Schur's inequality

Schur's inequality states that for nonnegative real numbers x, y, z and a positive integer r, the cyclic sum of xʳ(x−y)(x−z) is nonnegative. It is a symmetric inequality connecting sums of powers and products.

Schur's inequality in degree three: For nonnegative x, y, z, x³+y³+z³+3xyz is at least the sum of the six terms x²y. The case r = 1 is the most familiar polynomial form of Schur's inequality.
Nonnegative real number: A real number greater than or equal to zero. Nonnegativity is the essential domain condition in Schur's inequality.
Rational inequality: An inequality involving ratios of polynomials, considered where its denominators are defined. Clearing positive denominators can reduce many rational inequalities to Schur-type polynomial expressions.
Chebyshev's sum inequality: An inequality comparing the average product of similarly ordered sequences with the product of their averages. Chebyshev concerns paired sequences, whereas Schur directly bounds a symmetric polynomial in three variables.
Symmetric polynomial: A polynomial unchanged when its variables are permuted. Schur's inequality is a nonnegative symmetric polynomial expression in three variables.
Integer exponent: A whole-number power formed by repeated multiplication, with exponent zero sometimes defined separately. The parameter r specifies which powers weight the three cyclic terms.
Polynomial positivity: The property that a polynomial takes only nonnegative values on a specified domain. Schur's inequality certifies nonnegativity of a family of symmetric polynomials on the nonnegative orthant.
Cauchy–Schwarz inequality: An inequality bounding the squared inner product of two vectors by the product of their squared lengths. Cauchy–Schwarz controls inner products; Schur exploits cyclic differences and nonnegative variables.
Rearrangement inequality: An inequality comparing sums of products under different pairings of ordered sequences. Ordering variables is a useful way to expose the sign structure in proofs of Schur's inequality.
Algebraic inequality: A comparison between algebraic expressions asserted to hold under stated conditions. Schur's inequality is a polynomial inequality with an explicit nonnegative-variable condition.

Linked from 7 pages