Distortion estimate and radius of starlikeness for generalized p−valent close-to-star functions

Abstract

Let A(p, n), n ≥ 1, p ≥ 1 be the class of all analytic functions in the open unit disc D = {z| |z| < 1} of the form f (z) = z p + a np+1 z np+1 + a np+2 z np+2 + • • •. Let g(z) be an element of A(p, n) such that g(z) satisfies the condition Re z g (z) g(z) > 0 for all z ∈ D, we then call g(z) the generalized p−valent starlike function in D. The class of such functions is denoted by S * (p, n). Let s(z) be an element of A(p, n) and g(z) be an element of S * (p, n). If the condition Re s(z) g(z) > 0 (z ∈ D), is satisfied then s(z) is called the generalized p−valent close-to-star function in D. The class of such functions is denoted by S * K(p, n). The aim of this paper is to give a distortion estimation and the radius of starlikeness of the class S * K(p, n).