On an Internal Characterization of Horocyclically Convex Domains in the Unit Disk

A proper subdomain $G$ of the unit disk $\mathbb{D}$ is horocyclically convex (horo-convex) if, for every $\omega \in \mathbb{D}\cap \partial G$, there exists a horodisk $H$ such that $\omega \in \partial H$ and $G\cap H=\emptyset$. In this paper we give an internal characterization of these domains, namely, that $G$ is horo-convex if and only if any two points can be joined inside $G$ by a $C^1$ curve composed with finitely many Jordan arcs with hyperbolic curvature in $(-2,2)$. We also give a lower bound for the hyperbolic metric of horo-convex regions and some consequences.