We generalize some classical results in the theory of extreme problems for nonoverlapping domains.
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References
M. A. Lavrent’ev, “On the theory of conformal mappings,” Trudy Fiz.-Mat. Inst. AN SSSR, 5, 159–245 (1934).
G. M. Goluzin, Geometric Theory of Functions of Complex Variable [in Russian], Nauka, Moscow (1966).
J. A. Jenkins, Univalent Functions and Conformal Mappings, Springer, Berlin (1958).
W. K. Hayman, Multivalent Functions, Cambridge Univ. Press, Cambridge (1994).
V. N. Dubinin, “On the product of inner radii of “partially nonoverlapping” domains,” in: Problems of the Metric Theory of Mappings and Its Applications [in Russian], Naukova Dumka, Kiev (1978), pp. 24–31.
V. N. Dubinin, “Disconnecting transformation of domains and the problems of extreme partition,” Zap. Nauch. Sem. Leningr. Otd. Mat. Inst. Akad. Nauk SSSR, 168, 48–66 (1988).
V. N. Dubinin, “Method of symmetrization in the geometric theory of functions of complex variable,” Uspekhi Mat. Nauk, 49, No. 1(295), 3–76 (1994).
V. N. Dubinin, “Asymptotics of the modulus of a degenerate condenser and some its applications,” Zap. Nauch. Sem. Leningr. Otd. Mat. Inst. Akad. Nauk SSSR, 237, 56–73 (1997).
E. G. Emel’yanov, “On the relationship between two problems of extreme partition,” Zap. Nauch. Sem. Leningr. Otd. Mat. Inst. Akad. Nauk SSSR, 160, 91–98 (1987).
P. P. Kufarev and A. E. Fales, “On one extreme problem for additional domains,” Dokl. Akad. Nauk SSSR, Ser. Mat., 81, No. 6, 995–998 (1951).
A. K. Bakhtin, “Estimates of functionals for open sets,” Nelin. Kolyv., 8, No. 2, 147–153 (2005).
A. K. Bakhtin, “On some extreme problems of the geometric theory of functions of complex variable,” Dop. Nats. Akad. Nauk Ukr., No. 9, 7–11 (2006).
A. K. Bakhtin, “Inequalities for the inner radii of nonoverlapping domains and open sets,” Dop. Nats. Akad. Nauk Ukr., No. 10, 7–13 (2006).
A. K. Bakhtin and V. E. V’yun, “Application of the disconnecting transformation to estimates of the inner radii of open sets,” Ukr. Mat. Zh., 59, No. 10, 1314–1322 (2007).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 5, pp. 596–610, May, 2009.
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Bakhtin, O.K. Inequalities for the inner radii of nonoverlapping domains and open sets. Ukr Math J 61, 716–733 (2009). https://doi.org/10.1007/s11253-009-0247-4
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DOI: https://doi.org/10.1007/s11253-009-0247-4