- Schur's inequality
mathematics, Schur's inequality, named after Issai Schur,establishes that for all non-negative real numbers"x", "y", "z" and a positive number"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".
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.
generalizationof 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:
2007, Romanian mathematician Valentin Vornicushowed that a yet further generalized form of Schur's inequality holds:
Consider , where , and either or . Let , and let be either convex or
The standard form of Schur's is the case of this inequality where "x" = "a", "y" = "b", "z" = "c", "k" = 1, ƒ("m") = "m""r". [Vornicu, Valentin; "Olimpiada de Matematica... de la provocare la experienta"; GIL Publishing House; Zalau, Romania.]
Wikimedia Foundation. 2010.