Schur's inequality


In mathematics, Schur's inequality, named after Issai Schur,
establishes that for all non-negative real numbers
x, y, z and 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.
When, the following well-known special case can be derived:

Proof

Since the inequality is symmetric in we may assume without loss of generality that. Then the inequality
clearly holds, since every term on the left-hand side of the inequality is non-negative. This rearranges to Schur's inequality.

Extensions

A generalization of Schur's inequality is the following:
Suppose a,b,c are positive real numbers. If the triples ' and ' 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, 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 = mr.
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