A1 Refereed original research article in a scientific journal
On quasi-inversions
Authors: Kalaj D, Vuorinen M, Wang GD
Publisher: SPRINGER WIEN
Publication year: 2016
Journal:: Monatshefte für Mathematik
Journal name in source: MONATSHEFTE FUR MATHEMATIK
Journal acronym: MONATSH MATH
Volume: 180
Issue: 4
First page : 785
Last page: 813
Number of pages: 29
ISSN: 0026-9255
DOI: https://doi.org/10.1007/s00605-016-0919-8
Abstract
Given a bounded domain strictly starlike with respect to we define a quasi-inversion w.r.t. the boundary We show that the quasi-inversion is bi-Lipschitz w.r.t. the chordal metric if and only if every "tangent line" of is far away from the origin. Moreover, the bi-Lipschitz constant tends to 1, when approaches the unit sphere in a suitable way. For the formulation of our results we use the concept of the -tangent condition due to Gehring and Vaisala (Acta Math 114:1-70,1965). This condition is shown to be equivalent to the bi-Lipschitz and quasiconformal extension property of what we call the polar parametrization of . In addition, we show that the polar parametrization, which is a mapping of the unit sphere onto , is bi-Lipschitz if and only if D satisfies the -tangent condition.
Given a bounded domain strictly starlike with respect to we define a quasi-inversion w.r.t. the boundary We show that the quasi-inversion is bi-Lipschitz w.r.t. the chordal metric if and only if every "tangent line" of is far away from the origin. Moreover, the bi-Lipschitz constant tends to 1, when approaches the unit sphere in a suitable way. For the formulation of our results we use the concept of the -tangent condition due to Gehring and Vaisala (Acta Math 114:1-70,1965). This condition is shown to be equivalent to the bi-Lipschitz and quasiconformal extension property of what we call the polar parametrization of . In addition, we show that the polar parametrization, which is a mapping of the unit sphere onto , is bi-Lipschitz if and only if D satisfies the -tangent condition.
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