The geometric mean is a Bernstein function

Abstract

In the paper, the authors establish, by using Cauchy integral formula in the theory of complex functions, an integral representation for the geometric mean of $n$ positive numbers. From this integral representation, the geometric mean is proved to be a Bernstein function and a new proof of the well known AG inequality is provided.

10 pages

References in corpus (2)

• The geometric mean is a Bernstein function
• A new proof of the geometric-arithmetic mean inequality by Cauchy's integral formula

Cited by in corpus (5)

• A logarithmically completely monotonic function involving the ratio of gamma functions
• The geometric mean is a Bernstein function
• A new proof of the geometric-arithmetic mean inequality by Cauchy's integral formula
• Some Bernstein functions and integral representations concerning harmonic and geometric means
• A ratio of many gamma functions and its properties with applications