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

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.