We study the problem of nonoverlapping domains with free poles on radial systems. Our main results strengthen and generalize several known results obtained in the investigation of this problem earlier.
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A. K. Bakhtin, G. P. Bakhtina, and Yu. B. Zelinskii, Topological-Algebraic Structures and Geometric Methods in Complex Analysis [in Russian], Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (2008).
A. K. Bakhtin, “Estimates for inner radii of mutually nonoverlapping domains,” in: Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv, 14, No. 1 (2017), pp. 25–33.
A. K. Bakhtin, V. E. V’yun, and A. L. Targonskii, “Inequalities for the inner radii of mutually nonoverlapping domains,” in: Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv, 12, No. 3, (2015), pp 38–46.
A. K. Bakhtin and I. V. Denega, “On one Dubinin’s problem,” in: Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv, 10, No. 4-5, (2013), pp. 396–406.
G. P. Bakhtina, “On the conformal radii of symmetric nonoverlapping domains,” in: Contemporary Problems of Real and Complex Analysis, Institute of Mathematics, Academy of Sciences of Ukr. SSR, Kiev (1984), pp. 21–27.
G. M. Goluzin, Geometric Theory of Functions of Complex Variable [in Russian], Nauka, Moscow (1966).
V. N. Dubinin, “On the product of inner radii of ‘partially nonoverlapping’ domains,” in: Problems of the Metric Theory of Mappings and Its Application [in Russian], Naukova Dumka, Kiev (1978), pp. 24–31.
V. N. Dubinin, “Separating transformation of domains and the problems of extreme decomposition,” Zap. Nauch. Sem. Leningrad. Otdel. Mat. Inst. Akad. Nauk SSSR, 168, 48–66 (1988).
V. N. Dubinin, “Symmetrization in the geometric theory of functions of complex variable,” Usp. Mat. Nauk, 49, No. 1, 3–76 (1994).
V. N. Dubinin, Capacitances of Capacitors and Symmetrization in the Geometric Theory of Functions of Complex Variable [in Russian], Dal’nauka, Vladivostok (2009).
J. A. Jenkins, Univalent Functions and Conformal Mapping [Russian translation], Inostrannaya Literatura, Moscow (1962).
E. G. Emel’yanov, “On the problem of maximum of the product of powers of conformal radii of nonoverlapping domains,” Zap. Nauch. Sem. POMI, 286, 103–114 (2002).
Ya. V. Zabolotnyi, “Application of the separating transformation in one problem of nonoverlapping domains,” Dop. Nats. Akad. Nauk Ukr., No. 9, 11–14 (2011).
Ya. V. Zabolotnyi, “Determination of the maximum of product of the inner radii of mutually nonoverlapping domains,” Dop. Nats. Akad. Nauk Ukr., No. 3, 7–13 (2016).
L. V. Kovalev, “On the problem of extreme decomposition with free poles on a circle,” Dal’nevost. Mat. Sb., 2, 96–98 (1996).
L. I. Kolbina, “Conformal mapping of a unit disk onto nonoverlapping domains,” Vestn. Leningrad. Univ., 5, 37–43 (1955).
G. V. Kuz’mina, “Methods of the geometric theory of functions. I, II,” Algebra Analiz, 9, No. 3, 41–103 (1997); No. 5, 1–50 (1997).
G. V. Kuz’mina, “Problems of extreme decomposition of the Riemannian sphere,” Zap. Nauch. Sem. POMI, 276, 253–275 (2001).
G. V. Kuz’mina, “Method of extreme metric in the problems of maximum of the product of powers of the conformal radii of nonoverlapping domains in the presence of free parameters,” Zap. Nauch. Sem. POMI, 302, 52–67 (2003).
M. A. Lavrent’ev, “On the theory of conformal mappings,” Tr. Fiz.-Mat. Akad. Nauk SSSR, 5, 159–245 (1934).
P. M. Tamrazov, “Extreme conformal mappings and the poles of quadratic differentials,” Izv. Akad. Nauk SSSR, Ser. Mat., 32, No. 5, 1033–1043 (1968).
A. Bakhtin, “Extremal decomposition of the complex plane with restrictions for free poles,” J. Math. Sci., 231, No. 1, 1–15 (2018).
A. Bakhtin, “Separating transformation and extremal problems on nonoverlapping simply connected domains,” J. Math. Sci., 234, No. 1, 1–13 (2018).
A. Bakhtin, L. Vyhivska, and I. Denega, “N-radial systems of points and problems for nonoverlapping domains,” Lobachevskii J. Math., 38, No. 2, 229–235 (2017).
A. Bakhtin, L. Vyhivska, and I. Denega, “Inequalities for the internal radii of nonoverlapping domains,” J. Math. Sci., 220, No. 5, 584–590 (2017).
L. Vyhivska, “Some inequalities for inner radii of partially overlapping domains,” Zbirn. Prats Inst. Math. Nat. Acad. Sci. Ukraine, 14, No. 1, 82–89 (2017).
Ya. Zabolotnii and I. Dvorak, “Some evaluation of maximum of the product of conformal radii for pairwise nonoverlapping domains,” Lobachevskii J. Math., 38, No. 3, 554–559 (2017).
E. S. Afanas’eva, “Generalized quasiisometries on smooth Riemannian manifolds,” Math. Notes, 102, No. 1-2, 12–21 (2017).
A. Golberg, R. Salimov, and E. Sevost’yanov, “Singularities of discrete open mappings with controlled p-module,” J. Anal. Math., 127, 303–328 (2015).
V. Ya. Gutlyanskii, V. I. Ryazanov, and E. Yakubov, “The Beltrami equations and prime ends,” J. Math. Sci., 210, No. 1, 22–51 (2015).
B. A. Klishchuk and R. R. Salimov, “Lower bounds for the area of the image of a circle,” Ufa Math. J., 9, No. 2, 55–61 (2017).
D. A. Kovtonyuk, V. I. Ryazanov, R. R. Salimov, and E. A. Sevost’yanov, “Toward the theory of Orlicz–Sobolev classes,” St.Petersburg Math. J., 25, No. 6, 929–963 (2014).
V. I. Ryazanov, R. R. Salimov, and E. A. Sevost’yanov, “On convergence analysis of space homeomorphisms,” Sib. Adv. Math., 23, No. 4, 263–293 (2013).
R. R. Salimov and E. A. Sevost’yanov, “Analogs of the Ikoma–Schwartz lemma and Liouville theorem for mappings with unbounded characteristic,” Ukr. Mat., Zh., 63, No. 10, 1368–1380 (2011); English translation:Ukr. Math. J., 63, No. 10, 1551–1565 (2012).
E. A. Sevost’yanov, “On the integral characterization of some generalized quasiregular mappings and the significance of the conditions of divergence of integrals in the geometric theory of functions,” Ukr. Mat., Zh., 61, No. 10, 1367–1380 (2009); English translation:Ukr. Math. J., 61, No. 10, 1610–1623 (2009).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 71, No. 10, pp. 1299–1320, October, 2019.
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Bakhtin, A.K. A Problem of Extreme Decomposition of the Complex Plane with Free Poles. Ukr Math J 71, 1485–1509 (2020). https://doi.org/10.1007/s11253-020-01727-8
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DOI: https://doi.org/10.1007/s11253-020-01727-8