An application of a moment problem to completely monotonic functions

TL;DR

This paper explores the relationship between completely monotonic functions and their representing functions via a moment problem, providing new insights into their characterization and applying these results to functions related to gamma ratios.

Contribution

It introduces a novel connection between complete monotonicity and the moment problem, offering new criteria for identifying completely monotonic functions.

Findings

Established a link between complete monotonicity and the moment problem.

Provided criteria for the non-negativity of the representing function.

Applied theoretical results to functions involving gamma ratios.

Abstract

We consider the following question: if a function of the form $\int_0^{\infty}\varphi(t)\, e^{-xt}dt$ is completely monotonic, is it then $\varphi\ge0$? It turns out that the question is related to a moment problem. In the end we apply those results to answer some questions concerning complete monotonicity of certain functions raised in F. Qi and R. Agarwal, \textit{On complete monotonicity for several classes of functions related to ratios of gamma functions}, J. Inequal. Appl. (2019).