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On Exceptional Values of Meromorphic Functions with the Set of Singularities of Capacity Zero1)

Published online by Cambridge University Press:  22 January 2016

Kikuji Matsumoto
Kikuji Matsumoto*
Affiliation:
Department of Mathematics, Hiroshima University
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Let E be a compact set in the z-plane and let Ω be its complement with respect to the extended z-plane. Suppose that E is of capacity zero. Then Ω is a domain and we shall consider a single-valued meromorphic function w = f(z) on Ω which has an essential singularity at each point of E. We shall say that a value w is exceptional for f(z) at a point ζ ∈ E if there exists a neighborhood of C where the function f(z) does not take this value w.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 18 , April 1961 , pp. 171 - 191
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1961

Footnotes

1)

In this paper, capacity is always logarithmic.

References

[1] Ahlfors, L. V. and Beurling, A.: Conformal invariants and function-theoretic null-sets, Acta Math., 83 (1950), pp. 101–129.Google Scholar
[2] Ahlfors, L. V. and Sario, L.: Riemann surfaces, Princeton Univ. Press (1960).Google Scholar
[3] Bohr, H. and Landau, E.: Über das Verhalten von ζ(s) und ζ k (s) in der Nähe der Geraden σ=1 , Göttinger Nachr., (1910).Google Scholar
[4] Hállström, G. af: Über meromorphe Funktionen mit mehrfach zusammenhängenden Existenzgebieten, Acta Acad. Aboensis, Math, et Phys., 12 (1939), pp. 1–100.Google Scholar
[5] Hersch, J.: Longueurs extrémales et théorie des fonctions, Comment. Math. Helv., 29 (1955), pp. 301–337.CrossRefGoogle Scholar
[6] Kuroda, T.: On analytic functions on some Riemann surfaces, Nagoya Math. Journ., 10 (1956), pp. 27–50.Google Scholar
[7] Matsumoto, K.: Exceptional values of meromorphic functions in a neighborhood of the set of singularities, Journ. Sci. Hiroshima Univ., A. 24 (1960), pp. 143–153.Google Scholar
[8] Noshiro, K.: Open Riemann surface with null boundary, Nagoya Math. Journ., 3 (1951), pp. 73–79.CrossRefGoogle Scholar
[9] Ohtsuka, M.: Selected topics in function theory, Kyȯritsu, Tokyo (1957), 132 pp., (Japanese).Google Scholar
[10] Ohtsuka, M.: Capacité d’ensembles de Cantor généralisés, Nagoya Math. Journ., 11 (1957), pp. 151–160.Google Scholar
[11] Sario, L.: Modular criterion on Riemann surfaces, Duke Math. Journ., 20 (1953), pp. 279–286.Google Scholar
[12] Tsuji, M.: On the behaviour of a meromorphic function in the neighbourhood of a closed set of capacity zero, Proc. Imp. Acad., 18 (1942), pp. 213–219.Google Scholar
[13] Tsuji, M.: Theory of meromorphic functions in a neighbourhood of a closed set of capacity zero, Jap. Journ. Math., 19 (1944), pp. 139–154.Google Scholar