A complete monotonicity property of the multiple gamma function

Abstract

We consider the following functions fn (x) = 1− ln x + lnGn (x +1) x and gn (x) = x Gn (x +1) x , x ∈ (0,∞), n ∈N, where Gn (z) = (Γn (z))(−1) and Γn is the multiple gamma function of order n. In this work, our aim is to establish that f (2n) 2n (x) and (ln g2n (x)) (2n) are strictly completely monotonic on the positive half line for any positive integer n. In particular, we show that f2(x) and g2(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. 2020 Mathematics Subject Classification. 33B15, 26D07. Manuscript received 2nd August 2020, revised and accepted 8th September 2020.