In this paper we alternatively prove that the function $x^\alpha
\big[\ln\frac{px}{x+p}-\psi_p(x)\big]$ is completely monotonic on $(0,\infty)$
if and only if $\alpha \le 1$, where $p\in\mathbb{N}$ and $\psi_p(x)$ is the
$p$-analogue of the classical psi function $\psi(x)$. This generalizes a known
result.