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Mathematics of Computation

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Monotonicity of some functions involving the gamma and psi functions


Author: Stamatis Koumandos
Journal: Math. Comp. 77 (2008), 2261-2275
MSC (2000): Primary 33B15; Secondary 26D20, 26D15
DOI: https://doi.org/10.1090/S0025-5718-08-02140-6
Published electronically: May 14, 2008
MathSciNet review: 2429884
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Abstract: Let $ L(x):=x-\frac{\Gamma(x+t)}{\Gamma(x+s)}\,x^{s-t+1}$, where $ \Gamma(x)$ is Euler's gamma function. We determine conditions for the numbers $ s,\,t$ so that the function $ \Phi(x):=-\frac{\Gamma(x+s)}{\Gamma(x+t)}\,x^{t-s-1}\,L^{\prime\prime}(x)$ is strongly completely monotonic on $ (0,\,\infty)$. Through this result we obtain some inequalities involving the ratio of gamma functions and provide some applications in the context of trigonometric sum estimation. We also give several other examples of strongly completely monotonic functions defined in terms of $ \Gamma$ and $ \psi:=\Gamma^{\prime}/\Gamma$ functions. Some limiting and particular cases are also considered.


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Additional Information

Stamatis Koumandos
Affiliation: Department of Mathematics and Statistics, University of Cyprus, P. O. Box 20537, 1678 Nicosia, Cyprus
Email: skoumand@ucy.ac.cy

DOI: https://doi.org/10.1090/S0025-5718-08-02140-6
Keywords: Gamma function, psi function, completely monotonic functions
Received by editor(s): June 5, 2007
Published electronically: May 14, 2008
Article copyright: © Copyright 2008 American Mathematical Society