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IJARM > IJARM Volume 4 > On the One Extremal Problem on the Riemann Sphere
On the One Extremal Problem on the Riemann Sphere
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On the One Extremal Problem on the Riemann Sphere

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Abstract:

Sharp estimates of product of inner radii for pairwise disjoint domains are obtained. In particular, we solve an extremal problem in the case of arbitrary finite number of rays containing arbitrary even number of free poles.

Info:

Periodical:
International Journal of Advanced Research in Mathematics (Volume 4)
Pages:
1-7
DOI:
https://doi.org/10.18052/www.scipress.com/IJARM.4.1
Citation:
A. L. Targonskii and I. Targonskaya, "On the One Extremal Problem on the Riemann Sphere", International Journal of Advanced Research in Mathematics, Vol. 4, pp. 1-7, 2016
Online since:
February 2016
Authors:
Andrey L. Targonskii, Irina Targonskaya
Keywords:
Green Function, Inner Radius of Domain, Logarithmic Capacity, Piecewise-Separating Transformation, Quadratic Differential, Radial Systems of Points, Variational Formula
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Open Access

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This work is licensed under a
Creative Commons Attribution 4.0 International License

References:

[1] Lavrent'ev M.A. On the theory of conformal mappings. Tr. Fiz. -Mat. Inst. Akad. Nauk SSSR. 1934, (5), 159-245. (in Russian).

[2] Goluzin G.M. Geometric theory of functions of a complex variable. Nauka, Moscow, 1966. (in Russian).

[3] Bakhtina G.P. Variational methods and quadratic differentials in problems for disjoint domains. PhD thesis, Kiev, 1975. (in Russian).

[4] Bakhtin A.K., Bakhtina G.P., Zelinskii Yu.B. Topological-algebraic structures and geometric methods in complex analysis. Inst. Math. NAS Ukraine, Kiev, 2008. (in Russian).

[5] Dubinin V.N. Separating transformation of domains and problems of extremal division. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Ros. Akad. Nauk. 1988, (168), 48-66. (in Russian).

[6] Dubinin V.N. Method of symmetrization in the geometric theory of functions of a complex variable. Usp. Mat. Nauk. 1994, 49 1(295), 3-76.

[7] Bakhtin A.K. Inequalities for the inner radii of nonoverlapping domains and open sets. Ukr. Math. J. 2009, 61 (5), 716-733.

[8] Bakhtin A.K., Targonskii A.L. Extremal problems and quadratic differentials. Nonlin. Oscillations. 2005, 8 (3), 296-301.

[9] Targonskii A.L. Extremal problems of partially nonoverlapping domains on a Riemann sphere. Dop. NAN Ukr. 2008, (9), 31-36. (in Russian).

[10] Targonskii A.L. Extremal problems on the generalized (n; d)-equiangular system of points. Anal. St. 2014, XXII (2), 239-251.

[11] Dubinin V.N. Asymptotic representation of the modulus of a degenerating condenser and some its applications. Zap. Nauchn. Sem. Peterburg. Otdel. Mat. Inst. 1997, (237), 56-73. (in Russian).

[12] Dubinin V.N. Capacities of condensers and symmetrization in geometric function theory of complex variables. Dal'nayka, Vladivostok, 2009. (in Russian).

[13] Hayman W.K. Multivalent functions. Cambridge University, Cambridge, (1958).

[14] Jenkins J.A. Univalent functions and conformal mapping. Springer, Berlin, (1958).

[15] Kolbina L.I. Conformal mapping of the unit circle into nonoverlapping domains. Vest. Lening. Univ. 1995, 5, 37-43. (in Russian).

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Cited By:

[1] A. Targonskii, I. Targonskaya, "On the one extremal problem with free poles system points on the rays", Lobachevskii Journal of Mathematics, Vol. 38, p. 525, 2017

DOI: https://doi.org/10.1134/S1995080217030234

[2] A. Targonskii, I. Targonskaya, "Extremal Problems on the Generalized (l, d)-Equiangular System Points in the Case of Arbitrary Multidimensional Complex Spaces", International Journal of Advanced Research in Mathematics, Vol. 9, p. 44, 2017

DOI: https://doi.org/10.18052/www.scipress.com/IJARM.9.44

[3] A. Targonskii, "An extremal problem for the nonoverlapping domains", Journal of Mathematical Sciences, Vol. 227, p. 98, 2017

DOI: https://doi.org/10.1007/s10958-017-3576-0

[4] A. Targonskii, I. Targonskaya, "Extreme problem for partially nonoverlapping domains on a Riemann sphere", Journal of Mathematical Sciences, 2018

DOI: https://doi.org/10.1007/s10958-018-4060-1

[5] A. Targonskii, "On the One Extremal Problem with the Free Poles on the Unit Circle", International Journal of Advanced Research in Mathematics, Vol. 6, p. 26, 2016

DOI: https://doi.org/10.18052/www.scipress.com/IJARM.6.26

[6] A. Targonskii, "About One Extremal Problem for the Projections of Points on a Unit Circle", Journal of Mathematical Sciences, 2019

DOI: https://doi.org/10.1007/s10958-019-04409-4

[7] A. Targonskii, I. Targonskaya, K. Vaschenko, "About one extremal problem for open sets and partially non-overlapping domains", Journal of Mathematical Sciences, 2019

DOI: https://doi.org/10.1007/s10958-019-04604-3

[8] A. Targonskii, I. Targonskaya, K. Vaschenko, "About one extremal problem for open sets and partially non-overlapping domains", Ukrainian Mathematical Bulletin, Vol. 16, p. 228, 2019

DOI: https://doi.org/10.37069/1810-3200-2019-16-2-5

[9] A. Targonskii, I. Targonskaya, "Extreme problem for a mosaic system of points", Journal of Mathematical Sciences, 2021

DOI: https://doi.org/10.1007/s10958-020-05180-7

[10] A. Targonskii, I. Targonskaya, "Extreme problem for a mosaic system of points", Ukrainian Mathematical Bulletin, Vol. 17, p. 437, 2020

DOI: https://doi.org/10.37069/1810-3200-2020-17-3-8

[11] A. Targonskii, "Estimates of generalized internal radii for polycylindrical domains", Journal of Mathematical Sciences, 2022

DOI: https://doi.org/10.1007/s10958-022-05812-0

[12] A. Targonskii, "Estimates of generalized internal radii for polycylindrical domains", Journal of Mathematical Sciences, Vol. 262, p. 222, 2022

DOI: https://doi.org/10.1007/s10958-022-05812-0

[13] A. Targonskii, "Estimates of generalized internal radii for polycylindrical domains", Ukrainian Mathematical Bulletin, Vol. 19, p. 121, 2022

DOI: https://doi.org/10.37069/1810-3200-2022-19-1-8
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