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On quasi-inversions

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Abstract

Given a bounded domain \(D \subset {\mathbb R}^n\) strictly starlike with respect to \(0 \in D\,,\) we define a quasi-inversion w.r.t. the boundary \(\partial D \,.\) We show that the quasi-inversion is bi-Lipschitz w.r.t. the chordal metric if and only if every “tangent line” of \(\partial D\) is far away from the origin. Moreover, the bi-Lipschitz constant tends to 1,  when \(\partial D\) approaches the unit sphere in a suitable way. For the formulation of our results we use the concept of the \(\alpha \)-tangent condition due to Gehring and Väisälä (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 \(\partial D\). In addition, we show that the polar parametrization, which is a mapping of the unit sphere onto \(\partial D\), is bi-Lipschitz if and only if D satisfies the \(\alpha \)-tangent condition.

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Acknowledgments

We are grateful to the anonymous referee for very constructive comments that have improved this paper. This paper was written during the visit of the first author to the University of Turku in the framework of programme JoinEU-SEE III, ERASMUS MUNDUS. The authors are indebted to Prof. T. Sugawa for bringing the papers [6] and [15] to our attention. The research of the second and the third authors was supported by the Academy of Finland, Project 2600066611. The third author was also supported by Science Foundation of Zhejiang Sci-Tech University (ZSTU).

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Authors and Affiliations

  1. Faculty of Natural Sciences and Mathematics, University of Montenegro, Cetinjski put b.b., 81000, Podgorica, Montenegro

    David Kalaj

  2. Department of Mathematics and Statistics, University of Turku, Turku, 20014, Finland

    Matti Vuorinen

  3. School of Sciences, Zhejiang Sci-Tech University, Hangzhou, 310018, China

    Gendi Wang

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  1. David Kalaj
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  2. Matti Vuorinen
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  3. Gendi Wang
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Correspondence to Gendi Wang.

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Communicated by A. Constantin.

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Kalaj, D., Vuorinen, M. & Wang, G. On quasi-inversions. Monatsh Math 180, 785–813 (2016). https://doi.org/10.1007/s00605-016-0919-8

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