  For any $k \in \mathbb N$, we discuss the problem of determining completely
alternating sequence of rational functions of the form
$\frac{p(x)}{\prod_{i=1}^{k}(x+b_i)}$, where $p$ is a polynomial of degree
at-most $k+1$ and $b_1,\ldots,b_k$ are distinct positive real numbers. In
particular, we also prove that the sequences corresponding to the rational
functions $\frac{\prod_{i=1}^{k}(x+a_i)}{\prod_{i=1}^{k}(x+b_i)}$ and
$\frac{\prod_{i=1}^{k+1}(x+a_i)}{\prod_{i=1}^{k}(x+b_i)}$ are completely
alternating if it satisfies $0< a_1 < b_1 < a_2 < b_2 < \ldots < a_k < b_k $
and $0< a_1 < b_1 < a_2 < b_2 < \ldots < a_k < b_k < a_{k+1}$, respectively.
Moreover, we also show a characterization of a class of joint completely
monotone nets in two variables with the help of certain completely alternating
sequences.
