The Dagum family of isotropic correlation functions

A function $\rho: [0, \infty) \to (0,1]$ is a completely monotonic function if and only if $\rho(||\xx||^2)$ is positive definite on $\RR^d$ 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 the characteristic function of spherically symmetric probability distributions.In this paper we analyse the function $\rho(\b,\gamma)(x)=1-(\frac{x^\b}{1+x^\b})^\gamma,\quad x\ge 0, \qquad \beta, \gamma >0$ called the Dagum function (\cite{Porcu}), 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.