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arXiv · 1301.0155

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

Feng Qi
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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 $\psi_q(x)$.

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Feng Qi. 2013-01-02. Complete monotonicity of functions involving the $q$-trigamma and $q$-tetragamma functions. https://doi.org/10.1007/s11425-012-4562-0

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