Normality and shared values concerning differential polynomials. (English) Zbl 1213.30055

Lei, ChunLin; Fang, MingLiang

Sci. China, Math. 53, No. 3, 749-754 (2010).

Let \({\mathcal F}\) be a family of meromorphic functions in a domain \(D\), and let \(P\) be a polynomial with either \(\deg P \geq 3\) or \(\deg P=2\), and \(P\) having only one zero. In this paper, the authors prove that if there is a non-zero complex number \(b\) such that, for any \(f \in {\mathcal F}\) and \(g \in {\mathcal F}\), \(P(f)f'\) and \(P(g)g'\) share \(b\) in \(D\), then \({\mathcal F}\) is normal in \(D\). The authors also give two examples showing that the theorem fails when \(\deg P=1\) or \(b=0\).

Reviewer: Zhuan Ye (DeKalb)

MSC:

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

Keywords:

meromorphic function; normal family; shared value

References:

[1]	Bergweiler W, Eremenko A. On the singularities of the inverse to a meromorphic function of finite order. Rev Mat Iberoamericana, 1995, 11: 355–373 · Zbl 0830.30016
[2]	Chang J M, Fang M L, Zalcman L. Normality and fixed-points of meromorphic functions. Ark Mat, 2005, 43: 307–321 · Zbl 1095.30026
[3]	Chen H H, Fang M L. On the value distribution of f n f’. Sci China Ser A, 1995, 38: 789–798 · Zbl 0839.30026
[4]	Chen H H, Gu Y X. Improvement of Marty’s criterion and its application. Sci China Ser A, 1993, 36: 647–681 · Zbl 0777.30018
[5]	Fang M L, Zalcman L. A note on normality and shared values. J Aust Math Soc, 2004, 76: 141–150 · Zbl 1074.30032
[6]	Hayman W K. Meromorphic Functions. Oxford: Clarendon Press, 1964
[7]	Pang X C, Zalcman L. Normal families and shared values. Bull London Math Soc, 2000, 32: 325–331 · Zbl 1030.30031
[8]	Pang X C, Zalcman L. Normality and shared values. Ark Mat, 2000, 76: 171–182 · Zbl 1079.30044
[9]	Schiff J. Normal Families. Berlin: Springer-Verlag, 1993
[10]	Schwick W. Sharing values and normality. Arch Math, 1992, 59: 50–54 · Zbl 0758.30028
[11]	Yang L. Value Distribution Theory. Berlin: Springer-Verlag, 1993 · Zbl 0790.30018
[12]	Zalcman L. Normal families: new perspectives. Bull Amer Math Soc, 1998, 35: 215–230 · Zbl 1037.30021
[13]	Zhang Q C. Some normality criteria of meromorphic functions. Complex Var Elliptic Equ, 2008, 53: 791–795 · Zbl 1152.30033