Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2025-01-03T18:33:16.292Z Has data issue: false hasContentIssue false
  • English
  • Français

Unicity theorems for meromorphic or entire functions II

Published online by Cambridge University Press:  17 April 2009

Hong-Xun Yi
Hong-Xun Yi
Affiliation:
Department of Mathematics, Shandong University, Jinan, Shandong 250100, Peoples Republic of China
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.

In 1976, Gross posed the question “can one find two (or possibly even one) finite sets Sj (j = 1, 2) such that any two entire functions f and g satisfying Ef(Sj) = Eg(Sj) for j = 1,2 must be identical?”, where Ef(Sj) stands for the inverse image of Sj under f. In this paper, we show that there exists a finite set S with 11 elements such that for any two non-constant meromorphic functions f and g the conditions Ef(S) = Eg(S) and Ef({∞}) = Eg({∞}) imply f ≡ g. As a special case this also answers the question posed by Gross.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 52 , Issue 2 , October 1995 , pp. 215 - 224
Copyright
Copyright © Australian Mathematical Society 1995

References

[1]Gross, F., ‘Factorization of meromorphic functions and some open problems’, in Complex Analysis, Lecture Notes in Math. 599, Proc. Conf. Univ. Kentucky, Lexington, Kentucky 1976 (Springer-Verlag, Berlin, Heidelberg, New York, 1977).Google Scholar
[2]Gross, F. and Yang, C.C., ‘On preimage range sets of meromorphic functions’,. Proc. Japan Acad. 58 (1982), 17–20.Google Scholar
[3]Hayman, W.K., Meromorphic functions (Clarendon Press, Oxford, 1964).Google Scholar
[4]Nevanlinna, R., Le théoréme de Picard-Borel et la théorie des fonctions méromorphes (Gauthier-Villars, Paris, 1929).Google Scholar
[5]Yang, C.C., ‘On deficiencies of differential polynomials II’, Math. Z. 125 (1972), 107–112.CrossRefGoogle Scholar
[6]Yi, Hong-Xun, ‘Meromorphic functions that share three values’, Chinese Ann. Math. Ser. A 9A (1988), 434–440.Google Scholar
[7]Yi, Hong-Xun, ‘Unicity theorems for meromorphic or entire functions’, Bull. Austral. Math. Soc. 49 (1994), 257–265.CrossRefGoogle Scholar