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Extremal partitions and distortion under the Montel bounded univalent maps

Part of: Geometric function theory

Published online by Cambridge University Press:  09 April 2009

Alexander Vasil'ev
Alexander Vasil'ev
Affiliation:
Departmento de Matemática, Universidad Técnica Federico Santa María, Casilla 110-V, Valparaíso, Chile e-mail: avassili@mat.utfsm.cl
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Abstract

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Extremal partitions of domains into configurations of certain to pological form are studied. The extremal value of the weighted sum of reduced moduli of circular domains and digons is obtained. These results are applied to some problems about distortion under bounded conformal maps of the unit disk with two preassigned values.

Keywords

moduli methodextremal problemunivalent functionMontel functiondistortion

MSC classification

Secondary: 30C75: Extremal problems for conformal and quasiconformal mappings, other methods 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 73 , Issue 3 , December 2002 , pp. 419 - 432
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Demin, S. E., ‘Isoperimetric distortion problem for univalent Montel functions’, Sibirsk. Mat. Zh. 37 (1996), 108–116; English translation: Siberian Math. J. 37 (1996) 94–101.Google Scholar
[2]Emel'yanov, E. G., ‘On extremal partitioning problems’, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. inst. Steklov. (LOMI) 154 (1986), 76–89; English translation: J. Soviet Math. 43 (1988), 2558–2566.Google Scholar
[3]Jenkins, J. A., ‘On the existence of certain general extremal metrics’, Ann. of Math. (2) 66 (1957), 440–453.CrossRefGoogle Scholar
[4]Krzyż, J., ‘On univalent functions with two preassigned values’, Ann. Univ. Mariae Curie Ski odowska Sect. A 15 (1961), 57–77.Google Scholar
[5]Krzyż, J., ‘Some remarks concerning my paper: ‘On univalent functions with two preassigned values’’, Ann. Univ. Mariae Curie-Sklodowska Sect. A 16 (1962), 129–136.Google Scholar
[6]Krzyż, J., ‘On the region of variability of the ratio f (z 1)/f (z 2) within the class S of univalent functions’, Ann. Univ. Mariae Curie-Sklodowska Sect. A 17 (1963), 55–64.Google Scholar
[7]Krzyż, J. and Zlotkiewicz, E., ‘Koebe sets for univalent functions with two preassigned values’, Ann. Acad. Sd. Fenn. Ser. Al Math. 487 (1971), 1–12.Google Scholar
[8]Kuz'mina, G. V., Moduli of families of curves and quadratic differentials, Trudy Mat. Inst. Steklov. 139 (1980), English translation: Proc. Steklov Inst. Math. (Amer. Math. Soc., Providence, 1982).Google Scholar
[9]Kuz'mina, G. V., ‘Extremal properties of quadratic differentials with strip domains in the structure of the trajectories’, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 154 (1986), 110–129; English translation: J. Soviet Math. 43 (1988) 2579–2591.Google Scholar
[10]Libera, R. J. and Zlotkiewicz, E., ‘Bounded Montel univalent functions’, Colloq. Math. 56 (1988), 169–177.CrossRefGoogle Scholar
[11]Mejía, D., Pommerenke, Ch. and Vasil'ev, A., ‘Distortion theorems for hyperbolically convex functions’, Complex Variables Theory Appl. 44 (2001), 117–130.Google Scholar
[12]Reade, M. O. and Zlotkiewicz, E., ‘On values omitted by univalent functions with two pre-assigned values’, Compositio Math. 24 (1972), 355–358.Google Scholar
[13]Solynin, A. Yu., ‘Moduli and extremal metric problems’, Algebra i Anatiz 11 (1999), 3–86; English translation: St. Petersburg Math. J. 11 (2000), 1–65.Google Scholar
[14]Vasil'ev, A. and Pronin, P., ‘Distortion theorem for the univalent Montel functions’, in: First Russian M. Souslin conference on Foundations of Math. and Function Theory, October 16–21 (Abstracts) (Saratov Pedagogical Institute, 1989) pp. 103–104.Google Scholar
[15]Vasil'ev, A. and Pronin, P., ‘On some extremal problems for bounded univalent functions with the Montel normalization’, in: Second Russian M. Souslin conference on Foundations of Math. and Function Theory, October 4–11 (Abstracts) (Saratov Pedagogical Institute, 1991) pp. 64–65.Google Scholar
[16]Vasil'ev, A. and Pronin, P., ‘On some extremal pro1ems for bounded univalent function with Montel's normalization’, Demonst ratio Math. 26 (1993), 703–707.Google Scholar
[17]Vasil'ev, A. and Pronin, P., ‘The range of a system of functionals for the Montel univalent functions’, Bol. Soc. Mat. Mexicana (3) 6 (2000), 177–190.Google Scholar