Comptes Rendus
no. 1
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

Caiyun Fang1 ; Yan Xu1
1 Institute of Mathematics, School of Mathematical Science, Nanjing Normal University, Nanjing 210023, PR China
Comptes Rendus. Mathématique, Volume 356 (2018) no. 1, pp. 56-62.

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.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2017.11.008
Caiyun Fang 1 ; Yan Xu 1

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

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

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

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

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

[6] G.M. Zhang; X.C. Pang; L. Zalcman 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).

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