Muirhead's inequality

In mathematics, Muirhead's inequality, also known as the "bunching" method, generalizes the inequality of arithmetic and geometric means.

Table of contents

1 Two preliminary definitions a-mean"">1.1 The "a-mean" 1.2 Doubly stochastic matrices 2 The inequality 2.3 Another equivalent condition 3 Reference

Two preliminary definitions

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The "a-mean"

For any real vector

define the "a-mean" [a] of nonnegative real numbers x₁, ..., x_n by

where the sum extends over all permutations σ of { 1, ..., n }.

In case a = (1, 0, ..., 0), this is just the ordinary arithmetic mean of x₁, ..., x_n. In case a = (1/n, ..., 1/n), it is the geometric mean of x₁, ..., x_n.

Doubly stochastic matrices

An n × n matrix P is doubly stochastic precisely if both P and its transpose P^T are stochastic matrices. A stochastic matrix is a square matrix of nonnegative real entries in which the sum of the entries in each column is 1. Thus, a doubly stochastic matrix is a square matric of nonnegative real entries in which the sum of the entries in each row and the sum of the entries in each column is 1.

The inequality

Muirhead's inequality states that [a] ≤ [b] for all x_i ≥ 0 if and only if there is some doubly stochastic matrix P for which a = Pb.

The proof makes use of the fact that every doubly stochastic matrix is a weighted average of permutation matrices.

Another equivalent condition

Because of the symmetry of the sum, no generality is lost by sorting the exponents into decreasing order:

Then the existence of a doubly stochastic matrix P such that a = Pb is equivalent to the following system of inequalities:

(The last one is an equality; the others are weak inequalities.)

Reference

Combinatorial Theory by John N. Guidi, based on lectures given by Gian-Carlo Rota in 1998, MIT Copy Technology Center, 2002.

Kiran Kedlaya's guide to solving inequalities at [1].

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