Difference between revisions of "AM-GM 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. | |
+ | |||
+ | 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>. | ||
+ | |||
+ | '''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 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 === | ||
+ | {{Main|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 <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. | ||
+ | |||
+ | === Power Mean Inequality === | ||
+ | {{Main|Power Mean Inequality}} | ||
+ | |||
+ | == Introductory examples == | ||
+ | WIP | ||
+ | |||
+ | == Intermediate examples == | ||
+ | WIP | ||
+ | |||
+ | == Olympiad examples | ||
+ | WIP | ||
+ | |||
+ | == More Problems == | ||
+ | WIP | ||
+ | |||
+ | OUTLINE: | ||
+ | |||
+ | * Proofs | ||
+ | ** Links to [[Proofs of AM-GM]] | ||
+ | ** | ||
+ | |||
+ | * Generalizations | ||
+ | ** Weighted AM-GM | ||
+ | ** QM-AM-GM-HM (with or without weights) | ||
+ | ** 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 , 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
[hide]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 and any list of weights such that , 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 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
WIP
Intermediate examples
WIP
== Olympiad examples WIP
More Problems
WIP
OUTLINE:
- Proofs
- Links to Proofs of AM-GM
- Generalizations
- Weighted AM-GM
- QM-AM-GM-HM (with or without weights)
- Power Mean (with or without weights)