Estimates of products of some powers of inner radii for multiconnected domains
Abstract
UDC 517.54
In this paper, we consider the problem of extreme partition of the complex plane, which is well-known in the geometric theory of functions.
We obtain estimates of the maximum value for the product of some powers of inner radii of $n$ disjoint domains in the complex plane with respect to $n$ arbitrary points of the plane.
Estimates found in this paper can be applied to various problems of the geometric theory of functions.
References
V. N. Duby`ny`n, Metod sy`mmetry`zacy`y` v geometry`cheskoj teory`y` funkcy`j kompleksnogo peremennogo, Uspexy` mat. nauk, 49, № 1(295), 3 – 76 (1994).
M. A. Lavrent`ev, K teory`y` konformnyx otobrazheny`j, Tr. Fy`z.-mat. y`n-ta AN SSSR, 5, 159 – 245 (1934).
G. M. Goluzy`n, Geometry`cheskaya teory`ya funkcy`j kompleksnogo peremennogo, Nauka, Moskva (1966).
G. V. Kuz`my`na, K zadache o maksy`mume proy`zvedeny`ya konformnyx rady`usov nenalegayushhy`x oblastej, Zap. nauch.sem. LOMY`, 131 – 145 (1980).
L. Y`. Kolby`na, Nekotorye ekstremal`nye zadachy` v konformnom otobrazheny`y`, Dokl. AN SSSR, ser. mat., 84, № 5, 865 – 868 (1952).
L. Y`. Kolby`na, Konformnoe otobrazheny`e edy`ny`chnogo kruga na nenalegayushhy`e oblasty`, Vestn. Leny`ngrad. un-ta, 5, 37 – 43 (1955).
Z. Nehari, Some inequalities in the theory of functions, Trans. Amer. Math. Soc., 75, № 2, 256 – 286 (1953), https://doi.org/10.2307/1990732 DOI: https://doi.org/10.1090/S0002-9947-1953-0056704-1
N. A. Lebedev, Pry`ncy`p ploshhadej v teory`y` odnoly`stnyx funkcy`j, Nauka, Moskva (1975).
L. L. Gromova, Nekotorye pry`lozheny`ya pry`ncy`pa ploshhadej, Vestn. Leny`ngrad. gos. un-ta, № 7, 31 – 40 (1968).
P. M. Tamrazov, Ekstremal`nye konformnye otobrazheny`ya y` polyusy kvadraty`chnyx dy`fferency`alov, Y`zv.AN SSSR, ser. mat., 32, № 5, 1033 – 1043 (1968).
V. N. Duby`ny`n, Emkosty` kondensatorov y` sy`mmetry`zacy`ya v geometry`cheskoj teory`y` funkcy`j kompleksnogo peremennogo, Dal`nauka DVO RAN, Vlady`vostok (2009).
V. N. Duby`ny`n, Razdelyayushhee preobrazovany`e oblastej y` zadachy` ob ekstremal`nom razby`eny`y`, Zap. nauch. sem. Leny`ngr. otd-ny`ya Mat. y`n-ta AN SSSR, 168, 48 – 66 (1988).
L. V. Kovalev, K zadache ob ekstremal`nom razby`eny`y` so svobodnymy` polyusamy` na okruzhnosty`, Dal`nevost. mat. sb., 2, 96 – 98 (1996).
L. V. Kovalev, O trex neperesekayushhy`xsya oblastyax, Dal`nevost. mat. zhurn., 1, № 1, 3 – 7 (2000).
A. K. Baxty`n, G. P. Baxty`na, Yu. B. Zely`nsky`j, Topologo-algebray`chesky`e struktury y` geometry`chesky`e metody v kompleksnom analy`ze, Praczi In-tu matematy`ky` NAN Ukrayiny`, 73 (2008).
A. K. Baxty`n, Y`. V. Denega, Neravenstva dlya vnutrenny`x rady`usov nenalegayushhy`x oblastej, Ukr. mat. zhurn., 71, № 7, 996 – 1002 (2019).
A. K. Baxty`n, Ocenky` vnutrenny`x rady`usov dlya vzay`mno neperesekayushhy`xsya oblastej, Zb. pracz` In-tu matematy`ky` NAN Ukrayiny`, 14, № 1, 25 – 33 (2017).
A. K. Bakhtin, Separating transformation and extremal problems on nonoverlapping simply connected domains, J. Math. Sci., 234, № 1, 1 – 13 (2018), https://doi.org/10.1007/s10958-018-3976-9 DOI: https://doi.org/10.1007/s10958-018-3976-9
A. K. Bakhtin, Extremal decomposition of the complex plane with restrictions for free poles, J. Math. Sci., 231, № 1, 1 – 15 (2018), https://doi.org/10.1007/s10958-018-3801-5 DOI: https://doi.org/10.1007/s10958-018-3801-5
O. K. Baxtin, I. Ya. Dvorak, Ya. V. Zabolotny`j, Oczinky` dobutku vnutrishnix radiusiv p'yaty` vzayemno neperety`nny`x oblastej, Ukr. mat. zhurn., 69, № 2, 261 – 267 (2017).
A. K. Baxty`n, Y`. V. Denega, Ekstremal`noe razby`eny`e kompleksnoj ploskosty` so svobodnymy` polyusamy`, Ukr. mat. visn., 16, № 3, 307 – 328 (2019). DOI: https://doi.org/10.37069/1810-3200-2019-16-3-1
A. K. Baxty`n, Y`. V. Denega, Ekstremal`noe razby`eny`e kompleksnoj ploskosty` so svobodnymy` polyusamy`, II, Ukr. mat. visn., 16, № 4, 477 – 495 (2019). DOI: https://doi.org/10.37069/1810-3200-2019-16-4-2
A. K. Bakhtin, L. V. Vyhivska, Problem on extremal decomposition of the complex plane with free poles, J. Math. Sci., 248, № 2, 145 – 165 (2020). DOI: https://doi.org/10.1007/s10958-020-04865-3
A. K. Baxty`n, G. P. Baxty`na, Razdelyayushhy`e preobrazovany`ya y` zadachy` o nenalegayushhy`x oblastyax, Zb. pracz` In-tu matematy`ky` NAN Ukrayiny`, 273 – 284 (2006).
A. K. Bakhtin, L. V. Vygivska, I. V. Denega, Inequalities for the internal radii of non-overlapping domains, J. Math. Sci., 220, № 5, 584 – 590 (2017). DOI: https://doi.org/10.1007/s10958-016-3201-7
O. K. Baxtin, Zadacha pro ekstremal`ne rozby`ttya kompleksnoyi ploshhy`ny` z vil`ny`my` polyusamy`, Ukr. mat. zhurn., № 10, 1299 – 1320 (2019).
O. K. Baxtin, I. V. Denega, Oczinky` maksy`mumu dobutkiv vnutrishnix radiusiv oblastej, shho vzayemno ne perety`nayut`sya, Ukr. mat. zhurn., 72, № 2, 173 – 183 (2020).
O. K. Baxtin, L. V. Vy`givs`ka, I. V. Denega, Zadacha pro ekstremal`ne rozby`ttya kompleksnoyi ploshhy`ny` z vil`ny`my` polyusamy` na koli, Ukr. mat. zhurn., 72, № 12, 1599 – 1620 (2020). DOI: https://doi.org/10.37863/umzh.v72i12.6216
A. L. Targonskii, An extremal problem for the nonoverlapping domains, J. Math. Sci., 227, № 1, 98 – 104 (2017). DOI: https://doi.org/10.1007/s10958-017-3576-0
A. L. Targonskii, About one extremal problem for the projections of points on a unit circle, J. Math. Sci., 241, № 1, 90 – 100 (2019). DOI: https://doi.org/10.1007/s10958-019-04409-4
Ya. V. Zabolotny`j, Problema V. M. Dubinina dlya sy`metry`chny`x bagatozv'yazny`x oblastej, Ukr. mat. zhurn., 72, № 11, 1502 – 1509 (2020). DOI: https://doi.org/10.37863/umzh.v72i11.6064
O. K. Baxtin, Ya. V. Zabolotny`j, Oczinky` dobutkiv vnutrishnix radiusiv bagatozv'yazny`x oblastej, Ukr. mat. zhurn., 73, № 1, 9 – 22 (2021), https://doi.org/10.37863/umzh.v73i1.6200 DOI: https://doi.org/10.37863/umzh.v73i1.6200
S. Y`. Fedorov, O maksy`mume odnogo konformnogo y`nvary`anta v zadache o nenalegayushhy`x oblastyax, Zap. nauch. sem. LOMY`, 112, 172 – 183 (1981).
G. Poly`a, G. Sege, Y`zopery`metry`chesky`e neravenstva v matematy`cheskoj fy`zy`ke, Fy`zmatgy`z, Moskva (1962).
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