On alpha-close-to-convex functions of order beta

Abstract

Let Mβ(α)[α≥0   and   β≥0] denote the class of all functionsf(z)=z+∑n=2∞anznanalytic in the unit disc U with f′(z)f(z)/z≠0 and which satisfy for z=reiθ∈U the condition ∫θ1θ2Re{(1−α)zf′(z)f(z)+α(1+zf″(z)f′(z))}dθ>−βπ for all θ2>θ1. In this note we show that each f∈Mβ(α) is close-to-star of order β when 0<β≤α.