Variance and the Inequality of Arithmetic and Geometric Means

TL;DR

This paper establishes sharp bounds for the geometric mean based on the arithmetic mean and variance, revealing that the maximum geometric mean occurs when all but one sequence element are equal and less than the mean.

Contribution

It proves a new inequality linking variance with the geometric mean, filling a gap in the understanding of AM-GM generalizations.

Findings

Maximal geometric mean occurs when all but one sequence element are equal and less than the arithmetic mean.

Provides sharp bounds for the geometric mean in terms of mean and variance.

Enhances the mathematical understanding of inequalities with applications in economics and finance.

Abstract

The classical AM-GM inequality has been generalized in a number of ways. Generalizations which incorporate variance appear to be the most useful in economics and finance, as well as mathematically natural. Previous work leaves unanswered the question of finding sharp bounds for the geometric mean in terms of the arithmetic mean and variance. In this paper we prove such an inequality. A particular consequence is easily described: among all positive sequences having given length, arithmetic mean and nonzero variance, the geometric mean is maximal when all terms in the sequence except one are equal to each other and are less than the arithmetic mean.