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.