Complete monotonicity of log-functions

TL;DR

This paper explores the complete monotonicity property of a family of multivariate log-functions, establishing a connection with non-negative polynomials and providing an algorithm to determine monotonicity.

Contribution

It introduces a linear isomorphism between the family of functions and polynomial space, linking complete monotonicity to non-negative polynomials and showing the cone is semi-algebraic.

Findings

Established a linear isomorphism between function family and polynomial space

Identified the cone of completely monotone functions with non-negative polynomials

Provided a finite algorithm to decide complete monotonicity

Abstract

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