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arXiv · 2411.10222

Hyperbolic convexity of holomorphic level sets

Iason Efraimidis·Pavel Gumenyuk
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Abstract

We prove that the sublevel set $\big\{z\in\mathbb D\colon k_{\mathbb D}\big(z,z_0\big)-k_{\mathbb D}\big(f(z),w_0\big)<μ\big\}$, ${μ\in\mathbb R}$, 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 ${μ\leqslant0}$. An analogous result is established also for the set $\{z\in\mathbb D \colon 1-|f(z)|^2<λ(1-|z|^2)\}$, ${λ>0}$. 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.

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Iason Efraimidis, Pavel Gumenyuk. 2024-11-15. Hyperbolic convexity of holomorphic level sets. https://doi.org/10.4171/rmi%2F1570

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