Muirhead inequality

The Muirhead inequality is a generalization of the inequality of the arithmetic and geometric mean .

Two definitions

The " a " means

For a given real vector

${\displaystyle a=(a_{1},\dots ,a_{n})}$

becomes the expression

${\displaystyle [a]={1 \over n!}\sum _{\sigma }x_{\sigma _{1}}^{a_{1}}\cdots x_{\sigma _{n}}^{a_{n}},}$

where σ of {1,…, n } is summed over all permutations , denoted as “ a- average” [ a ] of the nonnegative real numbers x ₁ ,…, x _n .

For the case a = (1, 0,…, 0), this gives exactly the arithmetic mean of the numbers x ₁ ,…, x _n ; for the case a = (1 / n ,…, 1 / n ) exactly the geometric mean results.

Double stochastic matrices

An n × n matrix P is called double stochastic if it consists of nonnegative numbers and both the sum of each row and the sum of each column are equal to one.

The Muirhead Inequality

The Muirhead inequality now states that [ a ] ≤ [ b ] for all x _i ≥ 0 if and only if a doubly stochastic matrix P exists, for which a = Pb holds.

A proof of the Muirhead inequality can be found, for example, in Godfrey Harold Hardy , John Edensor Littlewood , G. Polya : Inequalities , Cambridge University Press (1952), Chapters 2.18 and 2.19.

See also

• Average