On one problem of uniqueness of meromorphic functions concerning small functions

Abstract:

In this paper, we show that if two non-constant meromorphic functions $f$ and $g$ satisfy $\overline {E}(a_{j},k,f)=\overline {E}(a_{j},k,g)$ for $j=1,2,\dots ,5$, where $a_{j}$ are five distinct small functions with respect to $f$ and $g$, and $k$ is a positive integer or $\infty$ with $k\geq 14$, then $f\equiv g$. As a special case this also answers the long-standing problem on uniqueness of meromorphic functions concerning small functions.