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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 $\psi(x)$. Thisgeneralizes a known result.

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