Geometry of the Cassinian Metric and Its Inner Metric

  • Zair Ibragimov
  • Manas Ranjan Mohapatra
  • Swadesh Kumar SahooEmail author
  • Xiaohui Zhang


The Cassinian metric and its inner metric have been studied for subdomains of the n-dimensional Euclidean space \({\mathbb {R}}^n\) (\(n\ge 2\)) by the first named author. In this paper we obtain various inequalities between the Cassinian metric and other related metrics in some specific subdomains of \({\mathbb {R}}^n\). Also, a sharp distortion property of the Cassinian metric under Möbius transformations of the unit ball is obtained.


Möbius transformation The hyperbolic metric The Cassinian metric The distance ratio metric The visual angle metric Inner metric 

Mathematics Subject Classification

30C35 30C20 30F45 51M10 



The last author is supported by the Academy of Finland project 268009.


  1. 1.
    Anderson, G.D., Vamanamurthy, M.K., Vuorinen, M.K.: Conformal Invariants, Inequalities, and Quasiconformal Maps. Wiley, New York (1997)zbMATHGoogle Scholar
  2. 2.
    Beardon, A.F.: Geometry of Discrete Groups. Springer-Verlag, New York (1995)zbMATHGoogle Scholar
  3. 3.
    Beardon, A.F.: The Apollonian metric of a domain in \({\mathbb{R}}^n\). In: Duren, P., Heinonen, J., Osgood, B., Palka, B. (eds.) Quasiconformal Mappings and Analysis (Ann Arbor, MI, 1995), pp. 91–108. Springer-Verlag, New York (1998)CrossRefGoogle Scholar
  4. 4.
    Borovikova, M., Ibragimov, Z.: Convex bodies of constant width and the Apollonian metric. Bull. Malays. Math. Sci. Soc. 31(2), 117–128 (2008)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Chen, J., Hariri, P., Klén, R., Vuorinen, M.: Lipschitz conditions, triangular ratio metric and quasiconformal maps. Ann. Acad. Sci. Fenn. Math. 40, 683–709 (2015)Google Scholar
  6. 6.
    Ferrand, J.: A characterization of quasiconformal mappings by the behavior of a function of three points. In: Laine, I., Rickman, S., Sorvali, T. (eds.) Proceedings of the 13th Rolf Nevalinna Colloquium (Joensuu, 1987). Lecture Notes in Mathematics vol. 1351, Springer-Verlag, New York, pp. 110–123 (1988)Google Scholar
  7. 7.
    Gehring, F.W., Osgood, B.G.: Uniform domains and the quasihyperbolic metric. J. Anal. Math. 36, 50–74 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Gehring, F.W., Palka, B.P.: Quasiconformally homogeneous domains. J. Anal. Math. 30, 172–199 (1976)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Hästö, P.: A new weighted metric: the relative metric. I. J. Math. Anal. Appl. 274(1), 38–58 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Hästö, P.: The Apollonian metric: uniformity and quasiconvexity. Ann. Acad. Sci. Fenn. Math. 28(2), 385–414 (2003)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Hästö, P., Ibragimov, Z.: Apollonian isometries of planar domains are Möbius mappings. J. Geom. Anal. 15(2), 229–237 (2005)CrossRefzbMATHGoogle Scholar
  12. 12.
    Hästö, P., Ibragimov, Z.: Apollonian isometries of regular domains are Möbius mappings. Ann. Acad. Sci. Fenn. Math. 32(1), 83–98 (2007)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Hästö, P., Ibragimov, Z., Linden, H.: Isometries of relative metrics. Comput. Methods Funct. Theory 6(1), 15–28 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Hästö, P., Ibragimov, Z., Minda, D., Ponnusamy, S., Sahoo, S.K.: Isometries of some hyperbolic-type path metrics, and the hyperbolic medial axis. Tradi. Ahlfors-Bers IV Contemp. Math. 432, 63–74 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Herron, D., Ibragimov, Z., Minda, D.: Geodesics and curvature of Möbius invariant metrics. Rocky Mountain J. Math. 38(3), 891–921 (2008)CrossRefzbMATHGoogle Scholar
  16. 16.
    Ibragimov, Z.: On the Apollonian metric of domains in \(\overline{{\mathbb{R}}^n}\). Complex Var. Theory Appl. 48(10), 837–855 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Ibragimov, Z.: Conformality of the Apollonian metric. Comput. Methods Funct. Theory 3(1–2), 397–411 (2003)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Ibragimov, Z.: The Cassinian metric of a domain in \(\bar{{\mathbb{R}}}^n\). Uzbek. Mat. Zh. 1, 53–67 (2009)MathSciNetGoogle Scholar
  19. 19.
    Klén, R., Linden, H., Vuorinen, M., Wang, G.: The visual angle metric and Möbius transformations. Comput. Methods Funct. Theory 14(2–3), 577–608 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Kulkarni, R., Pinkall, U.: A canonical metric for Möbius structures and its applications. Math. Z. 216, 89–129 (1994)CrossRefzbMATHGoogle Scholar
  21. 21.
    Seittenranta, P.: Möbius-invariant metrics. Math. Proc. Camb. Philos. Soc. 125, 511–533 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Vuorinen, M.: Conformal invariants and quasiregular mappings. J. Anal. Math. 45, 69–115 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Vuorinen, M.: Conformal Geometry and Quasiregular Mappings. Lecture Note in Mathematics. Springer-Verlag, Berlin (1988)Google Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Universiti Sains Malaysia 2015

Authors and Affiliations

  • Zair Ibragimov
    • 1
  • Manas Ranjan Mohapatra
    • 2
  • Swadesh Kumar Sahoo
    • 2
    Email author
  • Xiaohui Zhang
    • 3
  1. 1.Department of MathematicsCalifornia State UniversityFullertonUSA
  2. 2.Discipline of MathematicsIndian Institute of Technology IndoreIndoreIndia
  3. 3.Department of Physics and MathematicsUniversity of Eastern FinlandJoensuuFinland

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