Characteristic functions of two meromorphic functions weakly sharing three small functions with bi-weights

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 two meromorphic functions $f$ and $g$ weakly share three small functions $a_i(1\le i\le 3)$ with bi-weights $(n_i,k)$ satisfying $n_1{n}_2{n}_3>n_1+n_2+n_3+2$ then $$(1-\epsilon -{\delta }_{\epsilon })T(r,f)\le (2+\epsilon +{\delta }_{\epsilon })T(r,g)+S(r,g)$$ for every positive number $\epsilon$, where ${\delta }_{\epsilon }$ is explicitly estimated depending only on $\epsilon$ and $k$, so that ${\delta }_{\epsilon }$ tends to zero as $k$ tends to $+\mathrm{\infty }$.