Schur's inequality

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In mathematics, Schur's inequality, named after Issai Schur, establishes that for all non-negative real numbers x, y, z and a positive number t $\ge 1$ ,

$x^t (x-y)(x-z) + y^t (y-z)(y-x) + z^t (z-x)(z-y) \ge 0$

with equality only if x = y = z or if two of them are equal and the other is zero. When t is an even positive integer, the inequality holds for all real numbers x, y and z.

A generalization of Schur's inequality is the following: Suppose a,b,c are positive real numbers. If the triples (a,b,c) and (x,y,z) are similarly sorted, then the following inequality holds:

$a (x-y)(x-z) + b (y-z)(y-x) + c (z-x)(z-y) \ge 0.$