This is a preview of subscription content, log in via an institution to check access.
References
J. A. Jenkins, Univalent Functions and Conformal Mappings [Russian translation], Izdat. Inostr. Liter., Moscow (1962).
W. K. Hayman, Multivalent Functions [Russian translation], Izdat. Inostr. Liter., Moscow (1960).
L. V. Ahlfors and A. Beurling, “Conformal invariants and function-theoretic null-sets,” Acta Math.,83, No. 1/2, 101–129 (1950).
H. Wittich, “Zur konformen Abbildung schlichter Gebiete,” Math. Nachr., No. 18, 226–234 (1958).
I. P. Mityuk, “The generalized reduced modulus and some of its applications,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 2, 110–119 (1964).
G. V. Kuz'mina, “Moduli of families of curves and quadratic differentials,” Trudy Mat. Inst. Steklov.,139, Leningrad, Nauka, 1980.
S. L. Sobolev, Some Applications of Functional Analysis to Mathematical Physics [in Russian], Izdat. Leningrad. Univ., Leningrad (1950).
G. M. Goluzin, Geometric Theory of Functions of Complex Variable [in Russian], Nauka, Moscow (1966).
S. Stoilov, Theory of Functions of Complex Variable. Vol. 2 [Russian translation], Izdat. Inostr. Lit., Moscow (1962).
V. G. Maz'ya, Sobolev Spaces [in Russian], Leningr. Univ., Leningrad (1985).
H. Renngli, “An inequality for logarithmic capacities,” Pacific J. Math.,11, 313–314 (1961).
I. P. Mityuk and A. Yu. Solynin, “Ordering systems of domains and capacitors in space,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 8, 54–59 (1991).
I. P. Mityuk, Symmetrization Methods and Their Application in Geometric Theory of Functions. Introduction to Symmetrization Methods [in Russian], Kuban. Univ., Krasnodar (1980).
J. A. Jenkins, “Some uniqueness results in the theory of symmetrization. II,” Ann. of Math. (2),75, No. 2, 223–230 (1962).
M. Ohtsuka, Dirichlet Problem, Extremal Length, and Prime Ends, Van Nostrand, New York (1970).
J. Krzyz, “Circular symmetrization and Green's function,” Bull. Acad. Polon. Sci. Sér. Math. Astr. Phys.,7, No. 6, 327–330 (1959).
A. Weitsman, “Symmetrization and the Poincaré metric,” Ann. of Math. (2),124, No. 1, 159–169 (1986).
V. N. Dubinin, “The question of constructing symmetrization transformations,” in: Mathematical Physics and Mathematical Modeling in Ecology. I [in Russian], Akad. Nauk SSSR, Dal'nevostochn. Otdel., Vladivostok, 1990, pp. 55–69.
V. N. Dubinin, “On transformation of the harmonic measure under symmetrization,” Mat. Sb.,124, No. 2, 272–279 (1984).
I. A. Aleksandrov and V. A. Andreev, “Extremal problems for systems of functions without common values,” Sibirsk. Mat. Zh.,19, No. 5, 970–982 (1978).
V. N. Dubinin, “The symmetrization method in problems on nonoverlapping domains,” Mat. Sb.,128, No. 1, 110–123 (1985).
L. V. Ahlfors, Conformal Invariants. Topics in Geometric Function Theory, McGraw-Hill Book Company, New York etc. (1973).
Author information
Authors and Affiliations
Additional information
The research was financially supported within the program “Universities of Russia” in the direction “Fundamental Problems of Mathematics and Mechanics” (Code 1.1.33).
Translated fromSibirskii Matematicheskii Zhurnal, Vol. 35, No. 4, pp. 774–792, July–August, 1994.
Rights and permissions
About this article
Cite this article
Dubinin, V.N. Some properties of the reduced inner modulus. Sib Math J 35, 689–705 (1994). https://doi.org/10.1007/BF02106612
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02106612