AoPSWiki

Art of Problem Solving's olympiad training program WOOT starts on Septebmer 8. Train with the top high school students in the the world! Click here to enroll today!

Personal tools

Log in / create account

Search

Toolbox

Vornicu-Schur Inequality

From AoPSWiki

The Vornicu-Schur Inequality is a generalization of Schur's Inequality discovered by the Romanian mathematician Valentin Vornicu.

Statement

Consider real numbers $a,b,c,x,y,z$ such that $a \ge b \ge c$ and either $x \geq y \geq z$ or $z \geq y \geq x$ . Let $k \in \mathbb{Z}_{> 0}$ be a positive integer and let $f:\mathbb{R} \rightarrow \mathbb{R}_{\geq 0}$ be a function from the reals to the nonnegative reals that is either convex or monotonic. Then

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

Schur's Inequality follows from Vornicu-Schur by setting $x=a$ , $y=b$ , $z=c$ , $k = 1$ , and $f(m) = m^r$ .

External Links

A full statement, as well as some applications can be found in this article.

References

Vornicu, Valentin; Olimpiada de Matematica... de la provocare la experienta; GIL Publishing House; Zalau, Romania.</ref>

Retrieved from "http://www.artofproblemsolving.com/Wiki/index.php/Vornicu-Schur_Inequality"

Categories: Theorems | Inequality

NEW! NEW! NEW!
Want to learn how to tackle those tough AMC/AIME/Olympiad algebra problems? Check out Art of Problem Solving's NEW Intermediate Algebra by Richard Rusczyk and Mathew Crawford. Over 1600 problems!

© Copyright 2007 AoPS Incorporated. All Rights Reserved. • Foundation • Privacy • Contact Us