Weakened problem on extremal decomposition of the complex plane

Author
A. K. Bakhtin1, I. V. Denega2
Institute of Mathematics of National Academy of Sciences of Ukraine, Department of Complex analysis and Potential Theory, Kyiv, Ukraine
Abstract
The paper deals with the problem of the maximum of the functional $$r^\gamma\left(B_0,0\right)\prod\limits_{k=1}^n r\left(B_k,a_k\right),$$ where $B_{0}$,..., $B_{n}$, $n\geqslant 2$, are pairwise disjoint domains in $\overline{\mathbb{C}},$ $a_0=0,$ $|a_{k}|=1$, $k\in\{1,\ldots,n\}$ and $\gamma\in (0, n]$ ($r(B,a)$ is the inner radius of the domain $B\subset\overline{\mathbb{C}}$ with respect to $a$). We show that the functional attains its maximum at a configuration of the domains $B_{k}$ and the points $a_{k}$ possessing rotational $n$-symmetry. The proof is due to Dubinin [1] for $\gamma = 1$ and to Kuz'mina [3] for $0 \le\gamma\le 1$. Subsequently, Kovalev [4] solved this problem for $n\geqslant5$ under the additional assumption that the angles between neighbouring line segments $[0, a_{k}]$ do not exceed $2\pi / \sqrt{\gamma}$. In the paper, we obtain some estimate of the functional for $\gamma\in(1,\,n]$.
Keywords
extremal problems with free poles; inner radius of domain; non-overlapping domains; complex plane; transfinite diameter; theorem on minimizing of the area; Cauchy inequality
DOI
doi:10.15330/ms.51.1.35-40
Reference
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Pages
35-40
Volume
51
Issue
1
Year
2019
Journal
Matematychni Studii
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