Extremal Bounds of Teichmüller-Wittich-Belinski? Type for Planar Quasiregular Mappings

摘要

The theorems of TWB (Teichmüller-Wittich-Belinski?) type imply the local conformality (or weaker properties) of quasiconformal mappings at a prescribed point under assumptions of the finiteness of appropriate integral averages of the quantity K_μ(z) -- 1, where K_μ(z) stands for the real dilatation coefficient. We establish the extremal bounds for distortions of the moduli of annuli in terms of integrals in TWB theorems under quasiconformal and quasiregular mappings and illustrate their sharpness by several examples. Some local conditions weaker than the conformality are also discussed.