Théorie des nombres
A complete monotonicity property of the multiple gamma function
Comptes Rendus. Mathématique, Tome 358 (2020) no. 8, pp. 917-922.

We consider the following functions

 ${f}_{n}\left(x\right)=1-lnx+\frac{ln{G}_{n}\left(x+1\right)}{x}\phantom{\rule{4pt}{0ex}}\text{and}\phantom{\rule{4pt}{0ex}}{g}_{n}\left(x\right)=\frac{\sqrt[x]{{G}_{n}\left(x+1\right)}}{x},\phantom{\rule{0.277778em}{0ex}}x\in \left(0,\infty \right),\phantom{\rule{0.277778em}{0ex}}n\in ℕ,$

where ${G}_{n}\left(z\right)={\left({\Gamma }_{n}\left(z\right)\right)}^{{\left(-1\right)}^{n-1}}$ and ${\Gamma }_{n}$ is the multiple gamma function of order $n$. In this work, our aim is to establish that ${f}_{2n}^{\left(2n\right)}\left(x\right)$ and ${\left(ln{g}_{2n}\left(x\right)\right)}^{\left(2n\right)}$ are strictly completely monotonic on the positive half line for any positive integer $n.$ In particular, we show that ${f}_{2}\left(x\right)$ and ${g}_{2}\left(x\right)$ are strictly completely monotonic and strictly logarithmically completely monotonic respectively on $\left(0,3\right]$. As application, we obtain new bounds for the Barnes G-function.

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DOI : https://doi.org/10.5802/crmath.115
Classification : 33B15,  26D07
@article{CRMATH_2020__358_8_917_0,
author = {Sourav Das},
title = {A complete monotonicity property of the multiple gamma function},
journal = {Comptes Rendus. Math\'ematique},
pages = {917--922},
publisher = {Acad\'emie des sciences, Paris},
volume = {358},
number = {8},
year = {2020},
doi = {10.5802/crmath.115},
language = {en},
}
Sourav Das. A complete monotonicity property of the multiple gamma function. Comptes Rendus. Mathématique, Tome 358 (2020) no. 8, pp. 917-922. doi : 10.5802/crmath.115. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.115/

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