Meromorphic functions bi-weighted weakly sharing pairs of small functions

Two meromorphic functions $f$ and $g$ are said to weakly share a small function $a$ with bi-weight $(n,k)$ if the functions $f-a$ and $g-a$ have the same zeros with multiplicities truncated at level $n+1$, while zeros whose multiplicities exceed $k$ are disregarded. In this article, we show that if $f$ and $g$ weakly share three distinct small functions with suitable bi-weights and are not related by a quasi-M\"obius transformation, then for every other small function $c$, the counting function $N(r,\nu_f^c)$ is asymptotically equivalent to the characteristic function $T(r,f)$. Moreover, the truncated counting function $N_{(3}(r,\nu_f^c)$, which counts only zeros of multiplicity at least $3$, is negligible. As an application, we further prove that $f$ and $g$ must be related by a quasi-M\"obius transformation provided that they satisfy an additional condition, which is weaker than the usual assumption that they share a fourth pair of small functions.