Journal of Applied Mathematics
Volume 2012 (2012), Article ID 312324, 7 pages
http://dx.doi.org/10.1155/2012/312324
Research Article

Normal Criterion Concerning Shared Values

Wei Chen,1 Yingying Zhang,1 Jiwen Zeng,2 and Honggen Tian1

1School of Mathematical Sciences, Xinjiang Normal University, Xinjiang, Urumqi 830054, China
2School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361105, China

Received 12 June 2012; Accepted 21 August 2012

Academic Editor: Yansheng Liu

Copyright © 2012 Wei Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study normal criterion of meromorphic functions shared values, we obtain the following. Let 𝐹 be a family of meromorphic functions in a domain 𝐷 , such that function 𝑓 ∈ 𝐹 has zeros of multiplicity at least 2, there exists nonzero complex numbers 𝑏 𝑓 , 𝑐 𝑓 depending on 𝑓 satisfying ( i ) 𝑏 𝑓 / 𝑐 𝑓 is a constant;   ( i i ) m i n { 𝜎 ( 0 , 𝑏 𝑓 ) , 𝜎 ( 0 , 𝑐 𝑓 ) , 𝜎 ( 𝑏 𝑓 , 𝑐 𝑓 ) ≥ 𝑚 } for some 𝑚 > 0 ;   ( i i i ) ( 1 / 𝑐 𝑓 𝑘 − 1 ) ( 𝑓 ) 𝑘 ( 𝑧 ) + 𝑓 ( 𝑧 ) ≠ 𝑏 𝑘 𝑓 / 𝑐 𝑓 𝑘 − 1 or ( 1 / 𝑐 𝑓 𝑘 − 1 ) ( 𝑓 ) 𝑘 ( 𝑧 ) + 𝑓 ( 𝑧 ) = 𝑏 𝑘 𝑓 / 𝑐 𝑓 𝑘 − 1 ⇒ 𝑓 ( 𝑧 ) = 𝑏 𝑓 , then 𝐹 is normal. These results improve some earlier previous results.