Hostname: page-component-7bb8b95d7b-qxsvm Total loading time: 0 Render date: 2024-09-27T02:10:09.168Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalizations of Montel-Lindelöf’s Theorem on Asymptotic Values

Published online by Cambridge University Press:  22 January 2016

Makoto Ohtsuka
Makoto Ohtsuka*
Affiliation:
Mathematical Institute Nagoya University
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Montel [10] proved in 1912 the following theorem: Let w = f(z) be an analytic function in the horizontal strip B : 0 < x < + ∞, 0 < y < 1 (z = x + iy) which is continuous on 0 < x < + ∞, 0 ≦ y < 1 and omits at least two values. If f (x) converges to a value w0 as x → + ∞, then f(z) converges to W0 as z tends to in 0 < x < + ∞, 0 ≦ y < 1 − ε for any ε such that 0 < ε < 1.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 10 , June 1956 , pp. 129 - 163
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1956

References

[ 1 ] Ahlfors, L.: On quasiconformal mappings, J. Anal. Math., 3 (1953-54), pp. 1–58; corrections, pp. 207–208.Google Scholar
[ 2 ] Brelot, M.: Points irréguliers et transformations continues en théorie du potentiel, J. Math., 19 (1940), pp. 310–337.Google Scholar
[ 3 ] Brelot, M. and Choquet, G.: Espaces et lignes de Green, Ann. Inst. Fourier, 3 (1952), pp. 199–263.Google Scholar
[ 4 ] Deny, J.: Sur les infinis d’un potentiel, C. R. Acad. Sci. Paris, 224 (1947), pp. 524–525.Google Scholar
[ 5 ] Gross, W.: Über die Singularitäten analytischer Funktionen, Monat. Math. Physik, 29 (1918), pp. 3–47.Google Scholar
[ 6 ] Gross, W.: Zum Verhalten der konformen Abbildung am Rande, Math. Zeit, 3 (1919), pp. 43–64.Google Scholar
[ 7 ] Kuramochi, Z.: On covering surfaces, Osaka Math. J., 5 (1953), pp. 155–201.Google Scholar
[ 8 ] Lehto, O.: Value distribution and boundary behaviour of a function of bounded characteristic and the Riemann surface of its inverse function, Ann. Acad. Sci. Fenn., A. I., (1955), No. 179, 46 pp.Google Scholar
[ 9 ] Lindelöf, E.: Sur un principe général de l’analyse et ses applications à la théorie de la représentation conforme, Acta Soc. Sci. Fenn., 46 (1915), No. 4, 35 pp.Google Scholar
[10] Montel, P.: Sur les familles de fonctions analytiques qui admettent des valeurs exceptionnelles dans un domaine, Ann. Sci. Ecole Norm. Sup. (3), 23 (1912), pp. 487–535.Google Scholar
[11] Mori, A.: On quasi-conformality and pseudo-analyticity, forthcoming.Google Scholar
[12] Nevanlinna, R.: Eindeutige analytische Funktionen, Berlin (1936).Google Scholar
[13] Ohtsuka, M.: Dirichlet problems on Riemann surfaces and conformai mappings, Nagoya Math. J., 3 (1951), pp. 91–137.Google Scholar
[14] Ohtsuka, M.: Théorèmes étoilés de Gross et leurs applications, Ann. Inst. Fourier, 5 (1955), pp. 1–28.Google Scholar
[15] Ohtsuka, M.: Sur un théorème étoile de Gross, Nagoya Math. J., 9 (1955), pp. 191–207.Google Scholar
[16] Pfluger, A.: Quelques théorèmes sur une classe de fonctions pseudo-analytiques, C. R. Acad. Sci. Paris, 231 (1950), pp. 1022–1023.Google Scholar
[17] Pfluger, A.: Extremallängen und Kapazität, Comment. Math. Helv., 29 (1955), pp. 120–131.Google Scholar
[18] Rado, T.: Length and area, New York (1948).Google Scholar
[19] Royden, H. L.: Harmonic functions on open Riemann surfaces, Trans. Amer. Math. Soc, 73 (1952), pp. 40–94.Google Scholar
[20] Saks, S.: Theory of the integral, 2nd ed., Warszawa-Lwów (1937).Google Scholar
[21] Strebel, K.: Die extremale Distanz zweier Enden einer Riemann’schen Fläche, Ann. Acad. Sci. Fenn., A. I., (1955), No. 179, 21 pp.Google Scholar
[22] Tsuji, M.: Theory of meromorphic functions on an open Riemann surface with null boundary, Nagoya Math. J., 6 (1953), pp. 137–150.Google Scholar
[23] Tsuji, M.: On the capacity of general Cantor sets, J. Math. Soc. Japan, 5 (1953), pp. 235–252.CrossRefGoogle Scholar
[24] Yûjôbô, Z.: On pseudo-regular functions, Comment. Math. Univ. St. Pauli, 1 (1953), pp. 67–80.Google Scholar
[25] Yûjôbô, Z.: On the quasi-conformal mapping from a simply connected domain on another one, ibid., 2 (1953), pp. 1–8.Google Scholar
[26] Yûjôbô, Z.: On absolutely continuous functions of two or more variables in the Tonelli sense and quasi-conformal mappings in the A. Mori sense, ibid., 4 (1955), pp. 67–92.Google Scholar