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 | ||
| − | + | <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> | |
| − | 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>. | + | [[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>. |
| − | The most widely used form of Vornicu-Schur is | + | 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 |
| − | |||
| − | |||
<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> | ||
==References== | ==References== | ||
| − | *Vornicu, Valentin; ''Olimpiada de Matematica... de la provocare la experienta''; GIL Publishing House; Zalau, Romania. | + | *Vornicu, Valentin; ''Olimpiada de Matematica... de la provocare la experienta''; GIL Publishing House; Zalau, Romania. |
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| − | [[Category: | + | [[Category:Algebra]] |
| − | [[Category: | + | [[Category:Inequalities]] |
Latest revision as of 15:58, 29 December 2021
The Vornicu-Schur Inequality is a generalization of Schur's Inequality discovered by the Romanian mathematician Valentin Vornicu.
Statement
Consider real numbers 
such that 
and either 
or 
. Let 
be a positive integer and let 
be a function from the reals to the nonnegative reals that is either convex or monotonic. Then
![\[f(x)(a-b)^k(a-c)^k+f(y)(b-a)^k(b-c)^k+f(z)(c-a)^k(c-b)^k \ge 0.\]](http://latex.artofproblemsolving.com/3/0/c/30c058138c623bea31c2169d77d8b18e9e77151f.png)
Schur's Inequality follows from Vornicu-Schur by setting 
, 
, 
, 
, and 
.
The most widely used form of Vornicu-Schur is in the case 
, , when we have for real numbers 
and nonnegative real numbers 
that if 
then
![\[x(a-b)(a-c) + y(b-c)(b-a) + z (c-a)(c-b) \geq 0 .\]](http://latex.artofproblemsolving.com/c/6/9/c69456a5cfa09a147703237b57667693b66dc380.png)
References
- Vornicu, Valentin; Olimpiada de Matematica... de la provocare la experienta; GIL Publishing House; Zalau, Romania.
