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Mathematica Slovaca

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Volume 63, Issue 3

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A completely monotonic function involving the tri- and tetra-gamma functions

Bai-Ni Guo / Jiao-Lian Zhao
  • Department of Mathematics and Informatics, Weinan Teachers University, Weinan City, Shaanxi Province, 714000, China
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/ Feng Qi
Published Online: 2013-06-28 | DOI: https://doi.org/10.2478/s12175-013-0109-2

Abstract

The di-gamma function ψ(x) is defined on (0,∞) by $\psi (x) = \frac{{\Gamma '(x)}} {{\Gamma (x)}} $ and ψ (i)(x) for i ∈ ℕ denote the polygamma functions, where Γ(x) is the classical Euler’s gamma function. In this paper we prove that a function involving the difference between [ψ′(x)]2 + ψ″(x) and a proper fraction of x is completely monotonic on (0,∞).

MSC: Primary 26A48, 33B15; Secondary 26A51, 26D10, 65R10

Keywords: completely monotonic function; tri-gamma function; tetra-gamma function; polygamma function; inequality

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About the article

Published Online: 2013-06-28

Published in Print: 2013-06-01


Citation Information: Mathematica Slovaca, Volume 63, Issue 3, Pages 469–478, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.2478/s12175-013-0109-2.

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© 2013 Mathematical Institute, Slovak Academy of Sciences. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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[1]
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[2]
Feng Qi and Qiu-Ming Luo
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