A complete monotonicity property of the multiple gamma function

Sourav Das

doi:10.5802/crmath.115

Number Theory

A complete monotonicity property of the multiple gamma function

Sourav Das ¹

¹ Department of Mathematics, National Institute of Technology Jamshedpur, Jharkhand-831014, India

Comptes Rendus. Mathématique, Volume 358 (2020) no. 8, pp. 917-922.

Abstract

We consider the following functions

f_{n} (x) = 1 - ln x + \frac{ln G_{n} (x + 1)}{x} and g_{n} (x) = \frac{\sqrt[x]{G_{n} (x + 1)}}{x}, x \in (0, \infty), n \in 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 $f_{2 n}^{(2 n)} (x)$ and ${(ln g_{2 n} (x))}^{(2 n)}$ 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.

Received: 2020-08-02
Revised: 2020-09-08
Accepted: 2020-09-08
Published online: 2020-12-03

DOI: 10.5802/crmath.115

Classification: 33B15, 26D07

Author's affiliations:

Sourav Das ¹

¹ Department of Mathematics, National Institute of Technology Jamshedpur, Jharkhand-831014, India

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Sourav Das},
     title = {A complete monotonicity property of the multiple gamma function},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {917--922},
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     volume = {358},
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     year = {2020},
     doi = {10.5802/crmath.115},
     language = {en},
}

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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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