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

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− | The '''Arithmetic Mean-Geometric Mean''' ('''AM-GM''') | + | 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. |

− | + | == 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. | ||

− | <math>\ | + | 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> | ||

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

+ | <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. | ||

− | The | + | The [[equality condition]] of this [[inequality]] states that the arithmetic mean and geometric mean are equal [[iff|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 [[United States of America Mathematics Olympiad | USAMO]] and [[International Mathematics Olympiad | IMO]]. | |

+ | === Proof === | ||

+ | |||

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

+ | |||

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

+ | |||

+ | The weighted form of AM-GM is given by using [[weighted average]]s. For example, the weighted arithmetic mean of <math>x</math> and <math>y</math> with <math>3:1</math> is <math>\frac{3x+1y}{3+1}</math> and the geometric is <math>\sqrt[3+1]{x^3y}</math>. | ||

+ | |||

+ | AM-GM applies to weighted averages. Specifically, the '''weighted AM-GM Inequality''' states that if <math>a_1, a_2, \dotsc, a_n</math> are nonnegative real numbers, and <math>\lambda_1, \lambda_2, \dotsc, \lambda_n</math> are nonnegative real numbers (the "weights") which sum to 1, then | ||

+ | <cmath> \lambda_1 a_1 + \lambda_2 a_2 + \dotsb + \lambda_n a_n \ge a_1^{\lambda_1} a_2^{\lambda_2} \dotsm a_n^{\lambda_n}, </cmath> | ||

+ | or, in more compact notation, | ||

+ | <cmath> \sum_{i=1}^n \lambda_i a_i \ge \prod_{i=1}^n a_i^{\lambda_i} . </cmath> | ||

+ | Equality holds if and only if <math>a_i = a_j</math> for all integers <math>i, j</math> such that <math>\lambda_i \neq 0</math> and <math>\lambda_j \neq 0</math>. | ||

+ | We obtain the unweighted form of AM-GM by setting <math>\lambda_1 = \lambda_2 = \dotsb = \lambda_n = 1/n</math>. | ||

+ | |||

+ | ==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 <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 == | ||

+ | |||

+ | * [[RMS-AM-GM-HM]] | ||

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

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

− | + | ||

− | + | ||

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

+ | [[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)