Normality of Composite Analytic Functions and Sharing an Analytic Function
1 Shaozhou Normal College, Shaoguan University, Shaoguan 512009, China
2 Department of Mathematics, Xinjiang Normal University, Urumqi 830054, China
3 School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China
Fixed Point Theory and Applications 2010, 2010:417480 doi:10.1155/2010/417480Published: 18 October 2010
A result of Hinchliffe (2003) is extended to transcendental entire function, and an alternative proof is given in this paper. Our main result is as follows: let be an analytic function, a family of analytic functions in a domain , and a transcendental entire function. If and share IM for each pair , and one of the following conditions holds: (1) has at least two distinct zeros for any ; (2) is nonconstant, and there exists such that has only one distinct zero , and suppose that the multiplicities and of zeros of and at , respectively, satisfy , for each , where ; (3) there exists a such that has no zero, and is nonconstant, then is normal in .