Characterizations of hyperbolically convex regions

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.