Characterizations of hyperbolically convex regions

Kim Seong-A

Sugawa Toshiyuki

Journal of Mathematical Analysis and Applications

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.

Mathematics [ 410 ]

eng

Elsevier Inc

Copyright (c) 2004 Elsevier Inc

[NCID] AA00252847

[ISSN] 0022-247X

[DOI] 10.1016/j.jmaa.2004.12.008

[DOI] http://dx.doi.org/10.1016/j.jmaa.2004.12.008 isVersionOf