Fekete-Szeg？ Inequalities for Starlike Functions with Respect to -Symmetric Points of Complex Order

Sharp upper bounds of for the function belonging to certain subclass of starlike functions with respect to -symmetric points of complex order are obtained. Also, applications of our results to certain functions defined through convolution with a normalized analytic function are given. In particular, Fekete-Szeg？ inequalities for certain classes of functions defined through fractional derivatives are obtained. 1. Introduction Let denote the class of analytic functions of the following form: And let be the subclass of , which are univalent functions. Let be given by the following: The Hadamard product (or convolution) of and is given by If and are analytic functions in , we say that is subordinate to , written if there exists a Schwarz function , which is analytic in with and for all , such that . Furthermore, if the function is univalent in , then we have the following equivalence (see [1, 2]): Sakaguchi [3] introduced a class of functions starlike with respect to symmetric points, which consists of functions satisfying the inequality Chand and Singh [4] introduced a class of functions starlike with respect to -symmetric points, which consists of functions satisfying the inequality where Al-Shaqsi and Darus [5] defined the linear operator as follows: and in general where In this paper, we define the following class ( ) as follows. Definition 1. Let be univalent starlike function with respect to which maps the unit disk onto a region in the right half plane which is symmetric with respect to the real axis. Let be a complex number and let . Then functions are in the class if where is defined by (9) and is defined by (7). We note that for suitable choices of , , , , and we obtain the following subclasses:(i) (see Al-Shaqsi and Darus [6]),(ii) (see Al-Shaqsi and Darus [6]),(iii) (see Al-Shaqsi and Darus [6]),(iv) (see Shanmugam et al. [7]),(v) (see Sakaguchi [3]),(vi) (see Shanthi et al. [8] and Al-Shaqsi and Darus [9]),(vii) (see Ma and Minda [10]),(viii) (see Janowski [11]),(ix) and (see Ravichandran et al. [12]),(x) (see Nasr and Aouf [13]),(xi) (see Nasr and Aouf [14] and Aouf et al. [15]),(xii) (see Libera [16]),(xiii) (see Chichra [17]),(xiv) and (see Aouf and Silverman [18]),(xv) (see Keogh and Markes [19]).Also, we note the following: In this paper, we obtain the Fekete-Szeg？ inequalities for the functions in the class . We also give application of our results to certain functions defined through convolution and, in particular, we consider the class defined by fractional derivatives. 2. Fekete-Szeg？ Problem To prove our results, we need the following