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Difference between revisions of "Vornicu-Schur Inequality"
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Vornicu-Schur refers to a generalized version of [[Schur's Inequality]].  
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The '''Vornicu-Schur'''' refers to a generalized version of [[Schur's Inequality]].  
  
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A full statement, as well as some applications can be found in [http://www.mathlinks.ro/portal.php?t=162684 this article].
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==Theorem==
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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 <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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:<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>.<ref>Vornicu, Valentin;  ''Olimpiada de Matematica... de la provocare la experienta''; GIL Publishing House; Zalau, Romania.</ref>
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==External Links==
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*A full statement, as well as some applications can be found in [http://www.mathlinks.ro/portal.php?t=162684 this article].
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==Notes==
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{{reflist}}
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[[Category:Theorems]]
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[[Category:Inequality]]

Revision as of 13:37, 30 March 2008

The Vornicu-Schur' refers to a generalized version of Schur's Inequality.

Theorem

In 2007, Romanian mathematician Valentin Vornicu showed that a generalized form of Schur's inequality exists:

Consider $a,b,c,x,y,z \in \mathbb{R}$, where $a \ge b \ge c$, and either $x \geq y \geq z$ or $>z \geq y \geq x$. Let $k \in \mathbb{Z}^{+}$, and let $f:\mathbb{R} \rightarrow \mathbb{R}_{0}^{+}$ be either convex or monotonic. Then,

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

The standard form of Schur's is the case of this inequality where $x=a$, $y=b$, $z=c$, $k = 1$, and $f(m) = m^r$.<ref>Vornicu, Valentin; Olimpiada de Matematica... de la provocare la experienta; GIL Publishing House; Zalau, Romania.</ref>

External Links

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

Notes

Template:Reflist

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