In mathematics, Schur's inequality, named after Issai Schur, establishes that for all non-negative real numbers x, y, z, and t>0, 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.
, the following well-known special case can be derived: Since the inequality is symmetric in
This rearranges to Schur's inequality.
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 further generalized form of Schur's inequality holds: Consider
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] Another possible extension states that if the non-negative real numbers
with and the positive real number t are such that x + v ≥ y + z then[2]