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In the paper, the authors introduce a new concept of geometrically r-convex functions and establish some inequalities of Hermite-Hadamard type for this class of functions.

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Authors: Feng Qi and Jun-Xiang Cheng
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Abstract

For a long time, rankings overused in evaluating Chinese universities’ research performance. The relationship between research production and research quality hasn't been taken seriously in ranking systems. Most university rankings in China put more weight on research production rather than research quality. Recently, the developmental strategy of Chinese universities has shifted from ‘quantity’ to ‘quality’. As a result, a two-dimensional approach was developed in this article to balance ‘quantity’ and ‘quality’. The research production index and the research quality index were produced to locate research universities (RU) from Mainland China, Hong Kong (HK) and Taiwan (TW) in the two-dimensional graph. Fifty-nine RU were classified into three categories according to their locations, which indicated the relevant level of research performance. University of Hong Kong, National Taiwan University, Tsing Hua University and Peking University appeared to be leading universities in research performance. The result showed that the mainland universities were generally of higher research production and lower research quality than HK and TW universities, and proved that the merging tides of Chinese universities enlarged their research production while causing a low level of research quality as well.

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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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