Abstract
We introduce a scale-invariant version of the Cassinian metric and study its basic properties. We establish connections between the new metric and other well-known metrics such as, j-metric, \(\tilde{j}\)-metric, the half-apollonian metric, and the hyperbolic metric. We also show that the density of the new metric is the same as the density of the quasihyperbolic metric.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anderson, G.D., M.K. Vamanamurthy, and M.K. Vuorinen. 1997. Conformal invariants, inequalities, and quasiconformal maps. Hoboken: Wiley.
Beardon, A.F. 1995. The geometry of discrete groups. New York: Springer.
Beardon, A.F. 1998. The Apollonian metric of a domain in \(\mathbb{R}^n\). In Quasiconformal mappings and analysis (Ann Arbor MI, 1995), ed. P. Duren, J. Heinonen, B. Osgood, and B. Palka, 91–108. New York: Springer.
Beardon, A.F. and D. Minda. 2007. The hyperbolic metric and geometric function theory, Proceedings of the International Workshop on Quasiconformal mappings and their applications (IWQCMA05), Narosa Publishing House, 9–56, New Delhi.
Ferrand, J. 1988. A characterization of quasiconformal mappings by the behavior of a function of three points. In: I. Laine, S. Rickman, and T. Sorvali (Eds.) Proceedings of the 13th Rolf Nevalinna Colloquium (Joensuu, 1987), 110–123. Lecture Notes in Mathematics Vol. 1351, Springer-Verlag, New York.
Gehring, F.W., and B.G. Osgood. 1979. Uniform domains and the quasihyperbolic metric. J. Analyse Math. 36: 50–74.
Gehring, F.W., and B.P. Palka. 1976. Quasiconformally homogeneous domains. J. Analyse Math. 30: 172–199.
Hästö, P. 2006. Gromov hyperbolicity of the \(j_G\) and \(\tilde{j}_G\) metrics. Proceedings of the American Mathematical Society 134: 1137–1142.
Hästö, P., Z. Ibragimov, and H. Lindén. 2006. Isometries of relative metrics. Computational Methods and Function Theory 6(1): 15–28.
Hästö, P., and H. Lindén. 2004. Isometries of the half-apollonian metric. Complex Variables, Theory and Application: An International Journal 49: 405–415.
Ibragimov, Z. 2003. On the Apollonian metric of domains in \(\overline{\mathbb{R}^n}\). Complex Variables, Theory and Application: An International Journal 48(10): 837–855.
Ibragimov, Z. 2009. The Cassinian metric of a domain in \(\overline{\mathbb{R}^n}\). Uzbek. Mat. Zh. 1: 53–67.
Ibragimov, Z. 2016. A Möbius invariant Cassinian metric. Ann. Acad. Sci. Fenn., Ser. A I Math
Lindén, H. 2007. Hyperbolic-type metrics, Proceedings of the International Workshop on Quasiconformal mappings and their applications (IWQCMA05), Narosa Publishing House, 151–164, New Delhi
Klén,R. 2009. On hyperbolic type metrics, Dissertation, University of Turku, Turku, 2009. Ann. Acad. Sci. Fenn. Math. Diss. No. 152, 49
Seittenranta, P. 1999. Möbius-invariant metrics. Math. Proc. Cambridge Philos. Soc. 125: 511–533.
Vuorinen, M. 1988. Conformal geometry and quasiregular mappings, vol. 1319., Lecture Notes in Math Berlin: Springer-Verlag.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to David Minda on the occasion of his 70th birthday.
Rights and permissions
About this article
Cite this article
Ibragimov, Z. A scale-invariant Cassinian metric. J Anal 24, 111–129 (2016). https://doi.org/10.1007/s41478-016-0018-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s41478-016-0018-1
Keywords
- Möbius transformations
- Cassinian metric
- j-metric
- \(\tilde{j}\)-metric
- Half-apollonian metric
- Hyperbolic metric
- Quasihyperbolic metric