Complete monotonicity of functions involving the $q$-trigamma and $q$-tetragamma functions

Abstract

Let ${\psi }_q(x)$, ${\psi }_q^{\prime }(x)$, and ${\psi }_q^{\prime \prime }(x)$ for $q>0$ stand respectively for the $q$-digamma, $q$-trigamma, and $q$-tetragamma functions. In the paper, the author proves along two different approaches that the functions $[{\psi }_q^{\prime }(x){]}^2+{\psi }_q^{\prime \prime }(x)$ for $q>1$ and $[{\psi }_q^{\prime }(x)-\ln q{]}^2+{\psi }_q^{\prime \prime }(x)$ for $0<q<1$ are completely monotonic on $(0,\infty )$. Applying these results, the author derives monotonic properties of four functions involving the $q$-digamma function ${\psi }_q(x)$ and two double inequalities for bounding the $q$-digamma function ${\psi }_q(x)$.

10 pages

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