# Jensen's Inequality

**Jensen's Inequality** is an inequality discovered by Danish mathematician Johan Jensen in 1906.

## Contents

## Inequality

Let be a convex function of one real variable. Let and let satisfy . Then

If is a concave function, we have:

## Proof

We only prove the case where is concave. The proof for the other case is similar.

Let . As is concave, its derivative is monotonically decreasing. We consider two cases.

If , then If , then By the fundamental theorem of calculus, we have Evaluating the integrals, each of the last two inequalities implies the same result: so this is true for all . Then we have as desired.

## Example

One of the simplest examples of Jensen's inequality is the quadratic mean - arithmetic mean inequality. Taking , which is convex (because and ), and , we obtain

Similarly, arithmetic mean-geometric mean inequality (AM-GM) can be obtained from Jensen's inequality by considering .

In fact, the power mean inequality, a generalization of AM-GM, follows from Jensen's inequality.

## Problems

### Introductory

#### Problem 1

Prove AM-GM using Jensen's Inequality

#### Problem 2

Prove the weighted AM-GM inequality. (It states that when )

### Intermediate

- Prove that for any , we have .
- Show that in any triangle we have

### Olympiad

- Let be positive real numbers. Prove that

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