In mathematics, Muirhead's inequality, named after Robert Franklin Muirhead, also known as the "bunching" method, generalizes the inequality of arithmetic and geometric means.
For any real vector 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
A stochastic matrix is a square matrix of nonnegative real entries in which the sum of the entries in each column is 1.
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.
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).
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.)
A success in reducing an inequality in this form means that the only condition for testing it is to verify whether one exponent sequence (
monomials, for instance: Let and We have Then which is yielding the inequality.
We seek to prove that x2 + y2 ≥ 2xy by using bunching (Muirhead's inequality).
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.