Muirhead's inequality

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

Preliminary definitions[edit]

The "a-mean"[edit]

For any real vector

define the "a-mean" [a] of nonnegative real numbers x1, ..., xn 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, ..., xn. In case a = (1/n, ..., 1/n), it is the geometric mean of x1, ..., xn. (When n = 2, this is the Heinz mean.)

Notice that the "a"-mean as defined above only has the usual properties of a mean (e.g., if the mean of equal numbers is equal to them) if . In the general case, one can consider instead , which is called a Muirhead mean.[1]

Doubly stochastic matrices[edit]

An n × n matrix P is doubly stochastic precisely if both P and its transpose PT 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 matrix 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[edit]

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

The latter condition can be expressed in several equivalent ways; one of them is given below.

The proof makes use of the fact that every doubly stochastic matrix is a weighted average of permutation matrices (Birkhoff-von Neumann theorem).

Another equivalent condition[edit]

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.)

The sequence is said to majorize the sequence .

Symmetric sum-notation tricks[edit]

It is useful to use a kind of special notation for the sums. A success in reducing an inequality in this form means that the only condition for testing it is to verify whether one exponent sequence () majorizes the other one.

This notation requires developing every permutation, developing an expression made of n! monomials, for instance:

Deriving the arithmetic-geometric mean inequality[edit]

Let

we have

then

[aA] ≥ [aG]

which is

yielding the inequality.

Examples[edit]

Suppose you want to prove that x2 + y2 ≥ 2xy by using bunching (Muirhead's inequality): We transform it in the symmetric-sum notation:

The sequence (2, 0) majorizes the sequence (1, 1), thus the inequality holds by bunching. Again,

which yields

the sequence (3, 0, 0) majorizes the sequence (1, 1, 1), thus the inequality holds by bunching.

References[edit]

  1. ^ Bullen, P. S. Handbook of means and their inequalities. Kluwer Academic Publishers Group, Dordrecht, 2003. ISBN 1-4020-1522-4