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Difference between revisions of "AM-GM Inequality"

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== Generalizations ==
 
== Generalizations ==
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WIP
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The AM-GM Inequality has been generalized into several other inequalities, which either add weights or relate more means in the inequality.
  
 
=== Weighted AM-GM Inequality ===
 
=== Weighted AM-GM Inequality ===
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There exists an inequality similar to AM-GM that concerns weighted [[Weighted average | weighted]] arithmetic and geometric means. The '''Weighted AM-GM Inequality''' states that for any real numbers <math>x_1,  x_2, \ldots, x_n \geq 0</math> and any list of weights <math>\omega_1,  \omega_2, \ldots, \omega_n \geq 0</math> such that <math>\omega_1 + \omega_2 + \cdots + \omega_n = \omega</math>, <cmath>\frac{\omega_1 x_1 + \omega_2 x_2 + \cdots + \omega_n x_n}{\omega} \geq \sqrt[\omega]{x_1^{\omega_1} x_2^{\omega_2} \cdots x_n^{\omega_n}},</cmath> with equality if and only if <math>x_1 = x_2 = \cdots = x_n</math>. When <math>\omega_1 = \omega_2 = \cdots = \omega_n = 1/n</math>, the weighted form is reduced to the AM-GM Inequality. Several proofs of the Weighted AM-GM Inequality can be found in the [[proofs of AM-GM]] article.
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The '''Weighted AM-GM Inequality''' relates the [[Weighted average | weighted]] arithmetic and geometric means. It states that for any list of weights <math>\omega_1,  \omega_2, \ldots, \omega_n \geq 0</math> such that <math>\omega_1 + \omega_2 + \cdots + \omega_n = \omega</math>, <cmath>\frac{\omega_1 x_1 + \omega_2 x_2 + \cdots + \omega_n x_n}{\omega} \geq \sqrt[\omega]{x_1^{\omega_1} x_2^{\omega_2} \cdots x_n^{\omega_n}},</cmath> with equality if and only if <math>x_1 = x_2 = \cdots = x_n</math>. When <math>\omega_1 = \omega_2 = \cdots = \omega_n = 1/n</math>, the weighted form is reduced to the AM-GM Inequality. Several proofs of the Weighted AM-GM Inequality can be found in the [[proofs of AM-GM]] article.
  
 
=== Mean Inequality Chain ===
 
=== Mean Inequality Chain ===
 
{{Main|Mean Inequality Chain}}
 
{{Main|Mean Inequality Chain}}
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The '''Mean Inequality Chain,''' also called the '''RMS-AM-GM-HM Inequality''', relates the root mean square, arithmetic mean, geometric mean, and harmonic mean of a list of positive integers. In particular, it states that <cmath>\sqrt{\frac{x_1^2 + x_2^2 + \cdots + x_n^2}{n}} \geq \frac{x_1 + x_2 + \cdots + x_n}{n} \geq \sqrt[n]{x_1 x_2 \cdots x_n} \geq \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \cdots + \frac{1}{x_n}},</cmath> with equality if and only if <math>x_1 = x_2 = \cdots = x_n</math>. Like AM-GM, there also exists a weighted version of the Mean Inequality Chain.
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The '''Mean Inequality Chain''', also called the '''RMS-AM-GM-HM Inequality''', relates the root mean square, arithmetic mean, geometric mean, and harmonic mean of a list of nonnegative reals. In particular, it states that <cmath>\sqrt{\frac{x_1^2 + x_2^2 + \cdots + x_n^2}{n}} \geq \frac{x_1 + x_2 + \cdots + x_n}{n} \geq \sqrt[n]{x_1 x_2 \cdots x_n} \geq \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \cdots + \frac{1}{x_n}},</cmath> with equality if and only if <math>x_1 = x_2 = \cdots = x_n</math>. As with AM-GM, there also exists a weighted version of the Mean Inequality Chain.
  
 
=== Power Mean Inequality ===
 
=== Power Mean Inequality ===
 
{{Main|Power Mean Inequality}}
 
{{Main|Power Mean Inequality}}
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The '''Power Mean Inequality''' relates every power mean of a list of nonnegative reals. The power mean <math>M(p)</math> is defined as follows: <cmath>M(p) = \begin{cases} (\frac{x_1^p + x_2^p + \cdots + x_n^p}{n})^\frac{1}{p} &\text{if } p \neq 0 \\ \sqrt[n]{x_1 x_2 \cdots x_n} &\text{if } p = 0. \end{cases}</cmath> The Power Mean inequality then states that if <math>p>q</math>, then <math>M(p) \geq M(q)</math>, with equality holding if and only if <math>x_1 = x_2 = \cdots = x_n.</math> Plugging <math>p=1, 0</math> into this inequality reduces it to AM-GM, and <math>p=2, 1, 0, -1</math> gives the Mean Inequality Chain. As with AM-GM, there also exists a weighted version of the Power Mean Inequality.
  
 
== Introductory examples ==
 
== Introductory examples ==

Revision as of 20:23, 29 November 2021

In Algebra, the AM-GM Inequality, or more formally the Inequality of Arithmetic and Geometric Means, states that the arithmetic mean is greater than or equal to the geometric mean of any list of nonnegative reals; furthermore, equality holds if and only if every real in the list is the same.

In symbols, the inequality states that for any real numbers $x_1, x_2, \ldots, x_n \geq 0$, \[\frac{x_1 + x_2 + \cdots + x_n}{n} \geq \sqrt[n]{x_1 x_2 \cdots x_n}\] with equality if and only if $x_1 = x_2 = \cdots = x_n$.

NOTE: This article is a work-in-progress and meant to replace the Arithmetic mean-geometric mean inequality article, which is of poor quality.

Proofs

WIP

Generalizations

The AM-GM Inequality has been generalized into several other inequalities, which either add weights or relate more means in the inequality.

Weighted AM-GM Inequality

The Weighted AM-GM Inequality relates the weighted arithmetic and geometric means. It states that for any list of weights $\omega_1, \omega_2, \ldots, \omega_n \geq 0$ such that $\omega_1 + \omega_2 + \cdots + \omega_n = \omega$, \[\frac{\omega_1 x_1 + \omega_2 x_2 + \cdots + \omega_n x_n}{\omega} \geq \sqrt[\omega]{x_1^{\omega_1} x_2^{\omega_2} \cdots x_n^{\omega_n}},\] with equality if and only if $x_1 = x_2 = \cdots = x_n$. When $\omega_1 = \omega_2 = \cdots = \omega_n = 1/n$, the weighted form is reduced to the AM-GM Inequality. Several proofs of the Weighted AM-GM Inequality can be found in the proofs of AM-GM article.

Mean Inequality Chain

Main article: Mean Inequality Chain

The Mean Inequality Chain, also called the RMS-AM-GM-HM Inequality, relates the root mean square, arithmetic mean, geometric mean, and harmonic mean of a list of nonnegative reals. In particular, it states that \[\sqrt{\frac{x_1^2 + x_2^2 + \cdots + x_n^2}{n}} \geq \frac{x_1 + x_2 + \cdots + x_n}{n} \geq \sqrt[n]{x_1 x_2 \cdots x_n} \geq \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \cdots + \frac{1}{x_n}},\] with equality if and only if $x_1 = x_2 = \cdots = x_n$. As with AM-GM, there also exists a weighted version of the Mean Inequality Chain.

Power Mean Inequality

Main article: Power Mean Inequality

The Power Mean Inequality relates every power mean of a list of nonnegative reals. The power mean $M(p)$ is defined as follows: \[M(p) = \begin{cases} (\frac{x_1^p + x_2^p + \cdots + x_n^p}{n})^\frac{1}{p} &\text{if } p \neq 0 \\ \sqrt[n]{x_1 x_2 \cdots x_n} &\text{if } p = 0. \end{cases}\] The Power Mean inequality then states that if $p>q$, then $M(p) \geq M(q)$, with equality holding if and only if $x_1 = x_2 = \cdots = x_n.$ Plugging $p=1, 0$ into this inequality reduces it to AM-GM, and $p=2, 1, 0, -1$ gives the Mean Inequality Chain. As with AM-GM, there also exists a weighted version of the Power Mean Inequality.

Introductory examples

WIP

Intermediate examples

WIP

Olympiad examples

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More Problems

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See Also

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