Geometry of the Cassinian Metric and Its Inner Metric

  • Zair Ibragimov
  • Manas Ranjan Mohapatra
  • Swadesh Kumar Sahoo
  • Xiaohui Zhang
Article

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.

Keywords

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 

Notes

Acknowledgments

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

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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
  • 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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