On certain type of difference polynomials of meromorphic functions

TL;DR

This paper studies the zeros of certain difference polynomials of meromorphic functions, deriving inequalities and analogues of classical results, thereby advancing the understanding of their value distribution and zero behavior.

Contribution

It introduces new inequalities and difference analogues of Hayman's classical results, improving upon previous findings in the value distribution of difference polynomials.

Findings

Derived inequalities relating zero counting functions and characteristic functions.

Established difference analogues of Hayman's classical results.

Improved previous results on zeros of difference polynomials.

Abstract

In this paper, we investigate zeros of difference polynomials of the form $f(z)^nH(z, f)-s(z)$, where $f(z)$ is a meromorphic function, $H(z, f)$ is a difference polynomial of $f(z)$ and $s(z)$ is a small function. We first obtain some inequalities for the relationship of the zero counting function of $f(z)^nH(z, f)-s(z)$ and the characteristic function and pole counting function of $f(z)$. Based on these inequalities, we establish some difference analogues of a classical result of Hayman for meromorphic functions. Some special cases are also investigated. These results improve previous findings.