Schwarzian Derivatives and Uniform Local Univalence
Complex Variables
2007-07-16 v2
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.
Cite
@article{arxiv.0706.4296,
title = {Schwarzian Derivatives and Uniform Local Univalence},
author = {Martin Chuaqui and Peter Duren and Brad Osgood},
journal= {arXiv preprint arXiv:0706.4296},
year = {2007}
}