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 $n_1n_2n_3>n_1+n_2+n_3+2$ 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 $+\infty$.