A probabilistic interpretation of complete monotonicity

On some inequalities for the gamma and psi functions

“…In 1974, C. H. Kimberling [28] established the following property of completely monotonic functions: If f is continuous on [0, ∞) and completely monotonic on (0, ∞) and satisfies 0 < f(x) ≤ 1 for all x ≥ 0, then log(f ) is super-additive on [0, ∞).…”

Properties of hierarchical Archimedean copulas

“…(−1) i φ (i) (x) ≥ 0 for all i ≥ 0. The class of feasible generator functions we define by (see Kimberling (1974) …”

Length functions of lemniscates

“…This class was introduced and extensively studied by S. Bernstein [5] and D. Widder [30] in connection with the so-called completely (or absolutely) monotonic analytic functions (see the definition in Section 3.2). We only mention a deep penetration of the both classes into complex analysis, inequalities analysis [2], special functions [25], probability theory [19], radial-function interpolation [29], harmonic analysis on semigroups [3] (for further discussion and references, see recent survey [4]). …”

“…The latter limit does exist for every regular value τ of u(x) (even if u is only locally Lipschitz in D [15, § 3.2]) and (19) follows.…”

“…Thus, applying (19) to v = h∇u we obtain (15). Moreover, it follows from (17) that (15) can be written in the form…”