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This article is cited in 3 scientific papers (total in 3 papers)
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
$$
\left[r\left(B_0,0\right)r\left(B_\infty,\infty\right)\right]^{\gamma} \prod\limits_{k=1}^n r\left(B_k,a_k\right),
$$
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.
Received: 19.09.2018 Revised: 21.09.2018 Accepted: 28.12.2018
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
Linking options:
https://www.mathnet.ru/eng/pa255 https://www.mathnet.ru/eng/pa/v26/i1/p17
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