On an internal characterization of horocyclically convex domains in the unit disk

Abstract A proper subdomain G of the unit disk $$\mathbb {D}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>D</mml:mi> </mml:math> is horocyclically convex (horo-convex) if, for every $$\omega \in \mathbb {D}\cap \partial G$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ω</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>D</mml:mi> <mml:mo>∩</mml:mo> <mml:mi>∂</mml:mi> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> , there exists a horodisk H such that $$\omega \in \partial H$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ω</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>∂</mml:mi> <mml:mi>H</mml:mi> </mml:mrow> </mml:math> and $$G\cap H=\emptyset$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>∩</mml:mo> <mml:mi>H</mml:mi> <mml:mo>=</mml:mo> <mml:mi>∅</mml:mi> </mml:mrow> </mml:math> . 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 $$C^1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> curve composed with finitely many Jordan arcs with hyperbolic curvature in $$(-2,2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . We also give a lower bound for the hyperbolic metric of horo-convex regions as well as some of its consequences.