Difference between revisions of "AM-GM Inequality"
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The '''Power Mean Inequality''' relates all the different power means 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 all the different power means of a list of nonnegative reals. The power mean <math>M(p)</math> is defined as follows: <cmath>M(p) = | ||
+ | == Problems == | ||
− | == Introductory | + | === Introductory === |
− | + | * For nonnegative real numbers <math>a_1,a_2,\cdots a_n</math>, demonstrate that if <math>a_1a_2\cdots a_n=1</math> then <math>a_1+a_2+\cdots +a_n\ge n</math>. ([[Solution to AM - GM Introductory Problem 1|Solution]]) | |
+ | * Find the maximum of <math>2 - a - \frac{1}{2a}</math> for all positive <math>a</math>. ([[Solution to AM - GM Introductory Problem 2|Solution]]) | ||
− | == Intermediate | + | === Intermediate === |
− | + | * Find the minimum value of <math>\frac{9x^2\sin^2 x + 4}{x\sin x}</math> for <math>0 < x < \pi</math>. | |
+ | ([[1983 AIME Problems/Problem 9|Source]]) | ||
− | == Olympiad | + | === Olympiad === |
− | + | * Let <math>a </math>, <math>b </math>, and <math>c </math> be positive real numbers. Prove that | |
− | + | <cmath> (a^5 - a^2 + 3)(b^5 - b^2 + 3)(c^5 - c^2 + 3) \ge (a+b+c)^3 . </cmath> | |
− | + | ([[2004 USAMO Problems/Problem 5|Source]]) | |
− | |||
== See Also == | == See Also == | ||
+ | * [[Proofs of AM-GM]] | ||
* [[Mean Inequality Chain]] | * [[Mean Inequality Chain]] | ||
* [[Power Mean Inequality]] | * [[Power Mean Inequality]] | ||
* [[Cauchy-Schwarz Inequality]] | * [[Cauchy-Schwarz Inequality]] | ||
* [[Inequality]] | * [[Inequality]] | ||
+ | |||
+ | [[Category:Algebra]] | ||
+ | [[Category:Inequalities]] | ||
+ | [[Category:Definition]] |
Revision as of 15:50, 29 December 2021
In algebra, the AM-GM Inequality, also known formally as the Inequality of Arithmetic and Geometric Means or informally as AM-GM, is an inequality that states that any list of nonnegative reals' arithmetic mean is greater than or equal to its geometric mean; furthermore, the two means are equal if and only if every number 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, more advanced inequalities. Its proof is far more complicated than its usage in introductory competitions; consequentially, learning it is not recommended to students new to proofs. The most elementary proof of AM-GM utilizes Cauchy Induction, a variant of induction that involves proving a result for two, then using induction to prove it for all powers of two, and then a backward step where implies .
Generalizations
The AM-GM Inequality has been generalized into several other inequalities. In addition to those listed, the Minkowski Inequality and Muirhead's Inequality are also generalizations 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 all the different power means 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.
Problems
Introductory
- For nonnegative real numbers , demonstrate that if then . (Solution)
- Find the maximum of for all positive . (Solution)
Intermediate
- Find the minimum value of for .
(Source)
Olympiad
- Let , , and be positive real numbers. Prove that
(Source)