# Difference between revisions of "Maclaurin's Inequality"

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− | with equality exactly when all the <math> | + | with equality exactly when all the <math>x_i </math> are equal. |

== Proof == | == Proof == | ||

− | By the lemma from [[Newton's Inequality]], it suffices to show that for any <math> | + | By the lemma from [[Newton's Inequality]], it suffices to show that for any <math>n </math>, |

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* [[Newton's Inequality]] | * [[Newton's Inequality]] | ||

* [[Symmetric sum]] | * [[Symmetric sum]] | ||

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+ | [[Category:Number Theory]] | ||

+ | [[Category:Inequality]] | ||

+ | [[Category:Definition]] |

## Revision as of 21:07, 14 October 2007

**Maclaurin's Inequality** is an inequality in symmetric polynomials. For notation and background, we refer to Newton's Inequality.

## Statement

For non-negative ,

,

with equality exactly when all the are equal.

## Proof

By the lemma from Newton's Inequality, it suffices to show that for any ,

.

Since this is a homogenous inequality, we may normalize so that . We then transform the inequality to

.

Since the geometric mean of is 1, the inequality is true by AM-GM.