AM-GM Inequality
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
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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 , 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
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Intermediate examples
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Olympiad examples
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More Problems
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