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On the One Extremal Problem on the Riemann Sphere

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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.


International Journal of Advanced Research in Mathematics (Volume 4)
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

[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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