Schwarzian Derivatives and Uniform Local Univalence

TL;DR

This paper establishes quantitative bounds on the valence of analytic functions with bounded Schwarzian derivatives, characterizes harmonic mappings with finite Schwarzian norm as uniformly locally univalent, and provides numerical bounds for univalent harmonic mappings.

Contribution

It generalizes Schwarz's classical result to harmonic mappings and offers new quantitative estimates for Schwarzian derivatives in the context of univalence.

Findings

Finite valence estimates for functions with bounded Schwarzian derivative

Characterization of harmonic mappings with finite Schwarzian norm as uniformly locally univalent

Numerical bounds for Schwarzian norms of univalent harmonic mappings

Abstract

Quantitative estimates are obtained for the (finite) valence of functions analytic in the unit disk with Schwarzian derivative that is bounded or of slow growth. A harmonic mapping is shown to be uniformly locally univalent with respect to the hyperbolic metric if and only if it has finite Schwarzian norm, thus generalizing a result of B. Schwarz for analytic functions. A numerical bound is obtained for the Schwarzian norms of univalent harmonic mappings.