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 a-mean 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 . In the general case, one can consider instead , which is called a Muirhead mean. 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. For a = (x, 1 − x), the a-mean is the Heinz mean. The Muirhead mean for a = (−1, 0, ..., 0) is the harmonic mean. Doubly stochastic matrices 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. Statement 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. 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 Because of the symmetry of the sum, no generality is lost by sorting the exponents into decreasing order: