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Meromorphic Functions Sharing the Same Zeros and Poles

Published online by Cambridge University Press:  20 November 2018

Günter Frank ,
Xinhou Hua  and
Rémi Vaillancourt
Günter Frank
Affiliation:
Technische Universität Berlin, Fachbereich 3 Mathematik, 10623 Berlin, Germany e-mail: hua@mathstat.uottawa.ca
Xinhou Hua
Affiliation:
Department of Mathematics and Statistics, University of Ottawa, Ottawa, K1N 6N5 e-mail: remi@uottawa.ca
Rémi Vaillancourt
Affiliation:
Department of Mathematics and Statistics, University of Ottawa, Ottawa, K1N 6N5 e-mail: remi@uottawa.ca
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Abstract

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In this paper, Hinkkanen's problem (1984) is completely solved, i.e., it is shown that any meromorphic function $f$ is determined by its zeros and poles and the zeros of ${{f}^{\left( j \right)}}$ for $j=1,2,3,4$.

Keywords

30D35Uniquenessmeromorphic functionsNevanlinna theory
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 56 , Issue 6 , 01 December 2004 , pp. 1190 - 1227
Copyright
Copyright © Canadian Mathematical Society 2004

References

[1] Barth, K.F., Brannan, D.A. and Hayman, W.K., Research problems in complex analysis. Bull. London Math. Soc. 16(1984), 490–517.Google Scholar
[2] Chuang, C.T., Sur la comparaison de la croissance d'une fonction méromorphe et de celle de sa dérivée. Bull. Sci. Math. 75(1951), 171–190.Google Scholar
[3] Clunie, J., On integral and meromorphic functions. J. LondonMath. Soc. 37(1962), 17–27.Google Scholar
[4] Frank, G. and Hennekemper, W., Einige Ergebnisse über dieWerteverteilung meromorpher Funktionen und ihrer Ableitungen. Resultate Math. 4(1981), 39–54.Google Scholar
[5] Frank, G., Hua, X.H. and Vaillancourt, R., Fonctions méromorphes aux zéroes et pôles communs / Meromorphic functions sharing the same zeroes and poles. C. R. Acad. Sci. Paris, Ser. I. 338(2004), 763–768.Google Scholar
[6] Gross, F., Factorization of meromorphic funcions. U.S. Government Printing Office, Washington DC, 1972.Google Scholar
[7] Gundersen, G.G.,When two entire functions and also their first derivatives have the same zeros. Indiana Univ.Math. J. 30(1981), 293–303.Google Scholar
[8] Hayman, W.K., Meromorphic functions, Oxford at the Clarendon Press, 1964.Google Scholar
[9] Hua, X.H., Some extensions of the Tumura–Clunie theorem. Complex Variables 16(1991), 69–77.Google Scholar
[10] Köhler, L.,Meromorphic functions sharing zeros and poles and also some of their derivatives sharing zeros. Complex Variables 11(1989), 39–48.Google Scholar
[11] Laine, I., Nevanlinna theory and complex differential equations. de Gruyter, Berlin, 1993.Google Scholar
[12] Nevanlinna, R., Analytic functions. Springer-Verlag, New York, 1970.Google Scholar
[13] Tohge, K., On a problem of Hinkkanen about Hadamard products. KodaiMath. J. 13(1990), 101–120.Google Scholar
[14] Yang, C.C., On two entire functions which together with their first derivatives have the same zeros. J. Math. Anal. Appl. 56(1976), 1–6.Google Scholar