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 if and only if x = y = z or 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.
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[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/3/f/d/3fde443ef31d67211cc035915691d5fc.png)
clearly holds, since every term on the left-hand side of the equation is non-negative. This rearranges to Schur's inequality.
[edit] Extension
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:
In 2007, Romanian mathematician Valentin Vornicu showed that a yet a further generalized form of Schur's inequality exists:
Consider 
, where 
, and either or 
. Let 
, and let 
be either convex or monotonic. Then,
The standard form of Schur's is the case of this inequality where x = a, y = b, z = c, k = 1, ƒ(m) = mr.[1]
[edit] Notes
- ^ Vornicu, Valentin; Olimpiada de Matematica... de la provocare la experienta; GIL Publishing House; Zalau, Romania.
