Muirhead's inequality

Muirhead's inequality compares normalized sums of monomials in nonnegative variables. The sum associated with an exponent vector is at least the sum associated with any vector it majorizes.

Majorization: A partial order on vectors comparing how unevenly their components are distributed. The exponent-vector ordering that determines which normalized symmetric sum is larger.
Integer partition: A way of writing a positive integer as a sum of positive integers, disregarding order. Sorted exponent vectors in Muirhead comparisons are partitions of the same degree.
AM–GM inequality: The arithmetic mean of nonnegative numbers is at least their geometric mean. Muirhead can prove inequalities whose terms reduce to arithmetic-versus-geometric mean comparisons.
Rearrangement inequality: An inequality comparing sums of products under different orderings of two sequences. It uses ordering in products rather than majorization between exponent vectors.
Symmetric polynomial: A polynomial unchanged when its variables are permuted. Muirhead's inequality compares particular normalized symmetric sums of monomials.
Monomial: A product of variables raised to nonnegative integer powers, multiplied by a coefficient. Each term being permuted in a Muirhead sum is a monomial.
Schur's inequality: A symmetric polynomial inequality relating sums of powers and pairwise differences of nonnegative variables. Schur is a standard alternative when a symmetric inequality has a specific degree-sensitive form.
Chebyshev's sum inequality: An inequality comparing the mean product of similarly ordered sequences with the product of their means. It offers a sequence-based route for some symmetric inequalities without expanding monomial sums.
Weighted AM–GM inequality: An inequality stating that a weighted arithmetic mean of nonnegative numbers is at least their weighted geometric mean. Averaging arguments reduce the majorization comparison to inequalities of this kind.
Permutation: A rearrangement of the elements of a finite set. The symmetric sum is formed by permuting the exponents among the variables.

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Karamata's inequalityRelated: Its exponent-vector ordering is an application of the same majorization principle.