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,
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:
[edit] Proof
Since the inequality is symmetric in x,y,z we may assume without loss of generality that 
. Then the inequality:
![(x-y)[x^t(x-z)-y^t(y-z)]+z^t(x-z)(y-z) \geq 0](../../../math/9/c/8/9c821ca3f14c34e4e23c81da8f1ecdbc.png)
clearly holds, since every term on the left-hand side of the equation is non-negative. This rearranges to Schur's inequality.
