Abstract
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.
Similar content being viewed by others
References
M. A. Lavrent’ev, “On the theory of conformal mappings,” Trudy Sci. Inst. AN SSSR. Otd. Mat., 5, 159–245 (1934).
G. M. Goluzin, Geometric Theory of Functions of a Complex Variable, Amer. Math. Soc., Providence, RI (1969).
G. V. Kuz’mina, “Methods of geometric function theory II,” St. Petersb. Math. J., 9, No. 5, 889–930 (1998).
V. N. Dubinin, “Symmetrization method in geometric function theory of complex variables,” Russian Math. Surveys, 1, No. 1, 1–79 (1994).
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).
J. A. Jenkins, Univalent Functions and Conformal Mappings, Springer, Berlin (1958).
K. Strebel, Quadratic Differentials, Springer, Berlin (1984).
N. A. Lebedev, The Area Principle in the Theory of Univalent Functions [in Russian], Nauka, Moscow (1975).
V. N. Dubinin, Condenser Capacities and Symmetrization in Geometric Function Theory, Birkhäuser/Springer, Basel (2014).
V. N. Dubinin, “Separating transformation of domains and problems on extremal decomposition,” J. Soviet Math., 53, No. 3, 252–263 (1991).
L. V. Kovalev, “On the problem of extremal decomposition with free poles on a circle,” Dal’nevost. Mat. Sb., No. 2, 96–98 (1996).
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).
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).
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).
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).
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).
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).
A. K. Bakhtin, “Estimates of inner radii for mutually disjoint domains,” Zb. Prats Inst. Mat. NANU, 14, No. 1, 1–9 (2017).
B. V. Shabat, Introduction to Complex Analysis, Part II, Amer. Math. Soc., Providence, RI (1992).
E. M. Chirka, Complex Analytic Sets [in Russian], Nauka, Moscow (1985).
B. A. Fuks, Introduction to the Theory of Analytic Functions of Several Complex Variables [in Russian], GIFML, Moscow (1962).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 15, No. 3, pp. 431–441, July–September, 2018.
Rights and permissions
About this article
Cite this article
Zabolotnii, Y., Denega, I. Extremal decomposition of a multidimensional complex space for five domains. J Math Sci 241, 101–108 (2019). https://doi.org/10.1007/s10958-019-04410-x
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-019-04410-x