Difference between revisions of "AM-GM Inequality"
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| − | In [[ | + | 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 <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>. | 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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=== Weighted AM-GM Inequality === | === Weighted AM-GM Inequality === | ||
| − | 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. | + | 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. |
=== Mean Inequality Chain === | === Mean Inequality Chain === | ||
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| − | + | == See Also == | |
| − | + | * [[Mean Inequality Chain]] | |
| − | * | + | * [[Power Mean Inequality]] |
| − | * | + | * [[Cauchy-Schwarz Inequality]] |
| − | * | + | * [[Inequality]] |
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| − | * | ||
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Revision as of 16:18, 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 
, ![\[\frac{x_1 + x_2 + \cdots + x_n}{n} \geq \sqrt[n]{x_1 x_2 \cdots x_n}\]](http://latex.artofproblemsolving.com/f/2/6/f262c59d120ea97b1e9c88ab410c9f0905494c9f.png)
with equality if and only if 
.
NOTE: This article is a work-in-progress and meant to replace the Arithmetic mean-geometric mean inequality article, which is of poor quality.
Contents
Proofs
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Generalizations
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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 and any list of weights 
such that 
, ![\[\frac{\omega_1 x_1 + \omega_2 x_2 + \cdots + \omega_n x_n}{\omega} \geq \sqrt[\omega]{x_1^{\omega_1} x_2^{\omega_2} \cdots x_n^{\omega_n}},\]](http://latex.artofproblemsolving.com/5/0/3/503e67fbce32fd9c201c6df50c1ef57a9c2c890d.png)
with equality if and only if . When 
, 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 ![\[\sqrt{\frac{x_1^2 + x_2^2 + \cdots + x_n^2}{n}} \geq \frac{x_1 + x_2 + \cdots + x_n}{n} \geq \sqrt[n]{x_1 x_2 \cdots x_n} \geq \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \cdots + \frac{1}{x_n}},\]](http://latex.artofproblemsolving.com/0/a/3/0a3b1a9ddcb951c831213da056f957896febffed.png)
with equality if and only if . Like AM-GM, there also exists a weighted version of the Mean Inequality Chain.
Power Mean Inequality
- Main article: Power Mean Inequality
Introductory examples
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Intermediate examples
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Olympiad examples
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More Problems
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