Sharp estimates of products of inner radii of non-overlapping domains in the complex plane
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 γ ∈ R + and find the maximum (and describe all extremals) of the functional [r (B 0 , 0) r (B ∞ , ∞)] γ Π n k=1 r (B k , a k ) , w...
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Petrozavodsk State University
2019-01-01
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doaj-5159c8eba72f4e5081ed096fa39a164d2021-07-02T06:36:09ZengPetrozavodsk State UniversityПроблемы анализа2306-34242306-34322019-01-018(26)1173110.15393/j3.art.2019.5452Sharp estimates of products of inner radii of non-overlapping domains in the complex planeBakhtin A. K.0Denega I. V.1Institute of Mathematics of the National Academy of Sciences of UkraineInstitute of Mathematics of the National Academy of Sciences of UkraineIn 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 γ ∈ R + and find the maximum (and describe all extremals) of the functional [r (B 0 , 0) r (B ∞ , ∞)] γ Π n k=1 r (B k , a k ) , where n ∈ N, n >= 2, a 0 = 0, |a k | = 1, B 0 , B ∞ , {B k } n k=1 is a system of mutually non-overlapping domains, a k ∈ B k ⊂ C, k = 0, n, ∞ ∈ B ∞ ⊂ C, (r(B, a) is an inner radius of the domain B ⊂ C at a ∈ 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 is obtained.http://issuesofanalysis.petrsu.ru/article/genpdf.php?id=5452&lang=ruinner radius of a domainnon-overlapping domainsradial system of pointsseparating transformationquadratic differentialGreen’s function |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Bakhtin A. K. Denega I. V. |
spellingShingle |
Bakhtin A. K. Denega I. V. Sharp estimates of products of inner radii of non-overlapping domains in the complex plane Проблемы анализа inner radius of a domain non-overlapping domains radial system of points separating transformation quadratic differential Green’s function |
author_facet |
Bakhtin A. K. Denega I. V. |
author_sort |
Bakhtin A. K. |
title |
Sharp estimates of products of inner radii of non-overlapping domains in the complex plane |
title_short |
Sharp estimates of products of inner radii of non-overlapping domains in the complex plane |
title_full |
Sharp estimates of products of inner radii of non-overlapping domains in the complex plane |
title_fullStr |
Sharp estimates of products of inner radii of non-overlapping domains in the complex plane |
title_full_unstemmed |
Sharp estimates of products of inner radii of non-overlapping domains in the complex plane |
title_sort |
sharp estimates of products of inner radii of non-overlapping domains in the complex plane |
publisher |
Petrozavodsk State University |
series |
Проблемы анализа |
issn |
2306-3424 2306-3432 |
publishDate |
2019-01-01 |
description |
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 γ ∈ R + and find the maximum (and describe all extremals) of the functional [r (B 0 , 0) r (B ∞ , ∞)] γ Π n k=1 r (B k , a k ) , where n ∈ N, n >= 2, a 0 = 0, |a k | = 1, B 0 , B ∞ , {B k } n k=1 is a system of mutually non-overlapping domains, a k ∈ B k ⊂ C, k = 0, n, ∞ ∈ B ∞ ⊂ C, (r(B, a) is an inner radius of the domain B ⊂ C at a ∈ 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 is obtained. |
topic |
inner radius of a domain non-overlapping domains radial system of points separating transformation quadratic differential Green’s function |
url |
http://issuesofanalysis.petrsu.ru/article/genpdf.php?id=5452&lang=ru |
work_keys_str_mv |
AT bakhtinak sharpestimatesofproductsofinnerradiiofnonoverlappingdomainsinthecomplexplane AT denegaiv sharpestimatesofproductsofinnerradiiofnonoverlappingdomainsinthecomplexplane |
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1721336996242128896 |