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 (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 , 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. 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 Notes Inequalities Articles containing proofs