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On the Asymptotic Behavior of Quasiconformal Mappings in Space

In: Quasiconformal Mappings and Analysis

Author

Listed:
  • V. Ya. Gutlyanskiǐ

    (NAS of Ukraine, Institute of Applied Mathematics and Mechanics
    University of Helsinki, Department of Mathematics)

  • O. Martio

    (NAS of Ukraine, Institute of Applied Mathematics and Mechanics
    University of Helsinki, Department of Mathematics)

  • V. I. Ryazanov

    (NAS of Ukraine, Institute of Applied Mathematics and Mechanics
    University of Helsinki, Department of Mathematics)

  • M. Vuorinen

    (NAS of Ukraine, Institute of Applied Mathematics and Mechanics
    University of Helsinki, Department of Mathematics)

Abstract

Conformality at a point of a space quasiconformal mapping is studied without the assumption of differentiability. First, we introduce a notion of asymptotic linearity for general mappings at a prescribed point. Then we prove that the simultaneous asymptotic linearity and analyticity of a mapping f at a point of discreteness imply that f preserves the angles between rays emanating from this point and the moduli of infinitesimal annuli centered at it and thus we strengthen an earlier result of Caraman. Finally, we give sufficient conditions for quasiconformal mappings to be asymptotically linear and analytic in terms of the distortion coefficient and, as a consequence, we generalize the well-known Belinskii weak conformality result to the n-dimensional case.

Suggested Citation

  • V. Ya. Gutlyanskiǐ & O. Martio & V. I. Ryazanov & M. Vuorinen, 1998. "On the Asymptotic Behavior of Quasiconformal Mappings in Space," Springer Books, in: Peter Duren & Juha Heinonen & Brad Osgood & Bruce Palka (ed.), Quasiconformal Mappings and Analysis, pages 159-180, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4612-0605-7_11
    DOI: 10.1007/978-1-4612-0605-7_11
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