Hostname: page-component-745bb68f8f-mzp66 Total loading time: 0 Render date: 2025-01-11T00:11:12.922Z Has data issue: false hasContentIssue false
  • English
  • Français

A SECTORIAL THEOREM ON COMPLETELY RAMIFIED RATIONAL FUNCTIONS

Part of: Entire and meromorphic functions, and related topics

Published online by Cambridge University Press:  01 February 2008

ANDREAS SAUER
ANDREAS SAUER*
Affiliation:
FH Dortmund, University of Applied Sciences and Arts, Sonnenstr. 96, 44047 Dortmund, Germany (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

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.

We prove that under a sharp growth condition meromorphic functions posses a direction such that at most four rational functions are completely ramified in any sector containing the direction.

Keywords

completely ramifiedJulia directionAhlfors theoryfilling discscercles de remplissagessingular direction

MSC classification

Secondary: 30D35: Distribution of values, Nevanlinna theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 84 , Issue 1 , February 2008 , pp. 131 - 138
Copyright
Copyright © 2008 Australian Mathematical Society

References

[1]Ahlfors, L. V., ‘Zur Theorie der Überlagerungsflächen’, Acta Math. 65 (1935), 157–194.CrossRefGoogle Scholar
[2]Fenton, P. C. and Rossi, J., ‘Cercles de remplissage for entire functions’, Bull. London Math. Soc. 31 (1999), 59–66.CrossRefGoogle Scholar
[3]Fenton, P. C. and Rossi, J., ‘Growth of functions in cercles de remplissage’, J. Aust. Math. Soc. 72 (2002), 131–136.Google Scholar
[4]Hayman, W. K., Meromorphic functions (Oxford University Press, London, 1975).Google Scholar
[5]Lehto, O., ‘The spherical derivative of meromorphic functions in the neighbourhood of an isolated singularity’, Comment. Math. Helv. 33 (1959), 196–205.CrossRefGoogle Scholar
[6]Lehto, O. and Virtanen, K. I., ‘On the behaviour of meromorphic functions in the neighbourhood of an isolated singularity’, Ann. Acad. Sci. Fenn. Ser. A. I. 240 (1957).Google Scholar
[7]Nevanlinna, R., Le théorème de Picard-Borel et a théorie des fonctions méromorphes (Gauthier-Villars, Paris, 1939). Reprint Chelsea, New York, 1974.Google Scholar
[8]Ostrowski, A., ‘Über Folgen analytischer Funktionen und einige Verschärfungen des Picardschen Satzes’, Math. Z. 24 (1924), 215–258.CrossRefGoogle Scholar
[9]Rossi, J., ‘A sharp result concerning cercles de remplissage’, Ann. Acad. Sci. Fenn. Ser. A I 20 (1995), 179–185.Google Scholar
[10]Sauer, A., ‘Deficient rational functions and Ahlfors’s theory of covering surfaces’, Ark. Mat. 39 (2001), 151–155.CrossRefGoogle Scholar
[11]Schiff, J. L., Normal families (Springer, Berlin, 1993).CrossRefGoogle Scholar
[12]Steinmetz, N., ‘The formula of Riemann–Hurwitz and iteration of rational functions’, Complex Var. Theory Appl. 22 (1993), 203–206.Google Scholar
[13]Toda, N., ‘Sur les directions de Julia et de Borel des fonctions algebroides’, Nagoya Math. J. 34 (1969), 1–23.CrossRefGoogle Scholar
[14]Yamanoi, K., ‘The second main theorem for small functions and related problems’, Acta Math. 192 (2004), 225–294.CrossRefGoogle Scholar
[15]Yang, L., ‘Meromorphic functions and their derivatives’, J. London Math. Soc. 25 (1982), 288–296.Google Scholar
[16]Yang, L., Value distribution theory (Springer, Berlin, 1993).Google Scholar
[17]Zalcman, L., ‘Normal families: new perspectives’, Bull. Amer. Math. Soc. 35 (1998), 215–230.CrossRefGoogle Scholar