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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>
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