Inequalities for inner radii of symmetric disjoint domains
We study the following problem: Let $a_0 = 0, | a_1| = ... = | a_n| = 1,\; a_k \in B_k {\subset C}$, where $B_0, ... ,B_n$ are disjoint domains, and $B_1, ... ,B_n$ are symmetric about the unit circle. It is necessary to find the exact upper bound for $r^{\gamma} (B_0, 0) \prod^n_{k=1} r(B_k, a_k...
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| Date: | 2018 |
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| Main Authors: | , , , , , |
| Format: | Article |
| Language: | Russian |
| Published: |
Institute of Mathematics, NAS of Ukraine
2018
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/1634 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | We study the following problem: Let $a_0 = 0, | a_1| = ... = | a_n| = 1,\; a_k \in B_k {\subset C}$, where $B_0, ... ,B_n$ are disjoint domains, and $B_1, ... ,B_n$ are symmetric about the unit circle. It is necessary to find the exact upper bound for
$r^{\gamma} (B_0, 0) \prod^n_{k=1}
r(B_k, a_k)$, where $r(B_k, a_k)$ is the inner radius of Bk with respect to $a_k$.
For $\gamma = 1$ and $n \geq 2$, the problem was solved by L. V. Kovalev. We solve this problem for $\gamma \in (0, \gamma_n], \gamma_n = 0,38 n^2$,
and $n \geq 2$ under the additional assumption imposed on the angles between the neighboring line segments $[0, a_k]$. |
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