New Subclasses of Analytic Functions with Respect to Symmetric and Conjugate Points

We introduce new subclasses of close-to-convex and quasiconvex functions with respect to symmetric and conjugate points. The coefficient estimates for functions belonging to these classes are obtained. 1. Introduction Let be the class of functions which are analytic and univalent in the open unit disk given by and satisfying the conditions , , . Let denote the class of functions which are analytic and univalent in of the form Let be the subclass of functions and satisfying the condition These functions are called starlike with respect to symmetric points and were introduced by Sakaguchi [1]. Also, let be the subclass of functions and satisfying the condition These functions are called starlike with respect to conjugate points and were introduced by El-Ashwah and Thomas [2]. Further results on starlike functions with respect to symmetric points or conjugate points can be found in [3–5]. Then, Das and Singh [6] introduced another class , namely, convex functions with respect to symmetric points and satisfying the condition Suppose that and are two analytic functions in . Then, we say that the function is subordinate to the function , and we write , , if there exists a Schwarz function with and such that . In view of subordination definition, Goel and Mehrok [7] introduced a subclass of denoted by . Let be the class of functions of the form (2) and satisfying the condition Following them, many authors introduced the analogue definitions by extension as follows (see [8, 9]). Definition 1. Let be the subclass of consisting of functions given by (2) satisfying the condition Let be the subclass of consisting of functions given by (2) satisfying the condition Let be the subclass of consisting of functions given by (2) satisfying the condition Motivated by the pervious classes, Selvaraj and Vasanthi [10] defined the following classes of functions with respect to symmetric and conjugate points. Definition 2. Let be the subclass of consisting of functions given by (2) satisfying the condition Let be the subclass of consisting of functions given by (2) satisfying the condition In this paper, we introduce the class consisting of analytic functions of the form (2) and satisfying where . In addition, we introduce the class consisting of analytic functions of the form (2) and satisfying where . We note that(i)for , ？ ？ (see Mehrok et al. [11]) and ？ ？ ;(ii)for and , (see Janteng and Halim [12]) and ;(iii)for and , and ;(iv)for , and ;(v)for and , (see Janteng and Halim [13]) and ;(vi)for and , and . By the definition of subordination, it follows that if and only if and