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 |
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<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 16:58, 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
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
References
- Vornicu, Valentin; Olimpiada de Matematica... de la provocare la experienta; GIL Publishing House; Zalau, Romania.