Abstract
One considers two problems of the type of the covering theorems. In one of them one investigates the covering of the trajectories of some quadratic differential by the union of a finite number of multiply connected domains. The second problem is concerned with the covering of radial segments by sets that are bounded by the level lines under a univalent mapping of the unit circle. Both problems are solved by the symmetrization method. The obtained theorems refine and generalize the known results of Nehari, Rengel, Reich-Schiffer, etc.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Literature cited
V. N. Dubinin, “The symmetrization method in problems on nonoverlapping domains,” Mat. Sb.,128 (170), No. 1, 110–123 (1985).
Z. Nehari, “Some inequalities in the theory of functions,” Trans. Am. Math. Soc.,75, No. 2, 256–286 (1953).
V. N. Dubinin, “On the variation of the harmonic measure at symmetrization,” Mat. Sb.,124 (166), No. 2, 272–279 (1984).
Yu. E. Alenitsyn, “On univalent functions in multiply connected domains,” Mat. Sb.,39 (81), No. 3, 315–336 (1956).
N. A. Lebedev, The Area Principle in the Theory of Univalent Functions [in Russian], Nauka, Moscow (1975).
N. A. Lebedev, “On the product of the powers of the conformal radii of nonoverlapping domains,” in: Mathematical Structures. Numerical Mathematics. Mathematical Modeling, Sofia (1975), pp. 113–129.
W. K. Hayman, Multivalent Functions, Cambridge Univ. Press (1958).
I. P. Mityuk, “The symmetrization principle for multiply connected domains,” Dokl. Akad. Nauk SSSR,157, No. 2, 268–270 (1964).
J. A. Jenkins, Univalent Functions and Conformal Mapping, Springer, Berlin (1958).
V. N. Dubinin, “The symmetrization method and transfinite diameter,” Sib. Mat. Zh.,27, No. 2, 39–46 (1986).
E. Reich and M. Schiffer, “Estimates for the transfinite diameter of a continuum,” Math. Z.,85, No. 1, 91–106 (1964).
E. Rengel, “Über einzige Schlitztheoreme der konformen Abbildung,” Schr. Math. Sem. u. Inst. f. Angew. Math., Univ. Berlin,1, No. 1, 141–162 (1933).
P. M. Tamrazov, “Theorems on the covering of lines under a conformal mapping,” Mat. Sb.,66 (108), No. 4, 502–524 (1965).
Additional information
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 154, pp. 67–75, 1986.
Rights and permissions
About this article
Cite this article
Dubinin, V.N. Covering theorems in the theory of analytic functions. J Math Sci 43, 2553–2558 (1988). https://doi.org/10.1007/BF01374985
Issue Date:
DOI: https://doi.org/10.1007/BF01374985