Monotonicity properties for a ratio of finite many gamma functions

Abstract

In the paper, the authors consider a ratio of finite many gamma functions and find its monotonicity properties such as complete monotonicity, the Bernstein function property, and logarithmically complete monotonicity.

Some logarithmically completely monotonic functions and inequalities for multinomial coefficients and multivariate beta functions

“…This paper is a revised version of the preprint [33] whose first version was announced almost at the same time as the preprint [21] which has been formally published as [20]. This paper is a companion of the papers [22,23,28,31,32].…”

Increasing property and logarithmic convexity of functions involving Riemann zeta function

“…In this paper, we use the notation For more information and recent developments of the gamma function Γ(z) and its logarithmic derivatives ψ (n) (z) for n ≥ 0, please refer to [1,Chapter 6], [25,Chapter 3], or recently published papers [14,18,20,21,31] and closely related references therein.…”

“…Recall from [11, Chapter XIII], [23, Chapter 1], [30, Chapter IV], and recently published papers [14,18,20,21] that (1) a function q(x) is said to be completely monotonic on an interval I if it is infinitely differentiable and (−1) n q (n) (x) ≥ 0 for n ≥ 0 on I. (2) a positive function q(x) is said to be logarithmically completely monotonic on an interval I ⊆ R if it is infinitely differentiable and its logarithm ln f (x) satisfies (−1) k [ln q(x)] (k) ≥ 0 for k ∈ N on I.…”

Complete Monotonicity for a New Ratio of Finitely Many Gamma Functions

“…This paper is motivated by a sequence of papers [2,11,12,18,20,30,34,35]. For a detailed review and survey, please consult the papers [20,30,34,35] and the closely related references therein.…”