Complex analysis
A counterexample of a normality criterion for families of meromorphic functions
Comptes Rendus. Mathématique, Volume 356 (2018) no. 1, pp. 56-62.

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.

Accepted:
Published online:
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, Volume 356 (2018) no. 1, pp. 56-62. doi : 10.1016/j.crma.2017.11.008. http://www.numdam.org/articles/10.1016/j.crma.2017.11.008/

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[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

Cited by 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).