Sufficient conditions for starlikeness associated with parabolic region

Abstract

An analytic function f(z) = z + a n+1 z n+1 + ⋯, defined on the unit disk △ = {z : |z| < 1}, is in the class S p if z f′(z)/f(z) is in the parabolic region Rew > |w − 1|. This class is closely related to the class of uniformly convex functions. Sufficient conditions for function to be in S p are obtained. In particular, we find condition on λ such that the function f(z), satisfying (1 − α)(f(z)/z) μ + αf′(z)(f(z)/z) μ−1≺1 + λz, is in S p.