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.