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

Re󶀧󶀧 EEff (EE) ff (EE) z ff (EE) 󶀷󶀷 > ωω EE ω EEz (4) ese functions are called starlike with respect to conjugate points and were introduced by El-Ashwah and omas [2]. Further results on starlike functions with respect to symmetric points or conjugate points can be found in [3–5]. en, Das and Singh [6] introduced another class CCss, namely, convex functions with respect to symmetric points and satisfying the condition


Introduction
Let  be the class of functions which are analytic and univalent in the open unit disk        given by  ()  ∞        (1) 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

Re 󶀧󶀧 𝐸𝐸𝑓𝑓 ′ (𝐸𝐸) 𝑓𝑓 (𝐸𝐸) − 𝑓𝑓 (−𝐸𝐸)
> ese 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 Re   ′ ()  ()   ()  >     (4) ese functions are called starlike with respect to conjugate points and were introduced by El-Ashwah and omas [2].Further results on starlike functions with respect to symmetric points or conjugate points can be found in [3][4][5].en, 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 .en, we say that the function  is subordinate to the function , and we write ()  ()   , if there exists a Schwarz function () with ()   and ()   such that ()  (())   .
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] de�ned the following classes of functions with respect to symmetric and conjugate points.
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 (   +  ∞ 2        (  . In addition, we introduce the class  *  (     consisting of analytic functions  of the form (2) and satisfying

Some Preliminary Lemmas
We will require the following lemmas for proving our main results.