- Schur's inequality
In
mathematics , Schur'sinequality , named afterIssai Schur ,establishes that for all non-negativereal number s"x", "y", "z" and apositive 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".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 equation is non-negative. This rearranges to Schur's inequality.
Extension
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)" aresimilarly sorted , then the following inequality holds::
In
2007 ,Romania n mathematicianValentin 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") = "m""r". [Vornicu, Valentin; "Olimpiada de Matematica... de la provocare la experienta"; GIL Publishing House; Zalau, Romania.]
Notes
ee also
*
Inequality
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