If and are sequences of real numbers where the following series converge, for a given and such that ,

and

Then we have the following inequality, know an Holder’s inequality.

**Proof**: Suppose and note the following, due to Young’s Inequality.

Summing over n, we have

Now, suppose we’re given sequences and where the associated series converge, but not to one. In that case we construct two new sequences, and as follows.

Now note that , so we can apply the previous result to obtain the following.

Multiplying both sides by the denominator on the left (which we can pull out of the sum since it’s constant), we obtain the desired inequality.

(It’s worth mentioning that there is another, more general, version of Holder’s Inequality that uses Lebesgue integrals. The Lebesgue integral version actually implies the version we’ve mentioned here, and I’ll post about that once I start posting my notes from measure theory, specifically Lebesgue integration.)

### Like this:

Like Loading...

*Related*

http://vankheakh.wordpress.com/2010/12/11/a-generalization-of-holders-inequality/

Comment by khea — December 18, 2010 @ 12:56 pm |