Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Vornicu-Schur Inequality"

(general cleanup)
m
Line 1: Line 1:
−
The '''Vornicu-Schur Inequality''' is a generalized version of [[Schur's Inequality]] discovered by the Romanian mathematician [[Valentin Vornicu]].
+
The '''Vornicu-Schur Inequality''' is a generalization of [[Schur's Inequality]] discovered by the Romanian mathematician [[Valentin Vornicu]].
  
 
==Statement==
 
==Statement==

Revision as of 17:51, 30 March 2008

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 "https://artofproblemsolving.com/wiki/index.php?title=Vornicu-Schur_Inequality&oldid=24417"