Variance and the Inequality of Arithmetic and Geometric Means

摘要

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