Extremal Bounds of Teichmüller-Wittich-Belinskiı̆ Type for Planar Quasiregular Mappings

Chapter
Part of the Fields Institute Communications book series (FIC, volume 81)

Abstract

The theorems of TWB (Teichmüller-Wittich-Belinskiı̆) 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.

Keywords

Module of families of curves Moduli of annuli Local conformality Local weak conformality Quasiconformal and quasiregular mappings Extremal bounds 

Msc codes:

Primary 30C62, 30C75; Secondary 30C65 

Notes

Acknowledgements

The author gratefully thank the referee for his comments and suggestions.

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Authors and Affiliations

  1. 1.Department of MathematicsHolon Institute of TechnologyHolonIsrael

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