Far Eastern Mathematical Journal

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On I. P. Mityuk's reduced modulus

N. V. Eyrikh

2003, issue 2, P. 167–181

Abstract

We introduce the reduced modulus of an arbitrary open set with respect to several points of the set and some boundary arcs. This generalizes the reduced modulus introduced by I. P. Mityuk in 1964. We study the basic properties of this modulus — its behavior under extension, conformal mapping and the composition principles. As an application, the theorems on non-overlapping domains and the covering theorems under conformal mapping have been generalized.

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References

[1] I. P. Mityuk, “Obobshhennyj privedennyj modul' i nekotorye ego primeneniya”, Izv. vuzov. Matematika, 1964, № 2, 110–119. [2] Dzh. Dzhenkins, Odnolistnye funkcii i konformnye otobrazheniya, Izd-vo inostr. lit., M., 1962. [3] V. N. Dubinin, N. V. E'jrix, “Obobshhennyj privedennyj modul'”, Dal'nevost. mat. zhurn., 3:2 (2002), 150–165. [4] G. V. Vittix, Novejshie issledovaniya po odnoznachnym analiticheskim funkciyam, Fizmatgiz, M., 1960. [5] I. P. Mityuk, “Pro odnolisti konformni vidobrazhennya mnogosv"yaznix oblastej”, DAN URSR, 158–160:2 (1961). [6] V. A. Shlyk, “Emkost' kondensatora i modul' semejstva razdelyayushhix poverxnostej”, Zap. nauchn. sem. LOMI, 185, 1990, 168–182. [7] V. N. Dubinin, L. V. Kovalev, “Privedennyj modul' kompleksnoj sfery”, Zap. nauchn. semin. POMI, 254, 1998, 76–94. [8] Yu. E. Alenicyn, “Ob odnolistnyx funkciyax v mnogosvyaznyx oblastyax”, Mat. sb., 39(81):3 (1956), 315–336. [9] Yu. E. Alenicyn, “O funkciyax bez obshhix znachenij i vneshnej granice oblasti znachenij funkcij”, Mat. sb., 46(88):4 (1958), 373–388. [10] G. M. Goluzin, Geometricheskaya teoriya funkcij kompleksnogo peremennogo, Nauka, M., 1966. [11] V. N. Dubinin, “Simmetrizaciya v teorii funkcij kompleksnogo peremennogo”, UMN, 49:1 (1994), 3–76. [12] G. V. Kuz'mina, “Metody geometricheskoj teorii funkcij I, II”, Algebra i analiz, 9:3 (1997), 41–103; 5, 1–50. [13] L. I. Volkovyskij, G. L. Lunc, I. G. Aramanovich, Sbornik zadach po teorii funkcij kompleksnogo peremennogo, Nauka, M., 1970.

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