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Monotonicity of some functions involving the gamma and psi functions

Authors:	Stamatis Koumandos

Institution:	Department of Mathematics and Statistics, University of Cyprus, P. O. Box 20537, 1678 Nicosia, Cyprus

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.

Keywords:	Gamma function psi function completely monotonic functions

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