What is the definition of two meromorphic functions sharing a small function?

TL;DR

This paper discusses the definition of sharing a small function between meromorphic functions, proposing a modification at poles of the shared function, and highlights potential gaps in existing proofs.

Contribution

It suggests a revised definition for sharing small functions in meromorphic functions and addresses gaps in the current mathematical literature.

Findings

Proposes a modified definition at poles of the shared function.

Identifies gaps in existing proofs in the literature.

Highlights the importance of precise definitions in complex analysis.

Abstract

Two meromorphic functions $f(z)$ and $g(z)$ sharing a small function $\alpha(z)$ usually is defined in terms of vanishing of the functions $f-\alpha$ and $g-\alpha$. We argue that it would be better to modify this definition at the points where $\alpha$ has poles. Related to this issue we also point out some possible gaps in proofs in the published literature.