Muirhead's inequality

The Muirhead 's inequality is a generalization of the inequality of the arithmetic and geometric means.

Two definitions

The "a - means "

For a given real vector

The term

Wherein all the permutations σ from {1, ..., n} is summed, referred to as " A agent" [a] of the non-negative real numbers x1, ..., xn.

In the case of a = (1, 0, ..., 0), which gives exactly the arithmetic mean of the numbers x1, ..., xn; in the case of a = (1 / n, ... 1 / n ) is obtained exactly the geometric mean.

Double stochastic matrices

An n × n matrix P is doubly stochastic known if it consists of non-negative numbers and the sum of both as well as the sum of each row of each column is equal to one.

The Muirhead 's inequality

The Muirhead 's inequality now states that [a ] ≤ [b ] for all xi ≥ 0 if and only if a doubly stochastic matrix P exists, is valid for a = Pb.

A proof of Muirhead 's inequality is found for example in Godfrey Harold Hardy, John Edensor Littlewood, G. Polya: Inequalities, Cambridge University Press ( 1952 ), Chapter 2.18 and 2.19.