Skip to main content

Advertisement

SpringerLink
Log in

Geometry of the Cassinian Metric and Its Inner Metric

  • Published:
Bulletin of the Malaysian Mathematical Sciences Society Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Anderson, G.D., Vamanamurthy, M.K., Vuorinen, M.K.: Conformal Invariants, Inequalities, and Quasiconformal Maps. Wiley, New York (1997)

    MATH  Google Scholar 

  2. Beardon, A.F.: Geometry of Discrete Groups. Springer-Verlag, New York (1995)

    MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  4. Borovikova, M., Ibragimov, Z.: Convex bodies of constant width and the Apollonian metric. Bull. Malays. Math. Sci. Soc. 31(2), 117–128 (2008)

    MATH  MathSciNet  Google Scholar 

  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)

  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)

  7. Gehring, F.W., Osgood, B.G.: Uniform domains and the quasihyperbolic metric. J. Anal. Math. 36, 50–74 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  8. Gehring, F.W., Palka, B.P.: Quasiconformally homogeneous domains. J. Anal. Math. 30, 172–199 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  9. Hästö, P.: A new weighted metric: the relative metric. I. J. Math. Anal. Appl. 274(1), 38–58 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  10. Hästö, P.: The Apollonian metric: uniformity and quasiconvexity. Ann. Acad. Sci. Fenn. Math. 28(2), 385–414 (2003)

    MATH  MathSciNet  Google Scholar 

  11. Hästö, P., Ibragimov, Z.: Apollonian isometries of planar domains are Möbius mappings. J. Geom. Anal. 15(2), 229–237 (2005)

    Article  MATH  Google Scholar 

  12. Hästö, P., Ibragimov, Z.: Apollonian isometries of regular domains are Möbius mappings. Ann. Acad. Sci. Fenn. Math. 32(1), 83–98 (2007)

    MATH  MathSciNet  Google Scholar 

  13. Hästö, P., Ibragimov, Z., Linden, H.: Isometries of relative metrics. Comput. Methods Funct. Theory 6(1), 15–28 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  15. Herron, D., Ibragimov, Z., Minda, D.: Geodesics and curvature of Möbius invariant metrics. Rocky Mountain J. Math. 38(3), 891–921 (2008)

    Article  MATH  Google Scholar 

  16. Ibragimov, Z.: On the Apollonian metric of domains in \(\overline{{\mathbb{R}}^n}\). Complex Var. Theory Appl. 48(10), 837–855 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  17. Ibragimov, Z.: Conformality of the Apollonian metric. Comput. Methods Funct. Theory 3(1–2), 397–411 (2003)

    MATH  MathSciNet  Google Scholar 

  18. Ibragimov, Z.: The Cassinian metric of a domain in \(\bar{{\mathbb{R}}}^n\). Uzbek. Mat. Zh. 1, 53–67 (2009)

    MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  20. Kulkarni, R., Pinkall, U.: A canonical metric for Möbius structures and its applications. Math. Z. 216, 89–129 (1994)

    Article  MATH  Google Scholar 

  21. Seittenranta, P.: Möbius-invariant metrics. Math. Proc. Camb. Philos. Soc. 125, 511–533 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  22. Vuorinen, M.: Conformal invariants and quasiregular mappings. J. Anal. Math. 45, 69–115 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  23. Vuorinen, M.: Conformal Geometry and Quasiregular Mappings. Lecture Note in Mathematics. Springer-Verlag, Berlin (1988)

Download references

Acknowledgments

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

Author information

Authors and Affiliations

  1. Department of Mathematics, California State University, Fullerton, CA, 92834, USA

    Zair Ibragimov

  2. Discipline of Mathematics, Indian Institute of Technology Indore, Simrol, Khandwa Road, Indore, 452 020, India

    Manas Ranjan Mohapatra & Swadesh Kumar Sahoo

  3. Department of Physics and Mathematics, University of Eastern Finland, 80101, Joensuu, Finland

    Xiaohui Zhang

Authors
  1. Zair Ibragimov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Manas Ranjan Mohapatra
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Swadesh Kumar Sahoo
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Xiaohui Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Swadesh Kumar Sahoo.

Additional information

Communicated by See Keong Lee.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ibragimov, Z., Mohapatra, M.R., Sahoo, S.K. et al. Geometry of the Cassinian Metric and Its Inner Metric. Bull. Malays. Math. Sci. Soc. 40, 361–372 (2017). https://doi.org/10.1007/s40840-015-0246-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40840-015-0246-6

Keywords

Mathematics Subject Classification

Search

Navigation