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:
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 equation is non-negative. This rearranges to Schur's inequality.
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 x,y,z,v and the positive real number t are such that x + v ≥ y + z then