Оцінки максимума добутків внутрішніх радіусів областей, що взаємно не перетинаються

O. K. Bakhtin; I. V. Denega

O. K. Bakhtin Institute of Mathematics, National Academy of Sciences of Ukraine
I. V. Denega Institute of Mathematics, National Academy of Sciences of Ukraine, Kiev

Keywords: inner radius of domain, non-overlapping domains, the Green function, (n, m)-radial system of points, transfinite diameter, theorem on minimizing

Abstract

UDC 517.54

We establish effective upper estimates for the maximum products of the inner radii of mutually disjoint domains in the $(n,m)$-radial systems of points of the complex plane for all possible values of a certain parameter $\gamma.$ We also obtain conditions under which the structure of points and domains is not important for our investigations.

References

Bakhtin, A. K.; Bakhtina, G. P.; Zelinskiĭ, Yu. B. Тополого-алгебраические структуры и геометрические методы в комплексном анализе. (Russian) [[Topological-algebraic structures and geometric methods in complex analysis]] Працī Інституту Математики Нацīональноï Академīï Наук Украïни. Математика та ïï Застосування [Proceedings of Institute of Mathematics of NAS of Ukraine. Mathematics and its Applications], 73. Natsīonalʹna Akademīya Nauk Ukraïni, Īnstitut Matematiki, Kiev, 2008. 308 pp. ISBN: 978-966-02-4763-5 MR2467439

Dubinin, V. N. Symmetrization in the geometric theory of functions of a complex variable. (Russian) ; translated from Uspekhi Mat. Nauk 49 (1994), no. 1(295), 3--76 Russian Math. Surveys 49 (1994), no. 1, 1-79 doi: 10.1070/RM1994v049n01ABEH002002

Dubinin, Vladimir N. Condenser capacities and symmetrization in geometric function theory. Translated from the Russian by Nikolai G. Kruzhilin. Springer, Basel, 2014. xii+344 pp. ISBN: 978-3-0348-0842-2; 978-3-0348-0843-9 doi: 10.1007/978-3-0348-0843-9

Bakhtin, Aleksandr K. A separating transformation and extremal problems on nonoverlapping simply connected domains. (Russian) ; translated from Ukr. Mat. Visn. 14 (2017), no. 4, 456--471, 605 J. Math. Sci. (N.Y.) 234 (2018), no. 1, 1--13 doi: 10.1007/s10958-018-3976-9

Bakhtin, Aleksandr K. Extremal decomposition of the complex plane with restrictions for the free poles. (Russian) ; translated from Ukr. Mat. Visn. 14 (2017), no. 3, 309--329, 452 J. Math. Sci. (N.Y.) 231 (2018), no. 1, 1--15 doi: 10.1007/s10958-018-3801-5

G. Polia, G. Sege, Изопериметрические неравенства в математической физике. (Russian)[[Izoperimetricheskie neravenstva v matematicheskoj fizike]], Fizmatgiz., Moskva 1962.

Goluzin, G. M. Геометрическая теория функций комплексного переменного. (Russian) [[Geometrical theory of functions of a complex variable]] Second edition. Edited by V. I. Smirnov. With a supplement by N. A. Lebedev, G. V. Kuzmina and Ju. E. Alenicyn Izdat. ``Nauka'', Moscow 1966 628 pp. (1 plate; errata insert). MR0219714

G. P. Bakhtina, О конформных радиусах симметричных неналегающих областей. (Russian) [[O konformny`kh radiusakh simmetrichny`kh nenalegayushhikh oblastej]], Sovremeny`e voprosy` veshhestvennogo i kompleksnogo analiza, In-t matematiki AN USSR, Kiev (1984), s. 21 – 27.

A. L. Targons`kij, Екстремальнi задачi теорiї однолистих функцiй. (Russian) [[Ekstremal`ni zadachi teoriyi odnolistikh funkczij]], Dis. . . . kand. fiz.-mat. nauk, Kiyiv (2006).

A. K. Bakhtin, A. L. Targonskij, Обобщенные $(n, d)$-лучевые системы точек и неравенства для неналегающих областей и открытых множеств. (Russian) [[Obobshhenny`e $(n, d)$-luchevy`e sistemy` tochek i neravenstva dlya nenalegayushhikh oblastej i otkry`ty`kh mnozhestv]], Ukr. mat. zhurn., 63, № 7, 867 – 879 (2011).

I. V. Denega, A. L. Targonskii, Separating transformation in a problem on extremal decomposition of the complex plane, Zb. pracz` In-tu matematiki NAN Ukrayini, 14, № 1, 147 – 155 (2017).

Targonskii, Andrey L. About one extremal problem for projections of the points on unit circle. ; translated from Ukr. Mat. Visn. 15 (2018), no. 3, 418--430 J. Math. Sci. (N.Y.) 241 (2019), no. 1, 90--100 doi: 10.1007/s10958-019-04409-4

Targonskii, Andrey. An extremal problem for the non-overlapping domains. ; translated from Ukr. Mat. Visn. 14 (2017), no. 1, 126--134 J. Math. Sci. (N.Y.) 227 (2017), no. 1, 98--104 doi: 10.1007/s10958-017-3576-0

A. K. Bakhtin, I. V. Denega, Неравенства для внутренних радиусов неналегающих областей. (Russian) [[Neravenstva dlya vnutrennikh radiusov nenalegayushhikh oblastej]], Ukr. mat. zhurn., 71, № 7, 979 – 985 (2019).

Denega, Iryna. Estimates of the inner radii of non-overlapping domains. ; translated from Ukr. Mat. Visn. 16 (2019), no. 1, 46--56 J. Math. Sci. (N.Y.) 242 (2019), no. 6, 787–795 MR3967770

P. M. Tamrazov, Экстремальные конформные отображения и полюсы квадратичных дифференциалов. (Russian) [[E`kstremal`ny`e konformny`e otobrazheniya i polyusy` kvadratichny`kh differenczialov]], Izv. AN SSSR, ser. mat., 32, № 5, 1033 – 1043 (1968).

Kuzʹmina, G. V. Problems of the extremal decomposition of the Riemann sphere. (Russian) ; translated from Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 276 (2001), Anal. Teor. Chisel i Teor. Funkts. 17, 253–275, 352–353 J. Math. Sci. (N.Y.) 118 (2003), no. 1, 4880-4894 doi: 10.1023/A:1025580802209

Denega, Iryna V. Generalization of some extremal problems on non-overlapping domains with free poles. Ann. Univ. Mariae Curie-Skłodowska Sect. A 67 (2013), no. 1, 11–22. doi: 10.2478/v10062-012-0018-9

Denega, I. V.; Zabolotnii, Ya. V. Estimates of products of inner radii of non-overlapping domains in the complex plane. Complex Var. Elliptic Equ. 62 (2017), no. 11, 1611–1618. doi: 10.1080/17476933.2016.1265952

Targonskiĭ, AndreĭL.; Targonskaya, Irina I. An extremal problem for partially nonoverlapping domains on a Riemann sphere. (Russian) ; translated from Ukr. Mat. Visn. 15 (2018), no. 1, 94–102, 150–151 J. Math. Sci. (N.Y.) 235 (2018), no. 1, 74--80 MR3838843

Estimation of the maximum product of inner radii of mutually disjoint domains

Abstract

References