# AM-GM Inequality

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

## 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 .

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### Olympiad

- Let , , and be positive real numbers. Prove that

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