As I was reading through my linear analysis notes today, there was a small

passage where our lecturer Jim Wright provided a proof of a generalisation

of the inequality of arithmetic and geometric means:

*Let and . Then .*

I didn’t bother to read the proof as I thought I could probably come up with a

good one myself. Indeed, I am sufficiently happy with the proof I found that

I think it’s worth sharing. Here it is:

Proof:

If the proof is trivial. So I shall proceed by assuming the contrary:

- Let’s define and divide by . We then obtain:
- Now, let’s define the following differentiable functions:
- The derivatives of these functions are given by:
- Now, we can quickly reach the following conclusions:a) for

b) For

and both functions are monotonically increasing on so there can’t be such that unless equals zero or one.

It follows that on .

I must say that this was an easy problem but I liked the method I found for

solving it. Namely, reducing the number of variables and then replacing

variables with functions that can then be readily analysed. But, we can go a bit further

and show how Hölder’s inequality follows easily.

Hölder’s inequality:

For any two sequences , we have:

for where .

Proof:

If we define , and normalize these vectors we have:

Now, we may apply the generalised AM-GM inequality to deduce:

Interesting proof