Abstract
Problems on extremal decomposition of the complex plane with free poles located on an (n;m)-ray system of points are studied. A method that allowed us to obtain new upper bounds for the maximum of the products of the inner radii of mutually non-overlapping domains is proposed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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.
V. N. Dubinin, “Symmetrization method in geometric function theory of complex variables,” Russian Math. Surv., 1, 1–79 (1994).
V. N. Dubinin, Condenser Capacities and Symmetrization in Geometric Function Theory, Birkhäuser/Springer, Basel, 2014.
A. K. Bakhtin, “Separating transformation and extremal problems on nonoverlapping simply connected domains,” J. Math. Sci., 234, No. 1, 1–13 (2018).
G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Princeton Univ. Press, Princeton, 1951.
G. M. Goluzin, Geometric Theory of Functions of a Complex Variable, Amer. Math. Soc., Providence, R.I., 1969.
L. V. Kovalev, “On the problem of extremal decomposition with free poles on a circle,” Dal’nevost. Mat. Sb., No. 2, 96–98 (1996).
L. I. Kolbina, “Conformal mapping of a unit circle on nonoverlapping domains,” Vestn. Lenin. Univ., 5, 37–43 (1955).
A. K. Bakhtin and I. V. Denega, “The method of separating transformation in problems of maximum of the product of powers of the inner radii of non-overlopping domains,” Zbir. Prats’ Inst. Matem. NANU, 9, No. 2, 32–44 (2012).
A. K. Bakhtin, “Extremal decomposition of the complex plane with restrictions for free poles,” J. Math. Sci., 231, No. 1, 1–15 (2018).
G. V. Kuz’mina, “The method of extremal metric in extremal decomposition problems with free parameters,” J. 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. Soc. Sci. Lett. Lódź, Rech. Déform., 62, No. 2, 83–92 (2012).
A. K. Bakhtin and I. V. Denega, “Inequalities for inner radii of non-overlopping domains,” Ukr. Mat. Zh., 71, No. 7, 979–985 (2019).
I. Denega, “Estimates of the inner radii of non-overlapping domains,” J. Math. Sci., 242, No. 6, 787–795 (2019).
P. M. Tamrazov, “Extremal conformal mappings and poles of quadratic differentials,” Math. USSR-Izv., 2, No. 5, 987–996 (1968).
I. V. Denega, “Generalization of some extremal problems on non-overlapping domains with free poles,” Ann. Univ. Marie Curie–Sklodowska, 67, No. 1, 11–22 2013.
I. V. Denega and Ya. V. Zabolotnii, “Estimates of products of inner radii of non-overlapping domains in the complex plane,” Compl. Var. Ellipt. Equ., 62, No. 11, 1611–1618 (2018).
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to the century of the birthday of G. D. Suvorov
Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 16, No. 4, pp. 477–495 October–December, 2019.
Rights and permissions
About this article
Cite this article
Bakhtin, A.K., Denega, I.V. Extremal Decomposition of the Complex Plane with Free Poles II. J Math Sci 246, 602–616 (2020). https://doi.org/10.1007/s10958-020-04766-5
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-020-04766-5