (i) If we set in Definition 1, then we have the class which consists of functions satisfying This class was introduced by Bansal [13].(ii)If we set in Definition 1, then we have a new class which consists of functions satisfying Taking in (25), we have the class which consists of functions satisfying
%U http://www.hindawi.com/journals/ijanal/2014/490359/