On quasi-inversions



: Kalaj D, Vuorinen M, Wang GD

Publisher: SPRINGER WIEN

: 2016

: Monatshefte für Mathematik

: MONATSHEFTE FUR MATHEMATIK

: MONATSH MATH

: 180

: 4

: 785

: 813

: 29

: 0026-9255

DOI: https://doi.org/10.1007/s00605-016-0919-8


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.



Last updated on 2024-26-11 at 23:22