Nevanlinna's five-value theorem for algebroid functions

By using the second main theorem of the algebroid function, we study the following problem. Let $W_{1}(z)$ and $W_{2}(z)$ be two $\nu$-valued non-constant algebroid functions, $a_{j}\,(j=1,2,\ldots,q)$ be $q\geq 4\nu+1$ distinct complex numbers or $\infty$. Suppose that ${k_{1}\geq k_{2}\geq \ldots\geq k_{q},m}$ are positive integers or $\infty$, $1\leq m\leq q$ and $\delta_{j} \geq 0$, $j=1,2,\ldots,q$, are such that
\begin{equation*} \left(1+\frac{1}{k_{m}}\right)\sum_{j=m}^{q}\frac{1}{1+k_{j}}+3\nu +\sum_{j=1}^{q}\delta_{j}<(q-m-1)\left(1+\frac{1}{k_{m}}\right)+m. \end{equation*}

Let $B_{j}=\overline{E}_{k_{j}}(a_{j},f)\backslash\overline{E}_{k_{j}}(a_{j},g)$ for $j=1,2,\ldots,q.$ If
\begin{equation*} \overline{N}_{B_{j}}(r,\frac{1}{W_{1}-a_{j}})\leq \delta_{j}T(r,W_{1}) \end{equation*}
and
\begin{equation*} \liminf_{r\rightarrow \infty}^{}\frac{\sum\limits_{j=1}^{q} \overline{N}_{k_{j}}(r,\frac{1}{W_{1}-a_{j}})} {\sum\limits_{j=1}^{q}\overline{N}_{k_{j}}(r,\frac{1}{W_{2}-a_{j}})}> \frac{\nu k_{m}}{(1+k_{m})\sum\limits_{j=1}^{q} \frac{k_{j}}{k_{j}+1}-2\nu(1+k_{m}) +(m-2\nu-\sum\limits_{j=1}^{q}\delta_{j})k_{m}}, \end{equation*}
then $W_{1}(z)\equiv W_{2}(z).$ This result improves and generalizes the previous results given by Xuan and Gao.