Normality of Composite Analytic Functions and Sharing an Analytic Function

Qifeng Wu,Bing Xiao,Wenjun Yuan

doi:10.1155/2010/417480

Abstract

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 .

Highlights

Introduction and Main ResultsLet f z and g z be two nonconstant meromorphic functions in the whole complex plane C, and let a be a finite complex value or function
It is assumed that the reader is familiar with the standard notations and the basic results of Nevanlinna’s value-distribution theory
We denote by S r, f any function satisfying S r, f possibly outside of a set of finite measure

Summary

Introduction and Main Results

Let f z and g z be two nonconstant meromorphic functions in the whole complex plane C, and let a be a finite complex value or function. Let P z be a polynomial of degree p at least 2 and f z a transcendental entire function, and α z a nonconstant meromorphic function satisfying T r, α S r, f. Yuan et al 10 generalized Theorem G in another manner and proved the following result. Let α z be a nonconstant meromorphic function, F a family of analytic functions in a domain D, and R z a rational function of degree at least 2. If H ◦ f z and H ◦ g z share α z IM for each pair f z , g z ∈ F, and one of the following conditions holds:

Preliminary Lemmas

Proof of Theorem