  A function $\rho:[0,\infty)\to(0,1]$ is a completely monotonic function if
and only if $\rho(\Vert\mathbf{x}\Vert^2)$ is positive definite on
$\mathbb{R}^d$ 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 \[\rho(\beta ,\gamma)(x)=1-\biggl(\frac{x^{\beta}}{1+x^{\beta}}\biggr
)^{\gamma},\qquad x\ge 0, \beta,\gamma>0,\] 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.
