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

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 $$D$$.

### 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
Full Text:

### 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.