2019/02/19 by Fournier, Richard, Kraus, Daniela, Roth, Oliver
#30C80 #30D30 #35J15 #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.1902.07242
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.