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Möbius Invariant Cassinian Metric

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Abstract

We introduce a new Möbius invariant, \(\delta \)-hyperbolic metric \(\tau _D\) for domains D in \(\overline{\mathbb R^n}\), which can be considered as a Möbius invariant analogue of the scale-invariant Cassinian metric \(\tilde{\tau }_D\) recently introduced by the author. We establish basic properties of \(\tau _D\) including its connections with \(\tilde{\tau }_D\), the Apollonian metric, Seittenranta’s metric and the hyperbolic metric. We also show that \(\tau _D\) is monotonic with respect to domains, its density is the same as the density of Ferrand’s metric and that the \(\tau _D\)-isometries of twice-punctured spaces are Möbius maps.

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Authors and Affiliations

  1. Department of Mathematics, California State University, Fullerton, CA, 92831, USA

    Zair Ibragimov

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  1. Zair Ibragimov
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Correspondence to Zair Ibragimov.

Additional information

Communicated by Saminathan Ponnusamy.

Dedicated to academician Azimbay Sadullaev on the occasion of his 70th birthday.

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Ibragimov, Z. Möbius Invariant Cassinian Metric. Bull. Malays. Math. Sci. Soc. 42, 1349–1367 (2019). https://doi.org/10.1007/s40840-017-0550-4

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  • DOI: https://doi.org/10.1007/s40840-017-0550-4

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