Geodesic
Complex analysis
A counterexample of a normality criterion for families of meromorphic functions
[Un contre-exemple au critère de normalité pour les familles de fonctions méromorphes]

Présenté par : the Editorial Board

Fang, Caiyun1 ; Xu, Yan1
1 Institute of Mathematics, School of Mathematical Science, Nanjing Normal University, Nanjing 210023, PR China
Comptes Rendus. Mathématique, Tome 356 (2018) no. 1, pp. 56-62.

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Let A>1 be a constant, and let F be a family of meromorphic functions in a domain D. If, for every function f∈F, f has only zeros of multiplicity at least 2 and satisfies the following conditions: (1) f(z)=0⇒|f″(z)|≤A|z|, (2) f″(z)≠z, (3) all poles of f have multiplicity at least 4, then F is normal in D. In this paper, we first give an example to show that condition (3) is sharp, and prove that our counterexample is unique in some sense.

Soit A>1 une constante et F une famille de fonctions méromorphes dans un domaine D. Si toute fonction f∈F n'a que des zéros de multiplicité au moins 2 et satisfait les conditions suivantes : (1) f(z)=0⇒|f″(z)|≤A|z|, (2) f″(z)≠z, (3) tous les pôles de f ont multiplicité au moins 4, alors F est normale dans D. Dans cette Note, nous donnons un exemple montrant que la condition (3) est précise. Nous montrons ensuite que notre exemple est, en quelque sorte, unique.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2017.11.008

Fang, Caiyun 1 ; Xu, Yan 1

1 Institute of Mathematics, School of Mathematical Science, Nanjing Normal University, Nanjing 210023, PR China 
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Fang, Caiyun; Xu, Yan. A counterexample of a normality criterion for families of meromorphic functions. Comptes Rendus. Mathématique, Tome 356 (2018) no. 1, pp. 56-62. doi : 10.1016/j.crma.2017.11.008. http://geodesic.mathdoc.fr/articles/10.1016/j.crma.2017.11.008/

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[2] Pang, X.C.; Zalcman, L. Normal families and shared values, Bull. Lond. Math. Soc., Volume 32 (2000), pp. 325-331

[3] Schiff, J. Normal Families, Springer-Verlag, New York/Berlin, 1993

[4] Xu, Y. Normal families and fixed-points of meromorphic functions, Monatshefte Math., Volume 179 (2016), pp. 471-485

[5] Yang, L. Value Distribution Theory, Springer-Verlag & Science Press, Berlin, 1993

[6] Zhang, G.M.; Pang, X.C.; Zalcman, L. Normal families and omitted functions II, Bull. Lond. Math. Soc., Volume 41 (2009), pp. 63-71

Cité par Sources :

☆ C. Fang is supported by the NNSF of China (Grant Nos. 11401298, 11471163, 11501297). Y. Xu (corresponding author) is supported by the NNSF of China (Grant No. 11471163).