Difference between revisions of "AM-GM Inequality"

 
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#REDIRECT [[Arithmetic Mean-Geometric Mean Inequality]]
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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.
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In symbols, the inequality states that for any real numbers <math>x_1,  x_2, \ldots, x_n \geq 0</math>, <cmath>\frac{x_1 + x_2 + \cdots + x_n}{n} \geq \sqrt[n]{x_1 x_2 \cdots x_n}</cmath> with equality if and only if <math>x_1 = x_2 = \cdots = x_n</math>.
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'''NOTE''': This article is a work-in-progress and meant to replace the [[Arithmetic mean-geometric mean inequality]] article, which is of poor quality.
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== Proofs ==
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WIP
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== Generalizations ==
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WIP
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=== Weighted AM-GM Inequality ===
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There exists an inequality similar to AM-GM that concerns the weighted [[Weighted average | weighted]] arithmetic and geometric means of a list of nonnegative reals. 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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=== Mean Inequality Chain ===
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{{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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=== Power Mean Inequality ===
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{{Main|Power Mean Inequality}}
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== Introductory examples ==
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WIP
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== Intermediate examples ==
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WIP
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== Olympiad examples
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WIP
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== More Problems ==
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WIP
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OUTLINE:
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* Proofs
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** Links to [[Proofs of AM-GM]]
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**
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* Generalizations
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** Weighted AM-GM
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** QM-AM-GM-HM (with or without weights)
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** Power Mean (with or without weights)

Revision as of 15:03, 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

WIP

Weighted AM-GM Inequality

There exists an inequality similar to AM-GM that concerns the weighted weighted arithmetic and geometric means of a list of nonnegative reals. The Weighted AM-GM Inequality states that for any real numbers $x_1,  x_2, \ldots, x_n \geq 0$ and 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 positive integers. 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 $x_1 = x_2 = \cdots = x_n$. Like AM-GM, there also exists a weighted version of the Mean Inequality Chain.

Power Mean Inequality

Main article: Power Mean Inequality

Introductory examples

WIP

Intermediate examples

WIP

== Olympiad examples WIP

More Problems

WIP

OUTLINE:

  • Generalizations
    • Weighted AM-GM
    • QM-AM-GM-HM (with or without weights)
    • Power Mean (with or without weights)