The sharp estimates of the product of the inner radius for pairwise disjoint domains are obtained. In particular, we solve an extremal problem in the case of an arbitrary finite number of the free poles on the unit circle for the following functional (see formula in paper)

Periodical:

International Journal of Advanced Research in Mathematics (Volume 6)

Pages:

26-31

Citation:

A. L. Targonskii, "On the One Extremal Problem with the Free Poles on the Unit Circle", International Journal of Advanced Research in Mathematics, Vol. 6, pp. 26-31, 2016

Online since:

September 2016

Authors:

Keywords:

Distribution:

Open Access

This work is licensed under a

Creative Commons Attribution 4.0 International License

References:

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

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

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

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

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

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

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

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

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

[10] A. Targonskii, Extremal problems on the generalized (n; d)-equiangular system of points, An. St. Univ. Ovidius Constanta. 22(2) (2014) 239-251.

[11] A.L. Targonskii, Extremal problems for partially non-overlapping domains on equiangular systems of points, Bull. Soc. Sci. Lett. Lodz. 63(1) (2013) 57-63.

[12] A. Targonskii, I. Targonskaya, On the One Extremal Problem on the Riemann Sphere, International Journal of Advanced Research in Mathematics. 4 (2016) 1-7.

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

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

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

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

Cited By:

[1] 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