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Difference between revisions of "Vornicu-Schur Inequality"

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==Statement==
 
==Statement==
 
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
 
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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<cmath>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.</cmath>
  
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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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[[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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The most widely used form of Vornicu-Schur is  
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The most widely used form of Vornicu-Schur is in the case <math>f(x) = x</math>, <math>k = 1</math>, when we have for real numbers <math>a \geq b \geq c</math> and nonnegative real numbers <math>x, y, z</math> that if <math>x + z \geq y</math> then  
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Let <math>a \geq b \geq c</math> be real numbers, and let <math>x,y,z \geq 0</math>. Then if <math>x+z\geq y</math> then the following inequality is true
 
 
<cmath> x(a-b)(a-c) + y(b-c)(b-a) + z (c-a)(c-b) \geq 0 . </cmath>
 
<cmath> x(a-b)(a-c) + y(b-c)(b-a) + z (c-a)(c-b) \geq 0 . </cmath>
  

Revision as of 17:04, 8 September 2010

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.</ref>
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