This article is cited in 4 papers

Brief communications

An analog of the Schwarz lemma for locally quasiconformal automorphisms of a disk

S. Yu. Graf

Chair of Mathematical Analysis, Tver State University, 33 Zhelyabov str., Tver, 170000 Russia

Abstract: In the classes of normalized locally quasiconformal authomorphisms of the unit disk with the given majorant of M. A. Lavrentev's characteristic we obtain sharp estimates of the module of function as analogs of Schwarz's lemma and A. Mori's theorem. The classical growth theorems for quasiconformal authomorphisms of the disk follow from proved inequalities. In the classes of normalized locally quasiconformal homeomorphisms of the unit disk with the given majorant of M. A. Lavrentev's characteristic we prove sharp estimates of the conformal radius and radius of covering disk. The main results are obtained by the means of the methods of extremal lengths and symmetrization.

Keywords: locally quasiconformal mapping, growth theorem, Schwarz's lemma.

UDC: 517.54

Presented by the member of Editorial Board: L. A. Aksent'ev
Received: 17.11.2013

 English version: Russian Mathematics (Izvestiya VUZ. Matematika), 2014, 58:11, 74–79

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