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Some completely monotonic functions involving the gamma and polygamma functions

Part of: Elementary classical functions Inequalities

Published online by Cambridge University Press:  09 April 2009

Feng Qi ,
Bai-Ni Guo  and
Chao-Ping Chen
Feng Qi
Affiliation:
Research Institute of Mathematical Inequality Theory, Henan Polytechnic University, Jiaozuo City, Henan Province 454010, China, e-mail: qifeng@hpu.edu.cn, guobaini@hpu.edu.cn, chenchaoping@hpu.edu.cn
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Abstract

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The function [Γ(x + 1)]1/x(1 + 1/x)x/x is strictly logarithmically completely monotonic in (0, ∞). The function ψ″ (x + 2) + (1 + x2)/x2(1 + x)2 is strictly completely monotonic in (0, ∞).

Keywords

completely monotonic functionlogarithmically completely monotonic functiongamma functionpolygamma function.

MSC classification

Secondary: 33B15: Gamma, beta and polygamma functions 26D07: Inequalities involving other types of functions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 80 , Issue 1 , February 2006 , pp. 81 - 88
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Alzer, H. and Berg, C., ‘Some classes of completely monotonic functions’, Ann. Acad. Sci. Fenn. Math. 27 (2002), 445–460.Google Scholar
[2]Alzer, H. and Berg, C., ‘Some classes of completely monotonic functions, II’, Ramanujan J., to appear, available online at http://www.math.ku.dk/~berg.Google Scholar
[3]Berg, C. and Forst, G., Potential theory on locally compact Abelian groups, Ergebnisse der Math. 87 (Springer, Berlin, 1975).CrossRefGoogle Scholar
[4]Chen, Ch.-P. and Qi, F., ‘Inequalities relating to the gamma function’, Austral. J. Math. Anal. Appl. (1) 1 (2004), Art. 3.Google Scholar
[5]Qi, F. and Chen, Ch.-P., ‘A complete monotonicity property of the gamma function’, J. Math. Anal. Appl. 296 (2004), 603–607.CrossRefGoogle Scholar
[6]Qi, F. and Guo, B.-N., ‘Complete monotonicities of functions involving the gamma and digamma functions’, RGMIA Res. Rep. Coll. (1) 7 (2004), Art. 8.Google Scholar
[7]Qi, F., Guo, B.-N. and Chen, Ch.-P., ‘Some completely monotonic functions involving the gamma and polygamma functions’, RGMIA Res. Rep. Coll. (1) 7 (2004), Art. 5.Google Scholar
[8]Widder, D. V., The Laplace transform (Princeton University Press, Princeton, 1941).Google Scholar