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:
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
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.
For any two sequences , we have:
for where .
If we define , and normalize these vectors we have:
Now, we may apply the generalised AM-GM inequality to deduce: