Subclasses of close-to-convex functions

Abstract

Let 𝒦[C,D], −1≤D<C≤1, denote the class of functions g(z), g(0)=g′(0)−1=0, analytic in the unit disk U={z:|z|<1} such that 1+(zg″(z)/g′(z)) is subordinate to (1+Cz)/(1+Dz), z ϵ U. We investigate the subclasses of close-to-convex functions f(z), f(0)=f′(0)−1=0, for which there exists g ϵ 𝒦[C,D] such that f′/g′ is subordinate to (1+Az)/(1+Bz), −1≤B<A≤1. Distortion and rotation theorems and coefficient bounds are obtained. It is also shown that these classes are preserved under certain integral operators.