Abstract
A survey which highlights this theorem’s impact on value-distribution theory. An application is made to asymptotic values of entire functions, which shows some limitations of the method.
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Literature
Ahlfors, L: Über eine in der neueren Wertverteilungstheorie betrachte Klasse transzendenter Funktionen, Acta Math. 58 (1932), 375–406.
Ahlfors, L: Lectures on Quasiconformal Mappings, Van Nostrand, Princeton, 1966.
Ahlfors, L: Quasiconformal mappings, Teichmüller spaces and Kleinian groups, Proceedings of the ICM, Helsinki 1978 (1980), 71–84.
Belinskii, P. P.: Behavior of a quasiconformal mapping at an isolated point, Dokl. Akad. Sci. USSR 91(1953), 709–710 (in Russian).
Belinskii, P. P.: General Properties of Quasiconformal Mappings, Nauka, Novosibirsk, 1974 (in Russian).
Bers, L.: Quasiconformal mappings, with applications to differential equations, function theory and topology, Bull. Amer. Math. Soc. 83(1977), 1083–1100.
Bojarski, B. and T. Iwaniec: Analytical foundations of the theory of quasiconformal mappings in ℝn, Ann. Acad. Sci. Fenn. AI8 (1983), 257–324.
Douady, A. and J. Hubbard: The dynamics of polynomial-like mappings, Ann. Scient. Éc. Norm. Sup., 4e série, 18(1985), 287–343.
Drasin, D.: Value distribution of entire functions in regions of small growth, Ark. Math. 12(1974), 139–150.
Drasin, D: The inverse problem of the Nevanlinna theory, Acta Math. 138(1977), 83–151.
Drasin, D: Proof of a conjecture of F. Nevanlinna concerning functions with deficiency sum two, to appear in Acta Math.
Drasin, D and W. K. Hayman: Value distribution of functions meromorphic in an angle, Proc. London Math. Soc. (3), 48(1984), 319–340.
Drasin, D and A. Weitsman: Meromorphic functions with large sums of deficiencies, Advances in Math. 15(1975), 93–126.
Elfving, G.: Über eine Klasse von Riemannschen Flächen und ihre Uniformisierung, Acta. Soc. Sci. Fenn., N.S. 2 (1934), 1–60.
Eremenko, A. E.: On Valiron exceptional values of entire functions of completely regular growth, Teor. Funkts., Funkts. Anal. Prilozh. 44(1985), 48–52 (in Russian).
Fuchs, W. H. J. and W. K. Hayman: An entire function with assigned deficiencies, in Studies in Mathematical Analysis and Related Topics, D. Gilbarg, ed., Stanford U. Press, Stanford, 1962.
Goldberg, A. A. and I. V. Ostrovskii: The Distribution of Values of Meromorphic Functions, Nauka, Moscow, 1970 (in Russian).
Hayman, W. K.: Research Problems in Function Theory, Univ. of London, London, 1967.
Hayman, W. K: Meromorphic Functions, Oxford U., Oxford, 1964.
Hayman, W. K: On integral functions with distinct asymptotic values, Proc. Camb. Philos, Soc. 66(1969), 301–315.
Hyllengren, A.: Valiron deficient values for meromorphic functions in the plane, Acta Math. 124(1970), 1–7.
Kennedy, P. B.: On a conjecture of Heins, Proc. London Math. Soc. (3) 6(1956), 518–547.
Le-Van Thiem: Über das Umkehrproblem der Wertverteilungslehre, Comment. Math. Helv. 23(1949), 26–49.
Lehto, O. and K. I. Virtanen: Quasiconformal Mappings in the Plane, Springer, New York/Heidelberg/Berlin, 1973.
Nevanlinna, R.: Le Théorème de Picard-Borel et la Théorie des Fonctions Méromorphes, Gauthier-Villars, Paris, 1929.
Nevanlinna, R: Über Riemannschen Flächen mit eindlich vielen Windungspunkten, Acta. Math. 58(1932), 295–373.
Rickman, S.: A defect relation for quasimeromorphic mappings, Ann. of Math. 114(1981), 165–191.
Rickman, S.: The analogue of Picard’s theorem for quasiregular mappings in dimension three, Acta. Math. 154(1985), 195–242.
Sullivan, D.: Quasiconformal homeomorphisms and dynamics I, Solution of the Faton-Julia problem on wandering domains, to appear.
Teichmüller, O.: Untersuchungen über konforme und quasikonforme Abbildung, Deutsche Math. 3(1938), 621–678.
Ullrich, F. E.: Zum Umkehrproblem der Wertverteilungslehre, Nachr. Fes. Wiss. Göttingen Math.-Phys. Kl., Band 1(1936), 135–150.
Väisäla, J.: A survey of quasiregular maps in ℝn, Proceedings of the ICM, Helsinki 1978(1980), 685–691.
Wittich, H.: Über die konforme Abbildung einer Klasse Riemannscher Flächen, Math. Zeit. 45(1939), 642–668.
Wittich, H.: über die Wachstumsordnung einer ganzen transzendenten Funktion, Math. Zeit. 51(1947), 1–16.
Wittich, H: Zum Beweis eines Satzes über quasikonforme Abbildungen, Math. Zeit. 51(1947), 278–288.
Wittich, H: Über eine Klasse meromorpher Funktionen, Arch. Math 1(1948), 160–166.
Wittich, H: Neuere Untersuchungen über eindeutige analytische Funktionen, Springer, Berlin, 1955.
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Hans Wittich in memoriam
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Drasin, D. On the Teichmüller-Wittich-Belinskii Theorem. Results. Math. 10, 54–65 (1986). https://doi.org/10.1007/BF03322363
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DOI: https://doi.org/10.1007/BF03322363