Far Eastern Mathematical Journal

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Some applications of extremal decompositions in the geometric function theory


V. N. Dubinin, D. A. Kirillova

2010, issue 2, P. 130–152


Abstract
The applications of the extremal decompositions of the domains and condensers in the geometric function theory are considered. We prove new theorems for the families of meromorphic functions without common values, the multipoint distortion theorems and the estimates of the coefficients for univalent functions. Also, we get some new inequalities for polynomials. All results are obtained by the unified method using the suitable properties of the extremal decompositions. Previously, these properties were established by capacity approach and symmetrization.

Keywords:
meromorphic functions, Schwarzian derivative, distortion theorems, estimates of the coefficients, polynomials, extremal decompositions, condenser capacity.

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References

[1] G. V. Kuz'mina, “Metody geometricheskoj teorii funkcij. II”, Algebra i analiz, 9:5 (1997), 1–50.
[2] A. Yu. Solynin, “Moduli i e'kstremal'no-metricheskie problemy”, Algebra i analiz, 11:1 (1999), 3–86.
[3] L. V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Book Co, New York, 1973.
[4] V. N. Dubinin, “Obobshhennye kondensatory i asimptotika ix emkostej pri vyrozhdenii nekotoryx plastin”, Zap. nauchn. semin. POMI, 302, 2003, 38–51.
[5] V. N. Dubinin, N. V. E'jrix, “Nekotorye primeneniya obobshhennyx kondensatorov v teorii analiticheskix funkcij”, Zap. nauchn. semin. POMI, 314, 2004, 52–75.
[6] V. N. Dubinin, D. A. Kirillova, “K zadacham ob e'kstremal'nom razbienii”, Zap. nauchn. semin. POMI, 357, 2008, 54–74.
[7] V. N. Dubinin, L. V. Kovalev, “Privedennyj modul' kompleksnoj sfery”, Zap. nauchn. semin. POMI, 254, 1998, 76–94.
[8] G. M. Goluzin, Geometricheskaya teoriya funkcij kompleksnogo premennogo, Nauka, M., 1966.
[9] V. N. Dubinin, “Simmetrizaciya v geometricheskoj teorii funkcij kompleksnogo peremennogo”, Uspexi mat. nauk, 49:1 (1994), 3–76.
[10] D. A. Kirillova, “Ob odnolistnyx funkciyax bez obshhix znachenij”, Izvestiya vysshix uchebnyx zavedenij. Matematika, 2010, № 9, 86–89.
[11] V. N. Dubinin, “O kvadratichnyx formax, porozhdennyx funkciyami Grina i Robena”, Matematicheskij sbornik, 200:10 (2009), 25–38.
[12] N. A. Lebedev, Princip ploshhadej v teorii odnolistnyx funkcij, Nauka, M., 1975.
[13] A. K. Baxtin, G. P. Baxtina, Yu. B. Zelinskij, Topologo-algebraicheskie struktury i geometricheskie metody v kompleksnom analize, 73, In-t matematiki NAN Ukraini, Kiiv, 2008.
[14] A. K. Baxtin, “Neravenstva dlya vnutrennix radiusov nenalegayushhix oblastej i otkrytyx mnozhestv”, Ukrainskij metematicheskij zhurnal, 61:5 (2009), 596–610.
[15] E. Schippers, “Distortion theorems for higher-order Schwarzian derivatives of univalent functions”, Proc. Amer. Math. Soc., 128:11 (2000), 3241–3249.
[16] D. Kraus, O. Roth, “O Weighted distortion in conformal mapping in euclidean, hiperbolic and elliptic geometry”, Ann. Acad. Sci. Fenn. Math., 31 (2006), 111–130.
[17] V. N. Dubinin, “Konformnye otobrazheniya i neravenstva dlya algebraicheskix polinomov”, Algebra i analiz, 13:5 (2001), 16–43.
[18] V. N. Dubinin, “Emkosti kondensatorov, obobshheniya lemm Gretsha i simmetrizaciya”, Zap. nauchn. semin. POMI, 337, 2006, 73–100.
[19] A. Yu. Solynin, “Granichnoe iskazhenie i e'kstremal'nye zadachi v nekotryx klassax odnolistnyx funkcij”, Zap. nauchn. semin. POMI, 204, 1993, 115–142.
[20] G. M. Goluzin, “Nekotorye ocenki koe'fficientov odnolistnyx funkcij”, Matematicheskij sbornik, 3 (1938), 321–330.
[21] P. Borwein, T. Erdelyi, Polinomials and polynomial inequalities, Springer-Verlag, New York, 1995.

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