This article is cited in 2 papers

Real, complex and functional analysis

Normal families of light mappings of the sphere onto itself

V. V. Aseev, D. G. Kuzin

Sobolev Institute of Mathematics, pr. Koptyuga, 4, 630090, Novosibirsk, Russia

Abstract: Considering the class ${\mathcal D}$ of all continuous light mappings of the Riemann sphere $\bar{\mathbf C}$ onto itself, we introduce the notion of ${\mathcal D}$-normal family and prove that every mapping $f$ from a given Möbius invariant and ${\mathcal D}$-normal family ${\mathcal F}\subset {\mathcal D}$ is a composition of a $K$-quasiconformal automorphism of $\bar{\mathbf C}$ with the mapping, realized by a meromorphic function on $\bar{\mathbf C}$, where a constant $K$ is common for all $f\in {\mathcal F}$.

Keywords: quasiconformal mapping, mapping of bounded distortion, quasimeromorphic mapping, graph convergence, normal family, Möbius mapping, Möbius invariant family, Stoilov theorem, light mapping, open mapping.

UDC: 517.54

MSC: 30C62

Received December 16, 2013, published December 30, 2013

Language: English