Far Eastern Mathematical Journal

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The generalized reduced modulus

V. N. Dubinin, N. V. Eyrikh

2002, issue 2, P. 150–164

Abstract

The boundary reduced moduli of the digons and triangles are the essential part of the extremal metric method. They have many applications in the geometric theory of functions of a complex variable. In the present paper, we use the capacity approach to extend these concepts to the concepts of the boundary reduced modulus of the polygons, with any number of vertexs. Moreover, we connect the concept of the boundary reduced modulus with the inner reduced modulus. The correctness of the definition of the generalized reduced modulus is proved. We consider the special cases of reduced modulus, the behavior of the reduced modulus under the extension of the sets and under the conformal mappings of the sets. The principle of the symmetry and the formulae for some reduced moduli are obtained. We prove the new composition principles for the generalized reduced moduli. These principles generalize some theorems about the separating transformation and about the extremal partitioning of the domains.

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References

[1] L. V. Ahlfors, A. Beurling, Conformal invariants and function-theoretic null-sets, Acta Math., 83:1-2 (1950), 101–129. [2] V. K. Xejman, Mnogolistnye funkcii, Izd-vo inostr. lit., M., 1960. [3] Dzh. Dzhenkins, Odnolistnye funkcii i konformnye otobrazheniya, Izd-vo inostr. lit., M., 1962. [4] A. Pfluger, Extremalla?ngen und Kapazita?t, Comment. Math. Helv., 29 (1955), 120–131. [5] H. Wittich, Zur Konformen Abbildung schlichter Gebiete, Math. Nachr., 16 (1958), 226–234. [6] I.P. Mityuk, Obobshhennyj privedennyj modul' i nekotorye ego primeneniya, Izv. vuzov. Matematika, 1964, № 2, 110–119. [7] V. M. Miklyukov, O nekotoryx granichnyx zadachax teorii konformnyx otobrazhenij, Sib. matem. zhurn., 18:5 (1977), 1111–1124. [8] G. V. Kuz'mina, Moduli semejstv krivyx i kvadratichnye differencialy, Tr. Mat. in-ta im. V. A. Steklova, 139, Leningrad, 1980. [9] G. V. Kuz'mina, Ob e'kstremal'nyx svojstvax kvadratichnyx differencialov s polosoobraznymi oblastyami v strukture traektorij, Zap. nauch. semin. LOMI, 154, 1986, 110–129. [10] E. G. Emel'yanov, K zadacham ob e'kstremal'nom razbienii, Zap. nauch. semin. LOMI, 154, 1986, 76–89. [11] E. G. Emel'yanov, O svyazi dvux zadach ob e'kstremal'nom razbienii, Zap. nauch. semin. LOMI, 160, 1987, 91–98. [12] G. V. Kuz'mina, K voprosu ob e'kstremal'nyx svojstvax kvadratichnyx differencialov s koncevymi oblastyami v strukture traektorij, Zap. nauch. semin. LOMI, 168, 1988, 98–113. [13] V. N. Dubinin, Razdelyayushhee preobrazovanie oblastej i zadachi ob e'kstremal'nom razbienii, Zap. nauchn. semin. LOMI, 168, 1988, 48–66. [14] A. Yu. Solynin, Reshenie odnoj izoperimetricheskoj zadachi Polia–Sege, Zap. nauchn. semin. LOMI, 168, 1988, 140–153. [15] A. Baernstein II, A counterexample concerning integrability of derivatives of conformal mappings, J. Anal. Math., 53 (1989), 253–268. [16] D. Gaier, W. Hayman, On the computation of modules of long quadrilaterals, Constr. Approx., 7 (1991), 453–467. [17] Ch. Pommerenke, Boundary Behaviour of Conformal Maps, New York, 1992. [18] L. Carleson, N. G. Makarov, Some results connected with Brennan's conjecture, Ark. Mat., 32:1 (1994), 33–62. [19] V. N. Dubinin, Nekotorye svojstva vnutrennego privedennogo modulya, Sib. matem. zhurn., 35:4 (1994), 774–792. [20] V. N. Dubinin, Simmetrizaciya v geometricheskoj teorii funkcij kompleksnogo peremennogo, Uspexi matem. nauk, 49:1 (1994), 3–76. [21] V. N. Dubinin, Simmetrizaciya, funkciya Grina i konformnye otobrazheniya, Zap. nauchn. semin. POMI, 226, 1996, 80–92. [22] E. G. Emel'yanov, G. V. Kuz'mina, Teoremy ob e'kstremal'nom razbienii v semejstvax sistem oblastej razlichnyx tipov, Zap. nauchn. semin. POMI, 237, 1997, 74–104. [23] G. V. Kuz'mina, Metody geometricheskoj teorii funkcij I, II, Algebra i analiz, 9:3 (1997), 41–103; 5, 1–50. [24] V. N. Dubinin, Asimptotika modulya vyrozhdayushhegosya kondensatora i nekotorye ee primeneniya, Zap. nauchn. semin. POMI, 237, 1997, 56–73. [25] V. N. Dubinin, Privedennye moduli otkrytyx mnozhestv v teorii analiticheskix funkcij, Dokl. RAN, 363:6 (1998), 731–734. [26] V. N. Dubinin, L. V. Kovalev, Privedennyj modul' kompleksnoj sfery, Zap. nauchn. semin. POMI, 254, 1998, 76–94. [27] V. N. Dubinin, E. G. Prilepkina, Ob e'kstremal'nom razbienii prostranstvennyx oblastej, Zap. nauchn. semin. POMI, 254, 1998, 95–107. [28] A. Yu. Solynin, Moduli i e'kstremal'no-metricheskie problemy, Algebra i analiz, 11:1 (1999), 3–86. [29] V. N. Dubinin, Princip mazhoracii dlya p-listnyx funkcij, Matem. zametki, 65:4 (1999), 33–541. [30] G. V. Kuz'mina, K zadacham ob e'kstremal'nom razbienii v semejstvax sistem oblastej obshhego vida, Zap. nauch. semin. POMI, 263, 2000, 157–186. [31] N. V. E'jrix, Privedennye moduli n-ugol'nikov, Dal'nevostochnaya matem. shkola-seminar im. akademika E. V. Zolotova, Tez. Dokl., Dal'nauka, Vladivostok, 2000, 116–117. [32] J. Hersch, On the reflection principle and some elementary ratios of conformal radii, J. Anal. Math., 44 (1984/85), 251–268. [33] M. A. Lavrent'ev, B. V. Shabat, Metody teorii funkcij kompleksnogo peremennogo, Nauka, M., 1973. [34] V. N. Dubinin, Metod simmetrizacii i transfinitnyj diametr, Sib. matem. Zhurn., 27:2 (1986), 39–46.

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