Comptes Rendus
no. 8
Théorie des nombres
A complete monotonicity property of the multiple gamma function
Sourav Das1 
1 Department of Mathematics, National Institute of Technology Jamshedpur, Jharkhand-831014, India
Comptes Rendus. Mathématique, Volume 358 (2020) no. 8, pp. 917-922.

We consider the following functions

f n (x)=1-lnx+lnG n (x+1) xandg n (x)=G n (x+1) x x,x∈(0,∞),n∈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 : 10.5802/crmath.115
Classification : 33B15, 26D07
Sourav Das 1

1 Department of Mathematics, National Institute of Technology Jamshedpur, Jharkhand-831014, India
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     doi = {10.5802/crmath.115},
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Sourav Das. A complete monotonicity property of the multiple gamma function. Comptes Rendus. Mathématique, Volume 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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