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# Inequalities for Mathieu’s series and positive definiteness

Author: В. Заставный

## Резуме

Получены точные неравенства для рядов Матяе. Установлена связя между неравенствами для рядов Матяе и положителяно определенными и вполне монотонными функциями.

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# Complete monotonicity of two functions involving the tri-and tetra-gamma functions

Authors: Jiao-Lian Zhao, Bai-Ni Guo and Feng Qi

## Abstract

The psi function ψ(x) is defined by ψ(x) = Γ′(x)/Γ(x) and ψ (i)(x), for i ∈ ℕ, denote the polygamma functions, where Γ(x) is the gamma function. In this paper, we prove that the functions

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$[\psi '(x)]^2 + \psi ''(x) - \frac{{x^2 + 12}} {{12x^4 (x + 1)^2 }}$$ \end{document}
and
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\frac{{x + 12}} {{12x^4 (x + 1)}} - \{ [\psi '(x)]^2 + \psi ''(x)\}$$ \end{document}
are completely monotonic on (0,∞).

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# Inequalities involving gamma and digamma functions

Authors: Horst Alzer and Chao-Ping Chen

. 1997 66 373 389 Alzer, H. and Berg, C. , Some classes of completely monotonic functions, Ann. Acad. Sci

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