Difference between revisions of "Vornicu-Schur Inequality"

Revision as of 13:37, 30 March 2008

The Vornicu-Schur' refers to a generalized version of Schur's Inequality.

Theorem

In 2007, Romanian mathematician Valentin Vornicu showed that a generalized form of Schur's inequality exists:

Consider $a,b,c,x,y,z \in \mathbb{R}$ , where $a \ge b \ge c$ , and either $x \geq y \geq z$ or $>z \geq y \geq x$ . Let $k \in \mathbb{Z}^{+}$ , and let $f:\mathbb{R} \rightarrow \mathbb{R}_{0}^{+}$ be either convex or monotonic. Then,

$f(x)(a-b)^k(a-c)^k+f(y)(b-a)^k(b-c)^k+f(z)(c-a)^k(c-b)^k \ge 0$

The standard form of Schur's is the case of this inequality where $x=a$ , $y=b$ , $z=c$ , $k = 1$ , and $f(m) = m^r$ .<ref>Vornicu, Valentin; Olimpiada de Matematica... de la provocare la experienta; GIL Publishing House; Zalau, Romania.</ref>