Subordination criteria for starlikeness and convexity

Abstract

For functions p analytic in the open unit disc U={z:|z|<1} with the normalization p(0)=1, we consider the families 𝒫[A,−1], −1<A≤1, consisting of p such that p(z) is subordinate to (1+Az)/(1−z) in U and 𝒫(1,b), b>0, consisting of p, which have the disc formulation |p−1|<b in U. We then introduce subordination criteria for the choice of p(z)=zf′(z)/f(z), where f is analytic in U and normalized by f(0)=f′(0)−1=0. We also obtain starlikeness and convexity conditions for such functions f and consequently extend and, in some cases, improve the corresponding previously known results.