Some completely monotonic functions involving the gamma and polygamma functions

Abstract

The function [V(x + l)]

Two New Proofs of the Complete Monotonicity of a Function Involving the Psi Function

“…Utilization of (18) and (19) and combination of (23), (27) and (28) yield that lim x→∞ θ 1 (x) = 0. The inequality (29) means that lim x→∞ θ…”

“…For more information on the logarithmically completely monotonic functions, please refer to [5,10,19] and related references therein.…”

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.…”

Completely monotonic functions involving the gamma and $q$-gamma functions

“…We remind the reader that any function f (x) = e −h(x) is completely monotonic if h is completely monotonic. We call such functions f (x) logarithmically completely monotonic [8,24,25]. Berg [8] points out that these functions are the same as those studied by Horn [20] under the name infinitely divisible completely monotonic functions.…”