Subordination by convex functions

Abstract

For a fixed analytic functiong(z)=z+∑n=2∞gnzndefined on the open unit disk andγ<1, letTg(γ)denote the class of all analytic functionsf(z)=z+∑n=2∞anznsatisfying∑n=2∞|angn|≤1−γ. For functions inTg(γ), a subordination result is derived involving the convolution with a normalized convex function. Our result includes as special cases several earlier works.