Abstract
We consider an open extremal problem in geometric function theory of complex variables on the maximum of the functional
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. N. Dubinin, “Symmetrization method in geometric function theory of complex variables,” Russ. Math. Surveys, 1, 1–79 (1994).
V. N. Dubinin, Condenser Capacities and Symmetrization in Geometric Function Theory, Birkhäuser/Springer, Basel, 2014.
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).
L. V. Kovalev, “On the problem of extremal decomposition with free poles on a circle,” Dal’nevost. Mat. Sb.,, No. 2, 96–98 (1996).
A. K. Bakhtin and I. V. Denega, “Addendum to a theorem on extremal decomposition of the complex plane,” Bull. Soc. Sci. Lett. de Łódź, Rech. Déform., 62, No. 2, 83–92 (2012).
L. I. Kolbina, “Conformal mapping of a unit circle onto nonoverlapping domains,” Vestn. Lenin. Univ., 5, 37–43 (1955).
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 NASU, Kiev, 2008.
A. K. Bakhtin and I. V. Denega, “Inequalities for the inner radii of nonoverlapping domains,” Ukr. Mat. Zh., 71 (2019) (in press).
L. V. Kovalev, “On the inner radii of symmetric nonoverlapping domains,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 6, 80–81 (2000).
L. V. Kovalev, “On three disjoint domains,” Dal’nevost. Mat. Zh., 1, No. 1, 3–7 (2000).
A. K. Bakhtin and Y. V. Zabolotnyi, “Estimates of a product of the inner radii of nonoverlapping domains,” Ukr. Mat. Bull., 13, No. 2, 148–156 (2016).
Ya. V. Zabolotnii and L. V. Vyhivska, “On a product of the inner radii of symmetric multiply connected domains,” Ukr. Math. Bull., 14, No. 3, 440–451 (2017).
A. K. Bakhtin and L. V. Vyhivska, “Estimates of the inner radii of symmetric nonoverlapping domains,” Ukr. Math. Bull., 15, No. 3, 298–320 (2018).
I. Denega, “Estimates of the inner radii of non-overlapping domains,” Ukr. Math. Bull., 16, No. 1, 77–87 (2019).
A. Bakhtin, L. Vyhivska, and I. Denega, “Inequality for the inner radii of symmetric non-overlapping domains,” Bull. Soc. Sci. Lett. de Łódź, Rech. Déform., 68, No. 2, 37–44 (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.
P. M. Tamrazov, “Extremal conformal mappings and poles of quadratic differentials,” Math. of the USSRIzv., 2, No. 5, 987–996 (1968).
A. K. Bakhtin, “Separating transformation and extremal problems on nonoverlapping simply connected domains,” J. Math. Sci., 234, No. 1, 1–13 (2018).
A. K. Bakhtin, “Extremal decomposition of the complex plane with restrictions for free poles,” J. Math. Sci., 231, No. 1, 1–15 (2018).
I. V. Denega, “Generalization of some extremal problems on non-overlapping domains with free poles,” Ann. Univ. M. 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. 3, pp. 307–328 July–September, 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. J Math Sci 246, 1–17 (2020). https://doi.org/10.1007/s10958-020-04718-z
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-020-04718-z