Difference between revisions of "AM-GM Inequality"
Etmetalakret (talk | contribs) |
(Remove vandalism: Undo revision 215807 by Marianasinta (talk)) (Tag: Undo) |
||
(20 intermediate revisions by 3 users not shown) | |||
Line 1: | Line 1: | ||
− | In [[ | + | 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 <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>. | + | 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 condition | equality]] if and only if <math>x_1 = x_2 = \cdots = x_n</math>. |
− | + | The AM-GM Inequality is among the most famous inequalities in algebra and has cemented itself as ubiquitous across almost all competitions. Applications exist at introductory, intermediate, and olympiad level problems, with AM-GM being particularly crucial in proof-based contests. | |
== Proofs == | == Proofs == | ||
− | + | {{Main|Proofs of AM-GM}} | |
+ | All known proofs of AM-GM use [[induction]] or other, more advanced inequalities. Furthermore, they are all more complex than their usage in introductory and most intermediate competitions. AM-GM's most elementary proof utilizes [[Cauchy Induction]], a variant of induction where one proves a result for <math>2</math>, uses induction to extend this to all powers of <math>2</math>, and then shows that assuming the result for <math>n</math> implies it holds for <math>n-1</math>. | ||
== Generalizations == | == 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 === | === Weighted AM-GM Inequality === | ||
− | + | The '''Weighted AM-GM Inequality''' relates the [[Weighted average | weighted]] arithmetic and geometric means. It states that for 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 === | === Mean Inequality Chain === | ||
{{Main|Mean Inequality Chain}} | {{Main|Mean Inequality Chain}} | ||
− | The '''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 <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>. As with AM-GM, there also exists a weighted version of the Mean Inequality Chain. |
=== Power Mean Inequality === | === Power Mean Inequality === | ||
{{Main|Power Mean Inequality}} | {{Main|Power Mean Inequality}} | ||
+ | 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 === |
− | + | * 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 === |
− | + | * 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 === |
− | + | * 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]] |
Latest revision as of 14:32, 22 February 2024
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 .
The AM-GM Inequality is among the most famous inequalities in algebra and has cemented itself as ubiquitous across almost all competitions. Applications exist at introductory, intermediate, and olympiad level problems, with AM-GM being particularly crucial in proof-based contests.
Contents
[hide]Proofs
- Main article: Proofs of AM-GM
All known proofs of AM-GM use induction or other, more advanced inequalities. Furthermore, they are all more complex than their usage in introductory and most intermediate competitions. AM-GM's most elementary proof utilizes Cauchy Induction, a variant of induction where one proves a result for , uses induction to extend this to all powers of , and then shows that assuming the result for implies it holds for .
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)