A Schwarz lemma for locally univalent meromorphic functions
Complex Variables
2020-03-04 v2 Classical Analysis and ODEs
Abstract
We prove a sharp Schwarz-type lemma for meromorphic functions with spherical derivative uniformly bounded away from zero. As a consequence we deduce an improved quantitative version of a recent normality criterion due to Grahl & Nevo and Steinmetz, which is asymptotically best possibe. Based on a well--known symmetry result of Gidas, Ni & Nirenberg for nonlinear elliptic PDEs, we relate our Schwarz-type lemma to an associated nonlinear dual boundary extremal problem. As an application we obtain a generalization of Beurling's extension of the Riemann mapping theorem for the case of the spherical metric.
Cite
@article{arxiv.1902.07242,
title = {A Schwarz lemma for locally univalent meromorphic functions},
author = {Richard Fournier and Daniela Kraus and Oliver Roth},
journal= {arXiv preprint arXiv:1902.07242},
year = {2020}
}