Coefficient estimates for negative powers of the derivative of univalent functions

Abstract

Letf be a one-to-one analytic function in the unit disc with f′(0)=1. We prove sharp estimates for certain Taylor coefficients of the functions (f′)p, where p<0. The proof is similar to de Branges’ proof of Bieberbach’s conjecture, and thus relies on Löwner’s equation. A special case leads to a generalization of the usual estimate for the Schwarzian derivative of f. We use this to improve known estimates for integral means of the functions |f′|p for integers p⪯−2.