Characterizations of hyperbolically convex regions

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Abstract

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.

Journal

  • Journal of Mathematical Analysis and Applications

    Journal of Mathematical Analysis and Applications 309(1), 37-51, 2005-09-01

    Elsevier Inc

Codes

  • NII Article ID (NAID)
    120000878153
  • NII NACSIS-CAT ID (NCID)
    AA00252847
  • Text Lang
    ENG
  • Article Type
    journal article
  • ISSN
    0022-247X
  • Data Source
    IR 
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