Karamata's Inequality
Karamata's Inequality states that if majorizes and is a convex function, then
Proof
We will first use an important fact: If is convex over the interval , then and ,
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: .
Therefore,
Thus, we have proven Karamata's Theorem.
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