# Difference between revisions of "Arithmetic Mean-Geometric Mean Inequality"

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− | + | The '''Arithmetic Mean-Geometric Mean Inequality''' ('''AM-GM''' or '''AMGM''') is an elementary [[inequality]], and is generally one of the first ones taught in inequality courses. | |

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− | The '''Arithmetic Mean-Geometric Mean Inequality''' ('''AM-GM''' or '''AMGM''') is an elementary [[inequality]], generally one of the first ones taught in inequality courses. | ||

== Theorem == | == Theorem == | ||

− | + | AM-GM states that for any [[set]] of [[nonnegative]] [[real number]]s, the [[arithmetic mean]] of the set is greater than or [[equal]] to the [[geometric mean]] of the set. Algebraically, this is expressed as follows. | |

For a set of nonnegative real numbers <math>a_1,a_2,\ldots,a_n</math>, the following always holds: | For a set of nonnegative real numbers <math>a_1,a_2,\ldots,a_n</math>, the following always holds: | ||

<cmath> \frac{a_1+a_2+\ldots+a_n}{n}\geq\sqrt[n]{a_1a_2\cdots a_n} </cmath> | <cmath> \frac{a_1+a_2+\ldots+a_n}{n}\geq\sqrt[n]{a_1a_2\cdots a_n} </cmath> | ||

Using the shorthand notation for [[summation]]s and [[product]]s: | Using the shorthand notation for [[summation]]s and [[product]]s: | ||

− | <cmath> \sum_{i=1}^{n}a_i} | + | <cmath> \sum_{i=1}^{n}\frac{a_i}{n} \geq \prod\limits_{i=1}^{n}a_i^{\frac{1}{n}} . </cmath> |

For example, for the set <math>\{9,12,54\}</math>, the arithmetic mean, 25, is greater than the geometric mean, 18; AM-GM guarantees this is always the case. | For example, for the set <math>\{9,12,54\}</math>, the arithmetic mean, 25, is greater than the geometric mean, 18; AM-GM guarantees this is always the case. | ||

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=== Proof === | === Proof === | ||

− | + | See here: [[Proofs of AM-GM]]. | |

=== Weighted Form === | === Weighted Form === | ||

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=== 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]]) | ||

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=== 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>. | * Find the minimum value of <math>\frac{9x^2\sin^2 x + 4}{x\sin x}</math> for <math>0 < x < \pi</math>. | ||

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* [[Algebra]] | * [[Algebra]] | ||

* [[Inequalities]] | * [[Inequalities]] | ||

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[[Category:Inequality]] | [[Category:Inequality]] | ||

[[Category:Theorems]] | [[Category:Theorems]] |

## Latest revision as of 00:30, 6 March 2021

The **Arithmetic Mean-Geometric Mean Inequality** (**AM-GM** or **AMGM**) is an elementary inequality, and is generally one of the first ones taught in inequality courses.

## Contents

## Theorem

AM-GM states that for any set of nonnegative real numbers, the arithmetic mean of the set is greater than or equal to the geometric mean of the set. Algebraically, this is expressed as follows.

For a set of nonnegative real numbers , the following always holds: Using the shorthand notation for summations and products: For example, for the set , the arithmetic mean, 25, is greater than the geometric mean, 18; AM-GM guarantees this is always the case.

The equality condition of this inequality states that the arithmetic mean and geometric mean are equal if and only if all members of the set are equal.

AM-GM can be used fairly frequently to solve Olympiad-level inequality problems, such as those on the USAMO and IMO.

### Proof

See here: Proofs of AM-GM.

### Weighted Form

The weighted form of AM-GM is given by using weighted averages. For example, the weighted arithmetic mean of and with is and the geometric is .

AM-GM applies to weighted averages. Specifically, the **weighted AM-GM Inequality** states that if are nonnegative real numbers, and are nonnegative real numbers (the "weights") which sum to 1, then
or, in more compact notation,
Equality holds if and only if for all integers such that and .
We obtain the unweighted form of AM-GM by setting .

## Extensions

- The power mean inequality is a generalization of AM-GM which places the arithemetic and geometric means on a continuum of different means.
- The root-square-mean arithmetic-mean geometric-mean harmonic-mean inequality is special case 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)