Integral representations of the weighted geometric mean and the logarithmic mean

TL;DR

This paper demonstrates that the weighted geometric mean and the logarithmic mean are Bernstein functions and derives their integral representations using complex analysis techniques.

Contribution

It introduces integral representations of these means, revealing their properties as Bernstein functions, which was not previously established.

Findings

Both means are Bernstein functions.

Integral representations are derived using Cauchy's theorem.

Provides new insights into the mathematical properties of these means.

Abstract

In the paper, the authors show that the weighted geometric mean and the logarithmic mean are Bernstein functions and establish integral representations of these means by Cauchy's integral theorem in the theory of complex functions.