A CERTAIN SUBCLASS OF JANOWSKI TYPE FUNCTIONS ASSOCIATED WITH κ-SYMMETRIC POINTS

Abstract

Abstract. We introduce a subclass S (k)s (A,B) (−1 ≤ B < A ≤ 1)of functions which are analytic in the open unit disk and close-to-convexwith respect to k-symmetric points. We give some coefficient inequalities,integral representations and invariance properties of functions belongingto this class. 1. IntroductionLet A denote the class of functions which are analytic in the open unit diskUand normalized by f(0) = 0 and f ′ (0) = 1. Also let S denote the subclassof A consisting of all functions which are univalent in U.Let f(z) and F(z) be analytic in U. Then we say that the function f(z) issubordinate to F(z) in U, if there exists an analytic function w(z) in U suchthat |w(z)| ≤ 1 and f(z) = F(w(z)), denote by f ≺ F or f(z) ≺ F(z). IfF(z) is univalent in U, then the subordination is equivalent to f(0) = F(0) andf(U) ⊂ F(U).Now, we denote by S ∗ (A,B) and C(A,B) the subclasses of A as follows:(1) S ∗ (A,B) =ˆf ∈ A :zf ′ (z)f(z)≺1+Az1+Bz,z ∈ U˙and(2) C(A,B) =ˆf ∈ A : ∃g ∈ S ∗ (A,B) such thatzf