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A Theorem on the Cluster Sets of Pseudo-Analytic Functions

Published online by Cambridge University Press:  22 January 2016

Kiyoshi Noshiro
Kiyoshi Noshiro*
Affiliation:
Mathematical Institute, Nagoya University
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Let D be an arbitrary connected domain and w = f(z) = u(x,y) + iv(x,y), z = x + iy, be an interior transformation in the sense of Stoïlow in D. Denote by γ a set, in D, such that D and the derived set γ′ of γ have no point in common.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 1 , June 1950 , pp. 83 - 89
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1950

References

1) For the definition of pseudo-meromorphic functions, Cf. Kakutani, S.: Applications to the theory of pseudo-regular functions to the type-problem of Riemann surfaces, Jap. Journ. of Math. Vol. 13 (1937), pp. 375–392 CrossRefGoogle Scholar. Nevanlinna, R.: Eindeutige analytische Funktionen, Berlin, 1936, p. 343.CrossRefGoogle Scholar

2) “Capacity” means logarithmic capacity in this note.

3) Iversen, F.: Sur quelques propriétés des fonctions monogènes au voisinage d’un point singulier, Öfv. af Einska Vet-Soc. Förh. 58 (1916).Google Scholar

Kunugui, K.: Sur un théorème de M. M. Seidei-Beurling, Proc. Acad. Tokyo, 15 (1939)Google Scholar; Sur un problème de M. A. Beurling, Proc, Acad. Tokyo, 16 (1940); Sur l’allure d’une fonction analytique uniform au voisinage d’un point frontière de son domaine de définition, Jap. Journ. of Math. 18 (1942), pp. 1-39.

A. Beurling: Etudes sur un problème de majoration, Thèse de Upsal, 1933; Cf. pp. 100-103.

4) Beurling: 1. c 3); Kunugui: 1. c 3).

5) Tsuji, M.: On the cluster set of a meromorphic function, Proc. Acad. Tokyo, 19 (1943)Google Scholar; On the Riemann surface of an inverse function of a meromorphic function in the neighbourhood of a closed set of capacity zero, Proc. Acad. Tokyo, 19 (1943).

6) Tsuji: 1. c. 5). Kametani, S.: The exceptional values of functions with the set of capacity zero of essential singularities, Proc Acad. Tokyo, 17 (1941), pp. 429–433.CrossRefGoogle Scholar

7) Recently E. Sakai has obtained some interesting results concerning pseudo-meromorphic functions. Theorem 1 answers affirmatively a problem represented by him. Cf. E. Sakai: Note on pseudo-analytic functions, forthcoming Proc. Acad. Tokyo.

8) The special case where D is simply connected and w = f(z) is single-valued meromorphic in D has been treated by the writer in another note. Cf. K. Noshiro: Note on the cluster sets of analytic functions, forthcoming Journ. Math. Soc. Japan.

9) Evans, G. C.: Potentials and positively infinite singularities of harmonic functions, Monatsheft für Math, und Phys. 43 (1936), pp. 419–424.CrossRefGoogle Scholar

Noshiro, K.: Contributions to the theory of the singularities of analytic functions, Jap, Journ. of Math. 19 (1948), pp. 299–327.Google Scholar

10) R. Nevanlinna: 1. c. 1), pages 132 and 134.

11) Noshiro, K.: On the theory of the cluster sets of analytic functions, Journ. Fac. of Sci., Hokkaido Imp. Univ. 6 (1938), pp. 217–231 Google Scholar; Cf. theorem 4.

12) Ahlfors, L.: Zur Theorie der Überlagerungsflächen, Acta Math. 65 (1935), pp. 157–194 CrossRefGoogle Scholar. R. Nevanlinna: 1. c. 1), Cf. p. 323. K. Noshiro: 1. c. 8).

13) K. Noshiro: 1. c. 8).