Difference analogues of the logarithmic derivative lemma and their applications in value distribution

Abstract

Difference analogues of the logarithmic derivative lemma play an important part in investigating the value distribution of complex difference polynomials and determining the existence of meromorphic solutions of complex difference equations. In this paper, we establish several variants on the difference analogues of the logarithmic derivative lemma. Specifically, we obtain

$$m\left( r,\frac{f(z+c_{1})f(z+c_{2})\cdots f(z+c_{n})-a}{f(z)^{n}-a}\right) =S(r,f),$$

where f(z) is a transcendental meromorphic function belonging to a specific function class and n is a positive integer. Subsequently, we demonstrate some applications of the variants of difference analogues of the logarithmic derivative lemma by establishing uniqueness theorems of value sharing and value distribution properties of complex difference polynomials.