Difference "abc" theorem for entire functions and Difference analogue of truncated version of Nevanlinna second main theorem

TL;DR

This paper develops difference analogues of classical theorems for entire functions, including a difference abc theorem and a truncated Nevanlinna second main theorem, with applications to difference Fermat equations.

Contribution

It introduces new difference versions of key complex analysis theorems for entire functions, expanding the theoretical framework and applications.

Findings

Established a difference abc theorem for entire functions of order less than 1.

Derived a difference truncated Nevanlinna second main theorem with implications for meromorphic functions.

Applied results to entire solutions of difference Fermat functional equations.

Abstract

In this paper, we focus on the difference analogue of the Stothers-Mason theorem for entire functions of order less than 1, which can be seen as difference $abc$ theorem for entire functions. We also obtain the difference analogue of truncated version of Nevanlinna second main theorem which reveals that a subnormal meromorphic function $f(z)$ such that $\Delta f(z)\not\equiv 0$ cannot have too many points with long length in the complex plane. Both theorems depend on new definitions of the length of poles and zeros of a given meromorphic function in a domain. As for the application, we consider entire solutions of difference Fermat functional equations.