Completely monotone functions - a digest

TL;DR

This paper compiles and discusses lesser-known facts about completely monotone functions, emphasizing their Laplace transform representation and connection to log-convexity, with examples involving Gamma functions.

Contribution

It provides a curated collection of facts and methods for identifying and analyzing completely monotone functions, highlighting lesser-known properties and connections.

Findings

Representation of CM functions as Laplace transforms

Connection between CM functions and log-convexity

Examples involving Gamma functions

Abstract

This work has a purpose to collect selected facts about the completely monotone (CM) functions that can be found in books and papers devoted to different areas of mathematics. We opted for lesser known ones, and for those which may help determining whether or not a given function is completely monotone. In particular, we emphasize the role of representation of a CM function as the Laplace transform of a measure, and we present and discuss a little known connection with log-convexity. Some of presented methods are illustrated by several examples involving Gamma and related functions.