Complete monotonicity of a function involving the $p$-psi function and alternative proofs

In the paper the authors alternatively prove that the function $x^{\alpha }[\ln \frac{px}{x+p+1}-{\psi }_p(x)]$ is completely monotonic on $(0,\mathrm{\infty })$ 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.