## 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) dw of Ω to the hyperbolic metric λ_{X}(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 language | English |
---|---|

Pages (from-to) | 37-51 |

Number of pages | 15 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 309 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2005 Sep 1 |

Externally published | Yes |

## Keywords

- Hyperbolic metric
- Hyperbolically concave function
- Hyperbolically convex

## ASJC Scopus subject areas

- Analysis
- Applied Mathematics