The geometric mean is a Bernstein function
Feng Qi, Xiao-Jing Zhang, and Wen-Hui Li
TL;DR
This paper proves that the geometric mean of positive numbers is a Bernstein function using complex analysis, providing a new proof of the arithmetic-geometric mean inequality.
Contribution
It introduces an integral representation of the geometric mean and demonstrates its Bernstein function property, offering a novel proof of a classical inequality.
Findings
Geometric mean has a Bernstein function representation
New proof of the arithmetic-geometric mean inequality
Integral representation via Cauchy integral formula
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 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.
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Taxonomy
TopicsMathematical Inequalities and Applications · Mathematical functions and polynomials · Approximation Theory and Sequence Spaces