Extremal decomposition of a multidimensional complex space for five domains

  • Yaroslav ZabolotniiEmail author
  • Iryna Denega


The paper is devoted to one open extremal problem in the geometric function theory of complex variables associated with estimates of a functional defined on the systems of non-overlapping domains. We consider the problem of the maximum of a product of inner radii of n non-overlapping domains containing points of a unit circle and the power γ of the inner radius of a domain containing the origin. The problem was formulated in 1994 in Dubinin’s paper in the journal “Russian Mathematical Surveys” in the list of unsolved problems and then repeated in his monograph in 2014. Currently, it is not solved in general. In this paper, we obtained a solution of the problem for five simply connected domains and power γ (1; 2:57] and generalized this result to the case of multidimensional complex space.


Inner radius of the domain non-overlapping domains polycylindrical domain radial system of points separating transformation quadratic differential Green’s function 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M. A. Lavrent’ev, “On the theory of conformal mappings,” Trudy Sci. Inst. AN SSSR. Otd. Mat., 5, 159–245 (1934).Google Scholar
  2. 2.
    G. M. Goluzin, Geometric Theory of Functions of a Complex Variable, Amer. Math. Soc., Providence, RI (1969).Google Scholar
  3. 3.
    G. V. Kuz’mina, “Methods of geometric function theory II,” St. Petersb. Math. J., 9, No. 5, 889–930 (1998).Google Scholar
  4. 4.
    V. N. Dubinin, “Symmetrization method in geometric function theory of complex variables,” Russian Math. Surveys, 1, No. 1, 1–79 (1994).CrossRefzbMATHGoogle Scholar
  5. 5.
    A. K. Bakhtin, G. P. Bakhtina, and Yu. B. Zelinskii, Topological-Algebraic Structures and Geometric Methods in Complex Analysis [in Russian], Inst. of Math. of the NASU, Kiev (2008).Google Scholar
  6. 6.
    J. A. Jenkins, Univalent Functions and Conformal Mappings, Springer, Berlin (1958).Google Scholar
  7. 7.
    K. Strebel, Quadratic Differentials, Springer, Berlin (1984).CrossRefzbMATHGoogle Scholar
  8. 8.
    N. A. Lebedev, The Area Principle in the Theory of Univalent Functions [in Russian], Nauka, Moscow (1975).Google Scholar
  9. 9.
    V. N. Dubinin, Condenser Capacities and Symmetrization in Geometric Function Theory, Birkhäuser/Springer, Basel (2014).Google Scholar
  10. 10.
    V. N. Dubinin, “Separating transformation of domains and problems on extremal decomposition,” J. Soviet Math., 53, No. 3, 252–263 (1991).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    L. V. Kovalev, “On the problem of extremal decomposition with free poles on a circle,” Dal’nevost. Mat. Sb., No. 2, 96–98 (1996).Google Scholar
  12. 12.
    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,” J. of Math. Sci., 129, No. 3, 3843–3851 (2005).MathSciNetCrossRefGoogle Scholar
  13. 13.
    A. K. Bakhtin and I. V. Denega, “Addendum to a theorem on extremal decomposition of the complex plane,” Bull. Société Sci. et Lettres de Lódź, 62, No. 2, 83–92 (2012).MathSciNetzbMATHGoogle Scholar
  14. 14.
    Ja. V. Zabolotnij, “Determination of the maximum of a product of inner radii of pairwise nonoverlapping domains,” Dopov. Nac. Akad. Nauk Ukr., No. 3, 7–13 (2016).Google Scholar
  15. 15.
    A. K. Bakhtin, I. Ya. Dvorak, and Ya. V. Zabolotnyi, “Estimates of the product of inner radii of five nonoverlapping domains,” Ukr. Mat. Zh., 69, No. 2, 261–267 (2017).Google Scholar
  16. 16.
    I. V. Denega and Ya. V. Zabolotnii, “Estimates of products of inner radii of non-overlapping domains in the complex plane,” Complex Var. Ellipt. Equa., 62, No. 11, 1611–1618 (2017).Google Scholar
  17. 17.
    A. Bakhtin, L. Vygivska, and I. Denega, “N-radial systems of points and problems for non-overlapping domains,” Lobachevskii J. of Math., 38, No. 2, 229–235 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    A. K. Bakhtin, “Estimates of inner radii for mutually disjoint domains,” Zb. Prats Inst. Mat. NANU, 14, No. 1, 1–9 (2017).MathSciNetGoogle Scholar
  19. 19.
    B. V. Shabat, Introduction to Complex Analysis, Part II, Amer. Math. Soc., Providence, RI (1992).Google Scholar
  20. 20.
    E. M. Chirka, Complex Analytic Sets [in Russian], Nauka, Moscow (1985).zbMATHGoogle Scholar
  21. 21.
    B. A. Fuks, Introduction to the Theory of Analytic Functions of Several Complex Variables [in Russian], GIFML, Moscow (1962).Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of Mathematics of the National Academy of Sciences of UkraineKyivUkraine

Personalised recommendations