A complete monotonicity property of the multiple gamma function

Das, Sourav

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

We consider the following functions

$$f_n\left(x\right)=1-lnx+\frac{ln{G}_n(x+1)}{x}\text{and}{g}_n\left(x\right)=\frac{\sqrt[x]{G_n(x+1)}}{x},x\in (0,\infty ),n\in \mathbb{N},$$

where $G_n\left(z\right)={\left({\Gamma }_n\left(z\right)\right)}^{{(-1)}^{n-1}}$ 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 ${(ln{g}_{2n}\left(x\right))}^{\left(2n\right)}$ 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 $(0,3]$. As application, we obtain new bounds for the Barnes G-function.

Classification : 33B15, 26D07

[1] Adamchik, Victor S. The multiple gamma function and its application to computation of series, Ramanujan J., Volume 9 (2005) no. 3, pp. 271-288 | DOI | MR | Zbl

[2] Barnes, Ernest W. The theory of the G-function, Quart. J., Volume 31 (1900), pp. 264-314 | Zbl

[3] Barnes, Ernest W. On the theory of the multiple Gamma function, Trans. Camb. Philos. Soc., Volume 19 (1904), pp. 374-439 | Zbl

[4] Batir, Necdet Inequalities for the double gamma function, J. Math. Anal. Appl., Volume 351 (2009) no. 1, pp. 182-185 | DOI | MR | Zbl

[5] Choi, Junesang Determinant of Laplacian on $S^3$, Math. Japon., Volume 40 (1994) no. 1, pp. 155-166 | MR | Zbl

[6] Choi, Junesang Determinants of the Laplacians on the $n$-dimensional unit sphere ${\mathbf{S}}^n$, Adv. Differ. Equ., Volume 2013 (2013), 236, 12 pages | MR | Zbl

[7] Choi, Junesang Multiple gamma functions and their applications, Analytic number theory, approximation theory, and special functions, Springer, 2014, pp. 93-129 | DOI | Zbl

[8] Das, Sourav Inequalities involving the multiple psi function, C. R. Math. Acad. Sci. Paris, Volume 356 (2018) no. 3, pp. 288-292 | MR | Zbl

[9] Das, Sourav; Pedersen, Henrik L.; Swaminathan, Anbhu Pick functions related to the triple Gamma function, J. Math. Anal. Appl., Volume 455 (2017) no. 2, pp. 1124-1138 | MR | Zbl

[10] Das, Sourav; Swaminathan, Anbhu Bounds for triple gamma functions and their ratios, J. Inequal. Appl., Volume 2016 (2016), 210, 11 pages | MR | Zbl

[11] Qi, Feng; Chen, Chao-Ping A complete monotonicity property of the gamma function, J. Math. Anal. Appl., Volume 296 (2004) no. 2, pp. 603-607 | MR | Zbl

[12] Ueno, Kimio; Nishizawa, Michitomo The multiple gamma function and its $q$-analogue, Quantum groups and quantum spaces (Warsaw, 1995) (Banach Center Publications), Institute of Mathematics of the Polish Academy of Sciences, 1995, pp. 429-441

[13] Vardi, Ilan Determinants of Laplacians and multiple gamma functions, SIAM J. Math. Anal., Volume 19 (1988) no. 2, pp. 493-507 | DOI | MR | Zbl

[14] Vignéras, Marie-France L’équation fonctionnelle de la fonction zêta de Selberg du groupe modulaire $\mathrm{PSL}(2,\mathbf{Z})$, Journées Arithmétiques de Luminy (Colloq. Internat. CNRS, Centre Univ. Luminy, Luminy, 1978) (Astérisque), Volume 61, Société Mathématique de France (1979), pp. 235-249 | Numdam | Zbl