Subordination by convex functions

Abstract

For a fixed analytic function g(z) = z + ∑n=2∞gnzn defined on the open unit disk and γ < 1, let Tg (γ) denote the class of all analytic functions f (z) = z + ∑n=2∞anzn satisfying ∑n=2∞ an gn ≤ 1 - γ. For functions in Tg (γ), a subordination result is derived involving the convolution with a normalized convex function. Our result includes as special cases several earlier works.