Variance and the inequality of arithmetic and geometric means

Abstract

A number of recent papers have been devoted to generalizations of the classical AM-GM inequality. Those generalizations which incorporate \textit {variance} have been the most useful in applications to economics and finance. In this paper, we prove an inequality which yields the best possible upper and lower bounds for the geometric mean of a sequence solely in terms of its arithmetic mean and its variance. A particular consequence is the following: 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.