Subordination by convex functions

Abstract

Let K(α), 0≤α<1, denote the class of functions g(z)=z+Σn=2∞anzn which are regular and univalently convex of order α in the unit disc U. Pursuing the problem initiated by Robinson in the present paper, among other things, we prove that if f is regular in U,f(0)=0, and f(z)+zf′(z)<g(z)+zg′(z) in U, then (i) f(z)<g(z) at least in |z|<r0,r0=5/3=0.745… if f∈K; and (ii) f(z)<g(z) at least in |z|<r1,r1((51−242)/23)1/2=0.8612… if g∈K(1/2).