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Comptes Rendus. Mathématique

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

fn(x)=1-lnx+lnGn(x+1)xandgn(x)=Gn(x+1)xx,x(0,),n,

where G n (z)=Γ n (z) (-1) n-1 and Γ n is the multiple gamma function of order n. In this work, our aim is to establish that f 2n (2n) (x) and (lng 2n (x)) (2n) are strictly completely monotonic on the positive half line for any positive integer n. In particular, we show that f 2 (x) and g 2 (x) are strictly completely monotonic and strictly logarithmically completely monotonic respectively on (0,3]. 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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