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 and Nevo [J. Anal. Math. 117 (2012), 119–128] and Steinmetz [J. Anal. Math. 117 (2012), 129–132], which is asymptotically best possibe. Based on a well-known symmetry result of Gidas, Ni, and 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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