Juan Arango | Searcharxiv

arXiv2024

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.

math.CV↗

arXiv2019

Level sets of the Hyperbolic Derivative for analytic self-maps of the unit disk

Let the function $φ$ be holomorphic in the unit disk $\mathbb{D}$ of the complex plane $\mathbb{C}$ and let $φ(\mathbb{D})\subset \mathbb{D}$. We study the level sets and the critical points of the hyperbolic derivative of $φ$, $$|D_φ(z)|:=\frac{(1-|z|^2)|φ'(z)|}{1-|φ(z)|^2}.$$ In particular, we show how the Schwarzian derivative of $φ$ reveals the nature of the critical points.

math.CV↗