Complete monotonicity of log-functions

In this article we investigate the property of complete monotonicity within a special family ${\mathcal{F}}_s$ of functions in $s$ variables involving logarithms. The main result of this work provides a linear isomorphism between ${\mathcal{F}}_s$ and the space of real multivariate polynomials. This isomorphism identifies the cone of completely monotone functions with the cone of non-negative polynomials. We conclude that the cone of completely monotone functions in ${\mathcal{F}}_s$ is semi-algebraic. This gives a finite time algorithm to decide whether a function in ${\mathcal{F}}_s$ is completely monotone