Let ψq(x)\documentclass[12pt]{minimal}
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\begin{document}$$\psi _q(x)$$\end{document}, ψq′(x)\documentclass[12pt]{minimal}
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\begin{document}$$\psi _q'(x)$$\end{document}, and ψq′′(x)\documentclass[12pt]{minimal}
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\begin{document}$$\psi _q''(x)$$\end{document} for q>0\documentclass[12pt]{minimal}
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\begin{document}$$q>0$$\end{document} stand respectively for the q\documentclass[12pt]{minimal}
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\begin{document}$$q$$\end{document}-digamma, q\documentclass[12pt]{minimal}
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\begin{document}$$q$$\end{document}-trigamma, and q\documentclass[12pt]{minimal}
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\begin{document}$$q$$\end{document}-tetragamma functions. In the paper, the author proves along two different approaches that the functions [ψq′(x)]2+ψq′′(x)\documentclass[12pt]{minimal}
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\begin{document}$$[\psi '_q(x)]^2+\psi ''_q(x)$$\end{document} for q>1\documentclass[12pt]{minimal}
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\begin{document}$$q>1$$\end{document} and [ψq′(x)-lnq]2+ψq′′(x)\documentclass[12pt]{minimal}
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\begin{document}$$[\psi _{q}'(x)-\ln q]^2 +\psi ''_{q}(x)$$\end{document} for 0<q<1\documentclass[12pt]{minimal}
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\begin{document}$$0<q<1$$\end{document} are completely monotonic on (0,∞)\documentclass[12pt]{minimal}
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\begin{document}$$(0,\infty )$$\end{document}. Applying these results, the author derives monotonic properties of four functions involving the q\documentclass[12pt]{minimal}
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\begin{document}$$q$$\end{document}-digamma function ψq(x)\documentclass[12pt]{minimal}
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\begin{document}$$\psi _q(x)$$\end{document} and two double inequalities for bounding the q\documentclass[12pt]{minimal}
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\begin{document}$$q$$\end{document}-digamma function ψq(x)\documentclass[12pt]{minimal}
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\begin{document}$$\psi _q(x)$$\end{document}.
