This topic doesn’t get anywhere near the attention it deserves. I like it because it leads to a useful family of inequalities. Given a strictly convex function (i.e. and is monotone) we define its Legendre transform as
The maximum can be calculated by differentiating and requiring that . Hence we can write and use the fact that this function is invertible to write . With these we can check that this was indeed a maximisation because by assumption.
Now, with as above we get the Legendre transform:
Omitting the display of the functional dependency we get the symmetric relationship and this suggests (correctly) that the Legendre transform is its own inverse.
For more information you can check out the wikipedia page at http://en.wikipedia.org/wiki/Legendre_transform or the interesting article ‘Making Sense of the Legendre Transform’ available on the arxiv at http://arxiv.org/abs/0806.1147.
To finish we’ll use the Legendre transform to prove the following useful inequality.
Proposition. If satisfy then for all real .
Proof. Define and then gives since it is readily shown that . Therefore since . Therefore, by definition, implies that .