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Some Note on Exceptional Values of Meromorphic Functions

Published online by Cambridge University Press:  22 January 2016

Kikuji Matsumoto
Kikuji Matsumoto*
Affiliation:
Mathematical Institute, Nagoya University
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Let E be a totally-disconnected compact set in the z-plane and let Ω be its complement with respect to the extended z-plane. Then Ω is a domain and we can consider a single-valued meromorphic function w = f(z) on Ω which has a transcendental singularity at each point of E. Suppose that E is a null-set of the class W in the sense of Kametani [4] (the class NB in the sense of Ahlfors and Beurling [1]). Then the cluster set of f(z) at each transcendental singularity is the whole w-plane, and hence f(z) has an essential singularity at each point of E. We shall say that a value w is exceptional for f(z) at an essential singularity ζ ∈ E if there exists a neighborhood of ζ where the function f(z) does not take this value w.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 22 , June 1963 , pp. 189 - 201
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1963

References

[1] Ahlfors, L. V. and Beurling, A.: Conformai invariants and function-theoretic null-sets, Acta Math., 83 (1950), pp. 101–129.Google Scholar
[2] Bohr, H. and Landau, E.: Über das Verhalten von C(s) und Ck(s) in der Nähe der Geraden <r = l, Göttinger Nachr., (1910).Google Scholar
[3] Carleson, L.: A remark on Picard’s theorem, Bull. Amer. Math. Soc, 67 (1961), pp. 142–144.Google Scholar
[4] Kametani, S.: On Hausdorffs measures and generalized capacities with some of their applications to the theory of functions, Jap. Journ. Math., 19 (1944-48), pp. 217–257.Google Scholar
[5] Kuroda, T.: On analytic functions on some Riemann surfaces, Nagoyay Math. Journ., 10 (1956), pp. 27–50.Google Scholar
[6] 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
[7] Matsumoto, K.: On exceptional values of meromorphic functions with the set of singularities of capacity zero, Nagoya Math. Journ., 18 (1961), pp. 171–191.Google Scholar
[8] Mori, A.: A note on unramified abelian covering surfaces of a closed Riemann surface, Journ. Math. Soc. Japan, 6 (1954), pp. 162–176.Google Scholar
[9] Noshiro, K.: Open Riemann surface with null boundary, Nagoya Math. Journ., 3 (1951), pp. 73–79.Google Scholar
[10] Pfluger, A.: Sur l’existence de fonctions non constantes, analytiques, conformes et bornées sur une surface de Riemann ouverte, C. R. Paris, 230 (1950), pp. 166–168.Google Scholar