Schur's Inequality

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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, x^t (x-y)(x-z) + y^t (y-z)(y-x) + z^t (z-x)(z-y) ge 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. Since the inequality is symmetric in x,y,z we may assume without loss of generality that x geq y geq z. Then the inequality (x-y)[x^t(x-z)-y^t(y-z)]+z^t(x-z)(y-z) geq 0, clearly holds, since every term on the left-hand side of the equation is non-negative. This rearranges to Schur's inequality.