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

Let ${\psi }_q(x)$, ${\psi }_q^{\mathrm{\prime }}(x)$, and ${\psi }_q^{\mathrm{\prime }\mathrm{\prime }}(x)$ 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 $[{\psi }_q^{\mathrm{\prime }}(x){]}^2+{\psi }_q^{\mathrm{\prime }\mathrm{\prime }}(x)$ for $q>1$ and $[\psi_{q}‘(x)-\ln q]^2 +\psi’'_{q}(x)$for$0<q<1$arecompletelymonotonicon$(0,\infty)$. 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 ${\psi }_q(x)$.