Starlikeness and convexity of a class of analytic functions

Abstract

Let be the class of analytic functions in the unit disk that are normalized with f(0)=f′(0)−1=0 and let −1≤B<A≤1. In this paper we study the class Gλ,α={f∈:|(1−α+αzf″(z)/f′(z))/zf′(z)/f(z)−(1−α)|<λ,z∈},0≤α≤1, and give sharp sufficient conditions that embed it into the classes S∗[A,B]={f∈:zf′(z)/f(z)≺(1+Az)/(1+Bz)} and K(δ)={f∈:1+zf″(z)/f′(z)≺(1−δ)(1+z)/(1−z)+δ}, where “≺” denotes the usual subordination. Also, sharp upper bound of |a2| and of the Fekete-Szegö functional |a3−μa22| is given for the class Gλ,α.