Some normality criteria of meromorphic functions. (English) Zbl 1152.30033

Zhang, Qingcai

Complex Var. Elliptic Equ. 53, No. 8, 791-795 (2008).

Relations between the normality of a family \({\mathcal F}\) of meromorphic functions in a domain \(D\) and sharing values of functions in \({\mathcal F}\) is studied. It is proved the following
Theorem 1. Let \({\mathcal F}\) be a family of meromorphic functions in \(D\) such that all zeros and poles of \(f\in {\mathcal F}\) have multiplicities at least 3. If for each pair of functions \(f\) and \(g\) in \({\mathcal F}\), \(f^{\prime}\) and \(g^{\prime}\) share a non-zero value \(b\) in \(D\), then \({\mathcal F}\) is a normal family in \(D\).
As an application of Theorem 1. the following result is proved.
Theorem 2. Let \({\mathcal F}\) be a family of meromorphic functions in \(D\), and \(n\) be a positive integer. If \(n \geq 2\) and for each pair of functions \(f\) and \(g\) in \({\mathcal F}\), \(f^{\prime\prime} f^{\prime}\) and \(g^{\prime\prime} g^{\prime}\) share a non-zero value \(b\) in \(D\), then \({\mathcal F}\) is a normal family in \(D\).

Reviewer: Sergei V. Rogosin (Minsk)

MSC:

30D35	Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
30D45	Normal functions of one complex variable, normal families

Keywords:

normal family; sharing values

References:

[1]	DOI: 10.1007/BF01199014 · Zbl 0758.30028
[2]	Sun DC, J. Wuhan Univ. Natur. Sci. Ed. 3 pp 9– (1994)
[3]	Zhang Q, Kodai Math. J. 25 pp 8– (2002) · Zbl 1023.30036
[4]	DOI: 10.1112/S002460939900644X · Zbl 1030.30031
[5]	DOI: 10.1007/BF02384496 · Zbl 1079.30044
[6]	Fang M, J. Aust. Math. Soc. 76 pp 141– (2004) · Zbl 1074.30032
[7]	Wang Y, Acta Math. Sinica, New Series 14 pp 17– (1998) · Zbl 0909.30025
[8]	DOI: 10.1090/S0273-0979-98-00755-1 · Zbl 1037.30021