Schur's inequality
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,
with equality only if x = y = z or if 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) are similarly sorted, then the following inequality holds:
In 2007, Romanian mathematician Valentin Vornicu that a yet a further generalized form of Schur's inequality exists:
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 $x=a,\ y=b,\ z=c,\ k=1,\ f(m)=m^r$.[1]
Notes
- ^ Vornicu, Valentin; Olimpiada de Matematica... de la provocare la experienta; GIL Publishing House; Zalau, Romania.