Inequalities for the inner radii of nonorevlapping domains
UDC 517.54 We consider the problem of maximum of the functional $$ r^\gamma\left(B_0,0\right)\prod\limits_{k=1}^n r\left(B_k,a_k\right), $$ where $B_{0},\ldots,B_{n}$, $n\geq 2$, are pairwise disjoint domains in $\overline{\mathbb{C}},$ $a_0=0,$ $|a_{k}|=1$, $k=\overline{1,n},$ and $\gamma\in (0, n...
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| Datum: | 2019 |
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| Hauptverfasser: | , , , |
| Format: | Artikel |
| Sprache: | Russisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2019
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/1492 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
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Ukrains’kyi Matematychnyi Zhurnal| Zusammenfassung: | UDC 517.54
We consider the problem of maximum of the functional
$$
r^\gamma\left(B_0,0\right)\prod\limits_{k=1}^n r\left(B_k,a_k\right),
$$
where $B_{0},\ldots,B_{n}$, $n\geq 2$, are pairwise disjoint
domains in $\overline{\mathbb{C}},$ $a_0=0,$ $|a_{k}|=1$,
$k=\overline{1,n},$ and $\gamma\in (0, n]$ ($r(B,a)$ is the inner
radius of the domain $B\subset\overline{\mathbb{C}}$ with respect to
$a$).
Show that it attains its maximum at a configuration of domains $B_{k}$ and points $a_{k}$ possessing rotational $n$-symmetry.
This problem was solved by Dubinin for $\gamma=1$ and by Kuz’mina for
$0 |
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