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 fF, 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 fF 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.

Received:
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
@article{CRMATH_2018__356_1_56_0,
     author = {Fang, Caiyun and Xu, Yan},
     title = {A counterexample of a normality criterion for families of meromorphic functions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {56--62},
     publisher = {Elsevier},
     volume = {356},
     number = {1},
     year = {2018},
     doi = {10.1016/j.crma.2017.11.008},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2017.11.008/}
}
TY  - JOUR
AU  - Fang, Caiyun
AU  - Xu, Yan
TI  - A counterexample of a normality criterion for families of meromorphic functions
JO  - Comptes Rendus. Mathématique
PY  - 2018
SP  - 56
EP  - 62
VL  - 356
IS  - 1
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2017.11.008/
DO  - 10.1016/j.crma.2017.11.008
LA  - en
ID  - CRMATH_2018__356_1_56_0
ER  - 
%0 Journal Article
%A Fang, Caiyun
%A Xu, Yan
%T A counterexample of a normality criterion for families of meromorphic functions
%J Comptes Rendus. Mathématique
%D 2018
%P 56-62
%V 356
%N 1
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2017.11.008/
%R 10.1016/j.crma.2017.11.008
%G en
%F CRMATH_2018__356_1_56_0
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/

[1] Hayman, W.K. Meromorphic Functions, Clarendon Press, Oxford, UK, 1964

[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

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).