A complete monotonicity property of the gamma function

Abstract

A logarithmically completely monotonic function is completely monotonic. The function 1 − ln x + 1x ln Γ (x + 1) is strictly completely monotonic on (0, ∞). The function x √ Γ (x + 1)/x is strictly logarithmically completely monotonic on (0, ∞).

Some Logarithmically Completely Monotonic Functions Related to the Gamma Function

“…The results in Proposition 1 generalize and extend those of [29]. (27) and (21). Therefore, there must be a > b.…”

“…Recall also [5,27,28] that a positive function f is said to be logarithmically completely monotonic on an interval I if f has derivatives of all orders on I and (3) (−1)…”

“…Among other things, it is proved in [6,9,27] that a logarithmically completely monotonic function is always completely monotonic, that is, L[I] ⊂ C [I], but not conversely, since a convex function may not be logarithmically convex (see [24,p. 7,Remark.…”

Two logarithmically completely monotonic functions connected with gamma function

“…For more information on the logarithmically completely monotonic functions defined by Definition 2, please refer to [4,5,8,11,12,13], especially [7,10,15], and the references therein.…”

Schur-convexity of the Catalan–Qi function related to the Catalan numbers

“…Recall from [1,17,18] that an infinitely differentiable and positive function f is said to be logarithmically completely monotonic on an interval I if…”