On Completely Monotonic and Related Functions

Abstract

We deal with several classes of functions, such as, completely monotonic functions, absolutely monotonic functions, logarithmically completely monotonic functions, Stieltjes functions, and Bernstein functions. We give several interesting relations among theses classes of functions as well as various examples and applications. We show that several special functions belong to the aforementioned classes. We give a survey of recent results in this area and provide new proofs as well as additional remarks and comments.

Completely Monotonic Gamma Ratio and Infinitely Divisible H-Function of Fox

“…The result follows from Theorem 7 combined with [20, Definition 5.6] and [20,Theorem 5.9]. See also [12,Theorem 7]. Remark 5.…”

“…We plan to study its behavior in the neighborhood of the singular point log(1/ρ) in our future work. Corollary 2 Suppose µ > 0, a, b > 0 and conditions (12) and (16) are satisfied (in particular, (12) is true under Theorem 5). Then the function…”

“…Corollary 2 Suppose µ > 0, a, b > 0 and conditions (12) and ( 16) are satisfied (in particular, ( 12) is true under Theorem 5). Then the function…”

Nonparametric analysis of a duration model with stochastic unobserved heterogeneity

“…11 Note that the assumptions in Koumandos (2014) are expressed in terms of moments of Z (1), e.g. letting m k be the k th moment of Z (1), the assumption is stated as Lin (2017), this assumption is equivalent to requiring that the moment generating function of Z (1) exists.…”