Difference between revisions of "AM-GM Inequality"
Etmetalakret (talk | contribs) |
Etmetalakret (talk | contribs) |
||
Line 9: | Line 9: | ||
== Generalizations == | == Generalizations == | ||
− | The AM-GM Inequality has been generalized into several other inequalities, | + | 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 === | === Weighted AM-GM Inequality === | ||
Line 20: | Line 20: | ||
=== Power Mean Inequality === | === Power Mean Inequality === | ||
{{Main|Power Mean Inequality}} | {{Main|Power Mean Inequality}} | ||
− | The '''Power Mean Inequality''' relates every power mean of a list of nonnegative reals. The power mean <math>M(p)</math> is defined as follows: <cmath>M(p) = | + | The '''Power Mean Inequality''' relates every power mean of a list of nonnegative reals. The power mean <math>M(p)</math> is defined as follows: <cmath>M(p) = |
== Introductory examples == | == Introductory examples == |
Revision as of 12:21, 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
WIP
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
WIP
Intermediate examples
WIP
Olympiad examples
WIP
More Problems
WIP