Hyperbolic convexity of holomorphic level sets

TL;DR

This paper characterizes when certain holomorphic level sets in the unit disk are geodesically convex with respect to the Poincaré metric, extending previous results and addressing open problems in complex analysis.

Contribution

It provides a complete characterization of the geodesic convexity of specific holomorphic level sets in the unit disk, extending prior work and solving a known open problem.

Findings

Sublevel sets are geodesically convex if and only if μ ≤ 0.

Analogous convexity results for sets defined by |f(z)| and |z|.

Extension of Solynin's result and resolution of a problem posed in 2019.

Abstract

We prove that the sublevel set $\big\{z\in\mathbb D\colon k_{\mathbb D}\big(z,z_0\big)-k_{\mathbb D}\big(f(z),w_0\big)<\mu\big\}$, ${\mu\in\mathbb R}$, is geodesically convex with respect to the Poincar\'e distance $k_{\mathbb D}$ in the unit disk $\mathbb D$ for every ${z_0,w_0\in\mathbb D}$ and every holomorphic ${f:\mathbb D\to\mathbb D}$ if and only if ${\mu\leqslant0}$. An analogous result is established also for the set $\{z\in\mathbb D \colon 1-|f(z)|^2<\lambda(1-|z|^2)\}$, ${\lambda>0}$. This extends a result of Solynin (2007) and solves a problem posed by Arango, Mej\'{\i}a and Pommerenke (2019). We also propose several open questions aiming at possible extensions to more general settings.