A generalization of starlike functions of order alpha

Abstract

For every q ∈ (0, 1) and 0 ≤ A < 1 we define a class of analytic functions, the so-called q-starlike functions of order a, on the open unit disk. We study this class of functions and explore some inclusion properties with the well-known class of starlike functions of order A. The paper is also devoted to the discussion on the Herglotz representation formula for analytic functions zf′ (z)/f (z) when f (z) is q-starlike of order A. As an application we also discuss the Bieberbach conjecture problem for the q-starlike functions of order A.