define the "a-mean" [a] of positive real numbers x1, ..., xn by
where the sum extends over all permutations σ of { 1, ..., n }.
When the elements of a are nonnegative integers, the a-mean can be equivalently defined via the monomial symmetric polynomial as
where ℓ is the number of distinct elements in a, and k1, ..., kℓ are their multiplicities.
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]
Examples
For a = (1, 0, ..., 0), the a-mean is just the ordinary arithmetic mean of x1, ..., xn.
For a = (1/n, ..., 1/n), the a-mean is the geometric mean of x1, ..., xn.
The Muirhead mean for a = (−1, 0, ..., 0) is the harmonic mean.
Doubly stochastic matricesedit
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.
Statementedit
Muirhead's inequality states that [a] ≤ [b] for all x such that xi > 0 for every i ∈ { 1, ..., n } if and only if there is some doubly stochastic matrix P for which a = Pb.
Furthermore, in that case we have [a] = [b] if and only if a = b or all xi are equal.
The latter condition can be expressed in several equivalent ways; one of them is given below.
It is convenient to use a 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:
Examplesedit
Arithmetic-geometric mean inequalityedit
Let
and
We have
Then
[aA] ≥ [aG],
which is
yielding the inequality.
Other examplesedit
We seek 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.
Similarly, we can prove the inequality
by writing it using the symmetric-sum notation as
which is the same as
Since the sequence (3, 0, 0) majorizes the sequence (1, 1, 1), the inequality holds by bunching.