BOUNDED FUNCTIONS STARLIKE WITH RESPECT TO SYMMETRICAL POINTS

Let P[A,B], −1≤B<A≤1, be the class of functions p analytic in the unit disk E with p(0)=1 and subordinate to 1


INTRODUCTION
Let .,4denote the class of functions, analytic in E z Izl < 1} and normalized by the conditions f(0)--0 f'(0)-1 In [7] Sakaguchi defined the class of starlike functions with respect to symmetrical points as follows Let f E 4 Then f is said to be starlike with respect to symmetrical points in E if, and only if, Re >0, zEE.
(1 1) f(z) f z) We denote this class by S. Obviously, it forms a subclass of close-to-convex functions and hence univalent Moreover, this class includes the class of convex functions and odd starlike functions with respect to the origin, see [7] Janowski [4] introduced the classes P[A, B] and S*[A, B] as follows For A and B, I < B < A _< 1, a function p, analytic in E, with p(0) 1, belongs to the class P[A, B] ifp(z) is subordinate to l+Az l+Bz A function f .A is said to be in S*[A,B], if and only if, zf'(z) P [A, B]   We now define the following DEFINITION 1.1.LetfE.A ThenfES's[A,B], -1< B<A< l if and only if, forzE 2zf'(z) f(z)-f(-z) P[A, B].
(1 3) where p, p P[A, B], since f S [A, B] Using Lemma we have the required result REMARK 1.1.From Theorem and Definition we conclude that where K is the class of close-to-convex functions This implies that functions in S [A, B] are close-to- convex and hence univalent 2.

COEFFICIENT BOUNDS
In the following we will study the coefficients problem for the class S's[A, B], we need the following LEMMA 2.1 [1] z + ' b2,,-z 2'-Then n-----'2 Let 7-be an odd function and rS[1-2a,-1] and let "r(z)= This result is sharp as can be seen from the function [1] Let r be an odd function belonging to S'[A,B] and let z 2'-1 Put M (+B) the largest integer not greater th -.We have the follong The bounds in (2 1) and (2 2) are sharp LEMMA 2.3.
n=l Then Ic, I<_A-B.This result is sharp To solve the coefficient problem for the class S[1-2a,-1] we will use the technique of dominant power series which is defined as follows Let f and F be given by the power series n=0 n=0 convergent in some disk En "lzl < R, R > 0 We say that f is dominated by F (or F dominates f), and we write f << F if for each integer n > 0 THEOREM 2.1.Let f (5 S[1 20, 1] and be given by f(z) z + , a,z".Then (i) I1 _< (1-c,), Ial _< (1- (ii) la2l <_ (:;1 1 + 1"I ((1 a) +v) n > 2.