Problem on extremal decomposition of the complex plane

Authors

  • Iryna Denega Institute of Mathematics, National Academy of Sciences of Ukraine

DOI:

https://doi.org/10.26485/0459-6854/2018/68.3/6

Keywords:

inner radius of domain, non-overlapping domains, radial system of points, separating transformation, quadratic differential, Green’s function

Abstract

The paper is devoted to one extremal problem in geometric function theory of complex variables associated with estimates of functionals defined on the systems of non-overlapping domains. We consider Dubinin’s problem of the maximum of product of inner radii of n non-overlapping domains containing points of the unit circle and the power of the inner radius of a domain containing the origin. The problem was formulated in 1994 in the work of Dubinin and then repeated in his monograph in 2014. Currently it is not solved in general. In this paper we generalized it to the case of the more general system of points and obtained a solution of this problem for some concrete values of n and y.

References

M. A. Lavrentev, On the theory of conformal mappings, Tr. Sci. Inst An USSR, 5 (1934), 159-245. (in Russian)

G. M. Goluzin, Geometric theory of functions of a complex variable, Amer. Math. Soc. Providence, R.I. 1969.

G. V. Kuzmina,, Methods of geometric function theory, II, St. Petersbg. Math. J. 5 (1997), 889-930.

L. I. Kolbina, Conformal mapping of the unit circle onto non-overlapping domains, Vestnik Leningrad. Univ., 10(5) (1955), 37-43. (in Russian)

N. A. Lebedev, The area principle in the theory of univalent functions, Moscow: Science, 1975. (in Russian)

J. A. Jenkins, Univalent Functions and Conformal Mappings, Berlin: Springer, 1962.

Strebel K. Quadratic differentials, Springer-Verlag, 1984.

V. N. Dubinin, Symmetrization in the geometric theory of functions of a complex variable, Russian Mathematical Surveys, 49(1) (1994), 1-79.

V. N. Dubinin, Condenser capacities and symmetrization in geometric function theory, Birkhauser/Springer, Basel, 2014.

I. V. Denega, Generalization of some extremal problems on non-overlapping domains with free poles, Annales universitatis Mariae Curie-Skladovska, Lublin-Polonia, 67(1) (2013), 11-22.

A. K. Bakhtin, L. V. Vygivska, I. V. Denega, Inequalities for the internal radii of nonoverlapping domains, Journal of Mathematical Sciences, 220(5) (2017), 584-590.

A. Bakhtin, L. Vygivska and I. Denega, N-Radial Systems of Points and Problems for Non-Overlapping Domains, Lobachevskii Journal of Mathematics, 38(2) (2017), 229-235.

A. K. Bakhtin, I. V. Denega, About one problem of V.N. Dubinin, Zb. prats of the Inst. of Math. of NASU, 10(4-5) (2013), 401-411. (in Russian)

V. N. Dubinin, Separating transformation of domains and problems on extremal decomposition, J. Soviet Math., 53(3) (1991), 252-263.

L. V. Kovalev, To the ploblem of extremal decomposition with free poles on a circumference, Dalnevost. Mat. Sborn., 2 (1996), 96-98.

G. V. Kuzmina, Extremal metric method in problems of the maximum of product of powers of conformal radii of non-overlapping domains with free parameters, Journal of Mathematical Sciences, 129(3) (2005), 3843-3851.

A. K. Bakhtin, G.P. Bakhtina, Yu. B. Zelinskii, Topological-Algebraic Structures and Geometric Methods in Complex Analysis, K.: Inst. Math. of NAS of Ukraine, 2008.

Ja. V. Zabolotnij, Determination of the maximum of a product of inner radii of pairwise nonoverlapping domains, Dopov. Nac. akad. nauk Ukr., 3 (2016), 7-13.

A. Bakhtin, I. Dvorak and I. Denega, Separating transformation and extremal decomposition of the complex plane, Bulletin de la Societe des sciences et des lettres de Lodz, Recherches sur les deformations, 66(2) (2016), 13-20.

Downloads

Published

2019-04-26

Issue

Section

Articles