Seong-A KIM and Toshiyuki SUGAWA

*Characterizations of hyperbolically convex regions*

### 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 ρ_{Ω}(*w*)
of the hyperbolic metric of Ω to *X*, that is
the ratio of the hyperbolic metric λ_{Ω}(*w*)|d*w*| of Ω
to the hyperbolic metric λ_{X}(*w*)|d*w*| 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.

submission: 23 February 2004

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