A Schwarz lemma for locally univalent meromorphic functions

Fournier, Richard; Kraus, Daniela; Roth, Oliver

doi:10.1090/proc/15087

A Schwarz lemma for locally univalent meromorphic functions
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by Richard Fournier, Daniela Kraus and Oliver Roth PDF

Proc. Amer. Math. Soc. 148 (2020), 3859-3870 Request permission

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.

References

F. G. Avkhadiev and L. I. Shokleva, Generalizations of a theorem of Beurling and their applications to inverse boundary value problems, Izv. Vyssh. Uchebn. Zaved. Mat. 5 (1994), 80–83 (Russian); English transl., Russian Math. (Iz. VUZ) 38 (1994), no. 5, 78–81. MR 1301697
F. G. Avkhadiev, Conformal mappings that satisfy the boundary condition of the equality of metrics, Dokl. Akad. Nauk 347 (1996), no. 3, 295–297 (Russian). MR 1393054
Florian Bauer, Daniela Kraus, Oliver Roth, and Elias Wegert, Beurling’s free boundary value problem in conformal geometry, Israel J. Math. 180 (2010), 223–253. MR 2735064, DOI 10.1007/s11856-010-0102-1
Arne Beurling, An extension of the Riemann mapping theorem, Acta Math. 90 (1953), 117–130. MR 60027, DOI 10.1007/BF02392436
Peter Borwein and Tamás Erdélyi, Sharp extensions of Bernstein’s inequality to rational spaces, Mathematika 43 (1996), no. 2, 413–423 (1997). MR 1433285, DOI 10.1112/S0025579300011876
Miran Černe and Manuel Flores, On Beurling’s boundary differential relation, Israel J. Math. 199 (2014), no. 2, 831–840. MR 3219559, DOI 10.1007/s11856-013-0061-4
Richard Fournier and Stephan Ruscheweyh, Free boundary value problems for analytic functions in the closed unit disk, Proc. Amer. Math. Soc. 127 (1999), no. 11, 3287–3294. MR 1618666, DOI 10.1090/S0002-9939-99-04960-6
B. Gidas, Wei Ming Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), no. 3, 209–243. MR 544879
Jürgen Grahl and Shahar Nevo, Spherical derivatives and normal families, J. Anal. Math. 117 (2012), 119–128. MR 2944092, DOI 10.1007/s11854-012-0016-4
J. Gröhn, Converse growth estimates for ODEs with slowly growing solutions, arXiv:1811.08736
Reiner Kühnau, Längentreue Randverzerrung bei analytischer Abbildung in hyperbolischer und Sphärischer Metrik, Mitt. Math. Sem. Giessen 229 (1997), 45–53 (German). MR 1439207
F. Marty, Recherches sur la répartition des valeurs d’une fonction méromorphe, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. (3) 23 (1931), 183–261 (French). MR 1508418
Oliver Roth, A general conformal geometric reflection principle, Trans. Amer. Math. Soc. 359 (2007), no. 6, 2501–2529. MR 2286042, DOI 10.1090/S0002-9947-07-03942-6
Joel L. Schiff, Normal families, Universitext, Springer-Verlag, New York, 1993. MR 1211641, DOI 10.1007/978-1-4612-0907-2
Norbert Steinmetz, Normal families and linear differential equations, J. Anal. Math. 117 (2012), 129–132. MR 2944093, DOI 10.1007/s11854-012-0017-3
Lawrence Zalcman, A heuristic principle in complex function theory, Amer. Math. Monthly 82 (1975), no. 8, 813–817. MR 379852, DOI 10.2307/2319796