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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}
\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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Authors: Jun-Ying Fu, Xu Zhang, Yun-Hua Zhao, He-Feng Tong, Dar-Zen Chen, and Mu-Hsuan Huang


Acupuncture, the most important nonpharmacological therapy in traditional Chinese medicine, has attracted significant attention since its introduction to the Western world. This study employs bibliometric analysis to examine the profile of publication activity related to it. The data are retrieved from the database of Science Citation Index Expanded during 1980–2009, and 7,592 papers are identified for analysis. This study finds that almost 20 % of papers are published in CAM journals, and the average cited times per acupuncture paper is 8.69. While the most cited article has been cited 2,109 times, however, 38.15 % of total publications have never been cited. Europe has the largest amount of authored papers with high h-index values; the USA has the largest number of publications on and citations of acupuncture based on country distribution, and this has continued as a significant rising trend. The proportion of collaborative papers shows this upward trend on the worldwide scale while the percentage shares of national collaborations are the highest. The USA produces the most international collaborative documents, although South Korea occupies the highest percentage figure for international collaborative papers. International collaborative papers are the most frequently cited. The average number of authors per paper is 3.69 in the top eight countries/regions. Papers contributed by South Korea are authored by the most people. International collaboration papers are authored by more people, except in Taiwan. South Korea's Kyung Hee University is ranked first in terms of number of papers while Harvard University in the USA accounts for the largest proportion of citations. The University of Exeter, Harvard University and Karolinska Institute have the highest h-index values.

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