Characterizations of hyperbolically convex regions

Seong A. Kim, Toshiyuki Sugawa

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)


Let X be a simply connected and hyperbolic subregion of the complex plane ℂ. A proper subregion Ω of X is called hyperbolically convex in X if for any two points A and B in Ω, the hyperbolic geodesic arc joining A and B in X is always contained in Ω. We establish a number of characterizations of hyperbolically convex regions Ω in X in terms of the relative hyperbolic density ρΩ (w) of the hyperbolic metric of Ω to X, that is the ratio of the hyperbolic metric λΩ (w) dw of Ω to the hyperbolic metric λX(w) dw of X. Introduction of hyperbolic differential operators on X makes calculations much simpler and gives analogous results to some known characterizations for euclidean or spherical convex regions. The notion of hyperbolic concavity relative to X for real-valued functions on Ω is also given to describe some sufficient conditions for hyperbolic convexity.

Original languageEnglish
Pages (from-to)37-51
Number of pages15
JournalJournal of Mathematical Analysis and Applications
Issue number1
Publication statusPublished - 2005 Sept 1


  • Hyperbolic metric
  • Hyperbolically concave function
  • Hyperbolically convex


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