Value Sharing Results forq-Shifts Difference Polynomials

Yong Liu,Yinhong Cao,Hongxun Yi,Xiaoguang Qi

doi:10.1155/2013/152069

Abstract

We investigate the zero distribution ofq-shift difference polynomials of meromorphic functions with zero order and obtain some results that extend previous results of K. Liu et al.

Highlights

Many articles have focused on value distribution in difference analogues of meromorphic functions
There has been an increasing interest in studying the uniqueness problems related to meromorphic functions and their shifts or their difference operators
Our aim in this article is to investigate the uniqueness problems of q-difference polynomials

Summary

Preliminary Lemmas

The following lemma is a q-difference analogue of the logarithmic derivative lemma. Lemma 5 (see [14]). Let f(z) be a meromorphic function of zero order, and let c and q be two nonzero complex numbers. The following lemma is essential in our proof and is due to Heittokangas et al, see [12, Theorems 6 and 7]. Let f(z) be a meromorphic function of finite order, and let c ≠ 0 be fixed. Let f(z) be a meromorphic function with ρ(f) = 0, and let c and q be two nonzero complex numbers. For the case |q| ≤ 1, we can use the same method in the proof. Let f be a nonconstant meromorphic function of zero order, and let c and q be two nonzero complex numbers. In order to prove Theorem 2, we need the following lemma. Let F and G be two nonconstant meromorphic functions, and let F and G share 1 IM.

Proof of Theorem 1

Proof of Theorem 2

Proof of Theorem 3