Complete monotonicity of a function involving the p-psi function and alternative proofs - Krasniqi, Valmir; Qi, Feng (Global Journal of Mathematical Analysis, 2014)
In the paper, the authors 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 $\psi(x)$.Keywords: completely monotonic function; necessary and sufficient condition; p-gamma function; p-psi function; Inequality MSC: Primary 33D05; Secondary 26A48, 33B15, 33E50