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 Probl. Anal. Issues Anal., 2019, Volume 8(26), Issue 1, Pages 17–31 (Mi pa255)

Sharp estimates of products of inner radii of non-overlapping domains in the complex plane

A. K. Bakhtin, I. V. Denega

Institute of Mathematics of the National Academy of Sciences of Ukraine, Department of complex analysis and potential theory, 01004 Ukraine, Kiev-4, 3, Tereschenkivska st.

Abstract: In the paper we study a generalization of the extremal problem of geometric theory of functions of a complex variable on non-overlapping domains with free poles: Fix any $\gamma\in\mathbb{R^{+}}$ and find the maximum (and describe all extremals) of the functional
$$[r(B_0,0)r(B_\infty,\infty)]^{\gamma} \prod\limits_{k=1}^n r(B_k,a_k),$$
where $n\in \mathbb{N}$, $n\geqslant 2$, $a_{0}=0$, $|a_{k}|=1$, $B_0$, $B_\infty$, $\{B_{k}\}_{k=1}^{n}$ is a system of mutually non-overlapping domains, $a_{k}\in B_{k}\subset\overline{\mathbb{C}}$, $k=\overline{0, n}$, $\infty\in B_\infty\subset\overline{\mathbb{C}}$, ($r(B,a)$ is an inner radius of the domain $B\subset\overline{\mathbb{C}}$ at $a\in B$). Instead of the classical condition that the poles are on the unit circle, we require that the system of free poles is an $n$-radial system of points normalized by some "control" functional. A partial solution of this problem was is obtained.

Keywords: inner radius of a domain, non-overlapping domains, radial system of points, separating transformation, quadratic differential, Green's function.

DOI: https://doi.org/10.15393/j3.art.2019.5452

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UDC: 517.54
MSC: 30C75
Revised: 21.09.2018
Accepted:28.12.2018
Language:

Citation: A. K. Bakhtin, I. V. Denega, “Sharp estimates of products of inner radii of non-overlapping domains in the complex plane”, Probl. Anal. Issues Anal., 8(26):1 (2019), 17–31

Citation in format AMSBIB
\Bibitem{BakDen19} \by A.~K.~Bakhtin, I.~V.~Denega \paper Sharp estimates of products of inner radii of non-overlapping domains in the complex plane \jour Probl. Anal. Issues Anal. \yr 2019 \vol 8(26) \issue 1 \pages 17--31 \mathnet{http://mi.mathnet.ru/pa255} \crossref{https://doi.org/10.15393/j3.art.2019.5452} \elib{http://elibrary.ru/item.asp?id=38711937}