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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2303.08238 |
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| _version_ | 1866913209690095616 |
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| author | Hinkkanen, Aimo Vuorinen, Matti |
| author_facet | Hinkkanen, Aimo Vuorinen, Matti |
| contents | We prove that if $E$ is a compact subset of the unit disk ${\mathbb D}$ in the complex plane, if $E$ contains a sequence of distinct points $a_n\not= 0$ for $n\geq 1$ such that $\lim_{n\to\infty} a_n=0$ and for all $n$ we have $ |a_{n+1}| \geq \frac{1}{2} |a_n| $, and if $G={\mathbb D} \setminus E$ is connected and $0\in \partial G$, then there is a constant $c>0$ such that for all $z\in G$ we have $ λ_{G } (z) \geq c/|z| $ where $λ_{G } (z)$ is the density of the hyperbolic metric in $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_08238 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the hyperbolic metric of certain domains Hinkkanen, Aimo Vuorinen, Matti Complex Variables 30C80 We prove that if $E$ is a compact subset of the unit disk ${\mathbb D}$ in the complex plane, if $E$ contains a sequence of distinct points $a_n\not= 0$ for $n\geq 1$ such that $\lim_{n\to\infty} a_n=0$ and for all $n$ we have $ |a_{n+1}| \geq \frac{1}{2} |a_n| $, and if $G={\mathbb D} \setminus E$ is connected and $0\in \partial G$, then there is a constant $c>0$ such that for all $z\in G$ we have $ λ_{G } (z) \geq c/|z| $ where $λ_{G } (z)$ is the density of the hyperbolic metric in $G$. |
| title | On the hyperbolic metric of certain domains |
| topic | Complex Variables 30C80 |
| url | https://arxiv.org/abs/2303.08238 |