# A Schwarz lemma for locally univalent meromorphic functions

## TL;DR

This paper establishes a sharp Schwarz lemma for meromorphic functions with bounded spherical derivative, leading to improved normality criteria, a dual extremal problem, and a generalization of Beurling's theorem in the spherical metric.

## Contribution

It introduces a novel Schwarz-type lemma for meromorphic functions with bounded spherical derivative and connects it to extremal problems and classical theorems.

## Key findings

- Proves a sharp Schwarz lemma for meromorphic functions with bounded spherical derivative.
- Derives an improved normality criterion that is asymptotically optimal.
- Generalizes Beurling's extension of the Riemann mapping theorem in the spherical metric.

## 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.