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

TL;DR

This paper proves the complete monotonicity of certain functions involving the $q$-trigamma and $q$-tetragamma functions for different ranges of $q$, leading to new inequalities and properties of the $q$-digamma function.

Contribution

The paper introduces two approaches to establish the complete monotonicity of specific $q$-special functions, providing new bounds and properties.

Findings

Proves complete monotonicity of $[ ext{psi}_q'(x)]^2 + ext{psi}_q''(x)$ for $q>1$.

Establishes complete monotonicity of $[ ext{psi}_q'(x) - ext{ln} q]^2 + ext{psi}_q''(x)$ for $0<q<1$.

Derives new inequalities and monotonic properties of the $q$-digamma function.

Abstract

Let $\psi_q(x)$, $\psi_q'(x)$, and $\psi_q''(x)$ 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 $[\psi'_q(x)]^2+\psi''_q(x)$ for $q>1$ and $[\psi_{q}'(x)-\ln q]^2 +\psi''_{q}(x)$ 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 $\psi_q(x)$.