Vornicu-Schur Inequality

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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 $f(m) = m^r$.

The most widely used form of Vornicu-Schur is in the case $f(x) = x$, $k = 1$, when we have for real numbers $a \geq b \geq c$ and nonnegative real numbers $x, y, z$ that if $x + z \geq y$ then $$x(a-b)(a-c) + y(b-c)(b-a) + z (c-a)(c-b) \geq 0 .$$

References

• Vornicu, Valentin; Olimpiada de Matematica... de la provocare la experienta; GIL Publishing House; Zalau, Romania.