# Hölder's Inequality

## Elementary Form

If are nonnegative real numbers and are nonnegative reals with sum of 1, then

Note that with two sequences and , and , this is the elementary form of the Cauchy-Schwarz Inequality.

We can state the inequality more concisely thus: Let be several sequences of nonnegative reals, and let be a sequence of nonnegative reals such that . Then

## Proof of Elementary Form

We will use weighted AM-GM. We will disregard sequences for which one of the terms is zero, as the terms of these sequences do not contribute to the left-hand side of the desired inequality but may contribute to the right-hand side.

For integers , let us define Evidently, . Then for all integers , by weighted AM-GM, Hence But from our choice of , for all integers , Therefore since the sum of the is one. Hence in summary, as desired. Equality holds when for all integers , i.e., when all the sequences are proportional.

## Statement

If , , then and .

## Proof

If then a.e. and there is nothing to prove. Case is similar. On the other hand, we may assume that for all . Let . Young's Inequality gives us These functions are measurable, so by integrating we get

## Examples

- Prove that, for positive reals , the following inequality holds: