A complete monotonicity property of the multiple gamma function

Abstract

We consider the following functions \[ f_n(x)=1-\ln x+\frac{\ln G_n(x+1)}{x} \text{ and }g_n(x)=\frac{\@root x \of {G_n(x+1)}}{x},\; x\in (0,\infty ),\; n\in \mathbb{N}, \] where $G_n(z)=\left(\Gamma _n(z)\right)^{(-1)^{n-1}}$ and $\Gamma _n$ is the multiple gamma function of order $n$. In this work, our aim is to establish that $f_{2n}^{(2n)}(x)$ and $(\ln g_{2n}(x))^{(2n)}$ 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.