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Difference between revisions of "Vornicu-Schur Inequality"

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The '''Vornicu-Schur'''' refers to a generalized version of [[Schur's Inequality]].  
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The '''Vornicu-Schur Inequality''' is a generalized version of [[Schur's Inequality]] discovered by the Romanian mathematician [[Valentin Vornicu]].
  
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==Theorem==
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==Statement==
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In 2007, Romanian mathematician [[Valentin Vornicu]] showed that a generalized form of Schur's inequality exists:
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Consider [[real number]]s <math>a,b,c,x,y,z</math> such that <math>a \ge b \ge c</math> and either <math>x \geq y \geq z</math> or <math>z \geq y \geq x</math>.  Let <math>k \in \mathbb{Z}_{> 0}</math> be a [[positive integer]] and let <math>f:\mathbb{R} \rightarrow \mathbb{R}_{\geq 0}</math> be a [[function]] from the reals to the nonnegative reals that is either [[convex function|convex]] or [[monotonic]].  Then
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:<math>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.</math>
  
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Consider <math>a,b,c,x,y,z \in \mathbb{R}</math>, where <math>a \ge b \ge c</math>, and either <math>x \geq y \geq z</math> or <math>>z \geq y \geq x</math>.  Let <math>k \in \mathbb{Z}^{+}</math>, and let <math>f:\mathbb{R} \rightarrow \mathbb{R}_{0}^{+}</math> be either [[convex function|convex]] or [[monotonic]].  Then,
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Schur's Inequality follows from Vornicu-Schur by setting <math>x=a</math>, <math>y=b</math>, <math>z=c</math>, <math>k = 1</math>, and <math>f(m) = m^r</math>.
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:<math>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</math>
 
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The standard form of Schur's is the case of this inequality where <math>x=a</math>, <math>y=b</math>, <math>z=c</math>, <math>k = 1</math>, and <math>f(m) = m^r</math>.
 
 
==External Links==
 
==External Links==
 
*A full statement, as well as some applications can be found in [http://www.mathlinks.ro/portal.php?t=162684 this article].
 
*A full statement, as well as some applications can be found in [http://www.mathlinks.ro/portal.php?t=162684 this article].

Revision as of 17:50, 30 March 2008

The Vornicu-Schur Inequality is a generalized version 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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