Distortion theorems for higher order Schwarzian derivatives of univalent functions

Abstract

Let S ~ \tilde {\mathcal {S}} denote the class of functions which are univalent and holomorphic on the unit disc. We derive a simple differential equation for the Loewner flow of the Schwarzian derivative of a given f ∈ S ~ f \in \tilde {\mathcal {S}} . This is used to prove bounds on higher order Schwarzian derivatives which are sharp for the Koebe function. As well we prove some two-point distortion theorems for the higher order Schwarzians in terms of the hyperbolic metric.