Uniqueness theorems for meromorphic functions on annuli

In this paper, we discuss the uniqueness problems of meromorphic functions on annuli. We prove a general theorem on the uniqueness of meromorphic functions on annuli. An analogue of a famous Nevanlinna's five-value theorem is proposed. The main result in this paper is an analog of a result on the plane $\mathbb{C}$ obtained by H.S. Gopalkrishna and Subhas S. Bhoosnurmath for an annuli. That is, let $f_{1}(z)$ and $f_{2}(z)$ be two transcendental meromorphic functions on the annulus $\mathbb{A}=\left\{z:\frac{1}{R_{0}}<|z|<R_{0}\right\}$, where $1<R_{0}\leq +\infty.$ Let $a_{j}$, $j=1,2,\ldots,q)$, be $q$ distinct complex numbers in $\overline{\mathbb{C}}$, and $k_{j}$, $j=1,2,\ldots,q$ be positive integers or $\infty$ satisfying
\begin{equation*} k_{1}\geq k_{2}\geq \ldots \geq k_{q}. \end{equation*}
If
\begin{equation*} \overline{E}_{k_{j})}(a_{j},f_{1})=\overline{E}_{k_{j})}(a_{j},f_{2}), j=1,2,\ldots,q, \end{equation*}
and
\begin{equation*} \sum_{j=2}^{q}\frac{k_{j}}{k_{j}+1}-\frac{k_{1}}{k_{1}+1}>2, \end{equation*}
then $f_{1}(z)\equiv f_{2}(z).$