Karamata's Inequality states that if majorizes and is a convex function, then
We will first use an important fact:
This is proven by taking casework on . If , then
A similar argument shows for other values of .
Now, define a sequence such that:
Define the sequences such that and similarly.
Then, assuming and similarily with the 's, we get that . Now, we know: .
Thus we have proven Karamata`s Theorem This article is a stub. Help us out by expanding it.