  In the paper the authors alternatively prove that the function
$x^\alpha\big[\ln\frac{px}{x+p+1}-\psi_p(x)\big]$ 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.

Classical Analysis and ODEs