A function ρ:[0, ∞) → (0,1] is a completely monotonic function if and only if ρ(‖x ‖ 2) is positive definite on R d for all d and thus it represents the correlation function of a weakly stationary and isotropic Gaussian random field. Radial positive definite functions are also of importance as they represent characteristic functions of spherically symmetric probability distributions. In this paper, we analyze the function ( β x ρ(β,γ)(x) = 1 − 1 + xβ)γ, x ≥ 0, β,γ> 0, called the Dagum function, and show those ranges for which this function is completely monotonic, that is, positive definite, on any d-dimensional Euclidean space. Important relations arise with other families of completely monotonic and logarithmically completely monotonic functions.

( v2.0 released June 2011) We show that many functions containing the Lambert W function are Stieltjes functions. We extend the known properties of the set of Stieltjes functions and also prove a generalization of a conjecture of Jackson, Procacci & Sokal. In addition, we consider the relationship of functions of W to the class of completely monotonic functions and show that W is a complete Bernstein function.