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arXiv · 1902.07242

A Schwarz lemma for locally univalent meromorphic functions

Richard Fournier·Daniela Kraus·Oliver Roth
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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.

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Richard Fournier, Daniela Kraus, Oliver Roth. 2019-02-19. A Schwarz lemma for locally univalent meromorphic functions. https://arxiv.org/abs/1902.07242

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