Analyse mathématique
Fonctions méromorphes aux zéros et pôles communs
[Meromorphic functions sharing the same zeros and poles]
Comptes Rendus. Mathématique, Volume 338 (2004) no. 10, pp. 763-768.

Hinkkanen's problem (1984) is completely solved, i.e., it is shown that any meromorphic function f of one complex variable is determined by its zeros and poles and the zeros of f(j) for j=1,2,3,4.

On résout complètement le problème de Hinkkanen (1984) : les zéros et les pôles d'une fonction méromorphe quelconque f d'une variable complexe et les zéros de f(j), j=1,2,3,4, déterminent f.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2004.01.011
Frank, Günter 1; Hua, Xinhou 2; Vaillancourt, Rémi 2

1 Technische Universität Berlin, Fachbereich 3 Mathematik, 10623 Berlin, Allemagne
2 Département de mathématiques et de statistique, Université d'Ottawa, Ottawa ON, K1N 6N5, Canada
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Frank, Günter; Hua, Xinhou; Vaillancourt, Rémi. Fonctions méromorphes aux zéros et pôles communs. Comptes Rendus. Mathématique, Volume 338 (2004) no. 10, pp. 763-768. doi : 10.1016/j.crma.2004.01.011. http://www.numdam.org/articles/10.1016/j.crma.2004.01.011/

[1] Barth, K.F.; Brannan, D.A.; Hayman, W.K. Research problems in complex analysis, Bull. London Math. Soc., Volume 16 (1984), pp. 490-517

[2] Chuang, C.T. Sur la comparaison de la croissance d'une fonction méromorphe et de celle de sa dérivée, Bull. Sci. Math., Volume 75 (1951), pp. 171-190

[3] Clunie, J. On integral and meromorphic functions, J. London Math. Soc., Volume 37 (1962), pp. 17-27

[4] Frank, G.; Hennekemper, W. Einige Ergebnisse über die Werteverteilung meromorpher Funktionen und ihrer Ableitungen, Resultate Math., Volume 4 (1981), pp. 39-54

[5] Gross, F. Factorization of Meromorphic Functions, U.S. Government Printing Office, Washington, DC, 1972

[6] Gundersen, G.G. When two entire functions and also their first derivatives have the same zeros, Indiana Univ. Math. J., Volume 30 (1981), pp. 293-303

[7] Gundersen, G.G. Meromorphic functions that share four values, Trans. Amer. Math. Soc., Volume 277 (1983), pp. 545-567 (Correction Trans. Amer. Math. Soc., 304, 1987, pp. 847-850)

[8] G. Frank, X.H. Hua, R. Vaillancourt, Meromorphic functions sharing the same zeros and poles, J. Can. Math./Can. J. Math., sous presse

[9] Hayman, W.K. Meromorphic Functions, Oxford University Press, 1964

[10] Hua, X.H. Some extensions of the Tumura–Clunie theorem, Complex Variables, Volume 16 (1991), pp. 69-77

[11] Köhler, L. Meromorphic functions sharing zeros and poles and also some of their derivatives sharing zeros, Complex Variables, Volume 11 (1989), pp. 39-48

[12] Laine, I. Nevanlinna Theory and Complex Differential Equations, de Gruyter, Berlin, 1993

[13] Nevanlinna, R. Analytic Functions, Springer-Verlag, New York, 1970

[14] Tohge, K. On a problem of Hinkkanen about Hadamard products, Kodai Math. J., Volume 13 (1990), pp. 101-120

[15] Yang, C.C. On two entire functions which together with their first derivatives have the same zeros, J. Math. Anal. Appl., Volume 56 (1976), pp. 1-6

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