Hyperbolic convexity of holomorphic level sets

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.