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 \( [\psi '(x)]^2 + \psi ''(x) - \frac{{x^2 + 12}} {{12x^4 (x + 1)^2 }} \)and \( \frac{{x + 12}} {{12x^4 (x + 1)}} - \{ [\psi '(x)]^2 + \psi ''(x)\} \) are completely monotonic on (0,∞).
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References
M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Applied Mathematics Series 55, 9th printing, National Bureau of Standards, Washington, 1970.
H. Alzer, Sharp inequalities for the digamma and polygamma functions, Forum Math., 16 (2004), no. 2, 181–221.
H. Alzer and A. Z. Grinshpan, Inequalities for the gamma and q-gamma functions, J. Approx. Theory, 144 (2007), 67–83.
N. Batir, An interesting double inequality for Euler’s gamma function, J. Inequal. Pure Appl. Math., 5 (2004), no. 4, Art. 97.
N. Batir, On some properties of digamma and polygamma functions, J. Math. Anal. Appl., 328 (2007), no. 1, 452–465.
N. Batir, Some new inequalities for gamma and polygamma functions, J. Inequal. Pure Appl. Math., 6 (2005), no. 4, Art. 103.
B.-N. Guo and F. Qi, A class of completely monotonic functions involving divided differences of the psi and tri-gamma functions and some applications, J. Korean Math. Soc., 48 (2011), 655–667.
B.-N. Guo, F. Qi and H. M. Srivastava, Some uniqueness results for the nontrivially complete monotonicity of a class of functions involving the polygamma and related functions, Integral Transforms Spec. Funct., 21 (2010), 849–858.
D. K. Kazarinoff, OnWallis’ formula, Edinburgh Math. Notes, 40 (1956), 19–21.
D. S. Mitrinović, J. E. Pečarić and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht — Boston — London, 1993.
F. Qi, Bounds for the ratio of two gamma functions, J. Inequal. Appl., 2010 (2010), Article ID 493058, 84 pages.
F. Qi and B.-N. Guo, A class of completely monotonic functions involving divided differences of the psi and polygamma functions and some applications, http://arxiv.org/abs/0903.1430.
F. Qi and B.-N. Guo, A short proof of monotonicity of a function involving the psi and exponential functions, http://arxiv.org/abs/0902.2519.
F. Qi and B.-N. Guo, Completely monotonic functions involving divided differences of the di- and tri-gamma functions and some applications, Commun. Pure Appl. Anal., 8 (2009), 1975–1989.
F. Qi and B.-N. Guo, Necessary and sufficient conditions for a function involving divided differences of the di- and tri-gamma functions to be completely monotonic, http://arxiv.org/abs/0903.3071.
F. Qi and B.-N. Guo, Necessary and sufficient conditions for functions involving the tri- and tetra-gamma functions to be completely monotonic, Adv. Appl. Math., 44 (2010), 71–83.
F. Qi and B.-N. Guo, Sharp inequalities for the psi function and harmonic numbers, http://arxiv.org/abs/0902.2524.
F. Qi and Q.-M. Luo, Bounds for the ratio of two gamma functions-From Wendel’s and related inequalities to logarithmically completely monotonic functions, Banach J. Math. Anal., 6 (2012), 132–158.
D. V. Widder, The Laplace Transform, Princeton University Press, Princeton, 1946.
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Communicated by György Petruska
Partially supported by the Natural Science Basic Research Plan in Shaanxi Province of China (Program No. 2011JM1015).
Partially supported by the Science Foundation of Tianjin Polytechnic University.
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Zhao, JL., Guo, BN. & Qi, F. Complete monotonicity of two functions involving the tri-and tetra-gamma functions. Period Math Hung 65, 147–155 (2012). https://doi.org/10.1007/s10998-012-9562-x
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DOI: https://doi.org/10.1007/s10998-012-9562-x