Hyperbolic convexity of holomorphic level sets

Abstract

We prove that the sublevel set $\{z\in \mathbb{D}:k_{\mathbb{D}}(z,z_0)-k_{\mathbb{D}}(f(z),w_0)<\hat{I}¼\}$, $\hat{I}¼\in \mathbb{R}$, is geodesically convex with respect to the Poincaré 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 $\hat{I}¼⩽0$. An analogous result is established also for the set $\{z\in \mathbb{D}:1-|f(z){|}^2<\hat{I}»(1-|z{|}^2)\}$, $\hat{I}»>0$. This extends a result of Solynin (2007) and solves a problem posed by Arango, Mej\'ıa and Pommerenke (2019). We also propose several open questions aiming at possible extensions to more general settings.