On a subclass of α-convex λ-spiral functions

Abstract

Let H denote the class of functions f(z)=z+∑k=2∞akzk which are analytic in the unit disc Δ={z:|z|<1}. In this paper, we introduce the class Mαλ[A,B] of functions f∈H with f(z)f′(z)/z≠0, satisfying for z∈Δ:{(eiλ−αcosλ)(zf′(z)/f(z))+αcosλ(1+zf″(z)/f′(z))}≺cosλ((1+Az)/(1+Bz))+isinλ, where ≺ denotes subordination, α and λ are real numbers, |λ|<π/2 and −1≤B<A≤1. Functions in Mαλ[A,B] are shown to be λ-spiral like and hence univalent. Integral representation, coefficients bounds, and other results are given.