Difference between revisions of "AM-GM Inequality"
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== Proofs == | == Proofs == | ||
− | + | {{Main|Proofs of AM-GM}} | |
+ | All known proofs of AM-GM use either induction or other inequalities. Its proof is far more complicated than its usage in introductory competitions, which makes learning it not recommended for students new to proofs. Listed here is proof of AM-GM that utilizes [[Cauchy Induction]] | ||
+ | |||
+ | === Base Case === | ||
+ | The smallest nontrivial case of AM-GM is in two variables. By the [[Trivial Inequality]], <cmath>(x-y)^2 \geq 0,</cmath> with equality if and only if <math>x-y=0</math>, or <math>x=y</math>. Then because <math>x</math> and <math>y</math> are nonnegative, we can perform the following manipulations: <cmath>x^2 - 2xy + y^2 \geq 0</cmath> <cmath>x^2 + 2xy + y^2 \geq 4xy</cmath> <cmath>\frac{(x+y)^2}{4} \geq xy</cmath> <cmath>\frac{x+y}{2} \geq \sqrt{xy}.</cmath> This completes the proof of the base case. | ||
+ | |||
+ | === Powers of Two === | ||
+ | |||
+ | === Backwards Step === | ||
== Generalizations == | == Generalizations == |
Revision as of 13:04, 30 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
- Main article: Proofs of AM-GM
All known proofs of AM-GM use either induction or other inequalities. Its proof is far more complicated than its usage in introductory competitions, which makes learning it not recommended for students new to proofs. Listed here is proof of AM-GM that utilizes Cauchy Induction
Base Case
The smallest nontrivial case of AM-GM is in two variables. By the Trivial Inequality, with equality if and only if , or . Then because and are nonnegative, we can perform the following manipulations: This completes the proof of the base case.
Powers of Two
Backwards Step
Generalizations
The AM-GM Inequality has been generalized into several other inequalities. In addition to those listed, the Minkowski Inequality is another generalization of AM-GM.
Weighted AM-GM Inequality
The Weighted AM-GM Inequality relates the weighted arithmetic and geometric means. It states that for 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 nonnegative reals. In particular, it states that with equality if and only if . 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 is defined as follows: The Power Mean inequality then states that if , then , with equality holding if and only if Plugging into this inequality reduces it to AM-GM, and gives the Mean Inequality Chain. As with AM-GM, there also exists a weighted version of the 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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