A complete monotonicity property of the multiple gamma function

Abstract

We consider the following functions f n (x) = 1 − ln x + lnG n (x + 1) x and g n (x) = x G n (x + 1) 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 (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.