TY - JOUR

T1 - Characterizations of hyperbolically convex regions

AU - Kim, Seong A.

AU - Sugawa, Toshiyuki

PY - 2005/9/1

Y1 - 2005/9/1

N2 - 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.

AB - 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.

KW - Hyperbolic metric

KW - Hyperbolically concave function

KW - Hyperbolically convex

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U2 - 10.1016/j.jmaa.2004.12.008

DO - 10.1016/j.jmaa.2004.12.008

M3 - Article

AN - SCOPUS:22144468771

SN - 0022-247X

VL - 309

SP - 37

EP - 51

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 1

ER -