Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.10222 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908702552883200 |
|---|---|
| author | Efraimidis, Iason Gumenyuk, Pavel |
| author_facet | Efraimidis, Iason Gumenyuk, Pavel |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_10222 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hyperbolic convexity of holomorphic level sets Efraimidis, Iason Gumenyuk, Pavel Complex Variables Primary 30C80, 30F45, 52A55, secondary 30J99, 30H05, 51M10 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. |
| title | Hyperbolic convexity of holomorphic level sets |
| topic | Complex Variables Primary 30C80, 30F45, 52A55, secondary 30J99, 30H05, 51M10 |
| url | https://arxiv.org/abs/2411.10222 |