Notes on Generalization of Close-to-Star Functions

摘要

The close-to-star functions was introduced by Maxwell O. Reade [3]. Let f(z) = z+a_2z~2+a_3z~3+... be analytic in the open unit disc D = {z| |z| < 1}. We say that f(z) is close-to-star function if there exists a univalent star map g(z) = z + b_2z~2 + b_3z~3 + ... with respect to origin such that Re (f(z)/g(z)) > 0 holds for all z ∈ D. In this paper, we will introduce a generalization of close-to-star functions which, called generalized Janowski close-to-star functions, and we will give a distortion theorem, a growth theorem and the radius of starlikeness of these functions.