# Difference between revisions of "Hölder's Inequality"

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These functions are measurable, so by integrating we get | These functions are measurable, so by integrating we get | ||

<cmath> \frac{||fg||_1}{||f||_p||g||_q}\leq \frac{1}{p}\frac{||f(x)||^p}{||f||_p^p} + \frac{1}{q}\frac{||g(x)||^q}{||g||_q^q} = \frac{1}{p}+\frac{1}{q}=1 . </cmath> | <cmath> \frac{||fg||_1}{||f||_p||g||_q}\leq \frac{1}{p}\frac{||f(x)||^p}{||f||_p^p} + \frac{1}{q}\frac{||g(x)||^q}{||g||_q^q} = \frac{1}{p}+\frac{1}{q}=1 . </cmath> | ||

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+ | == Examples == | ||

+ | * Prove that, for positive reals <math>x,y,k</math>, the following inequality holds: | ||

+ | <center><math>\left(1 + \frac {x}{y}\right)^k + \left(1 + \frac {y}{x}\right)^k\geq 2^{k+1}</math></center> | ||

{{wikify}} | {{wikify}} |

## Revision as of 16:22, 1 July 2008

## 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: