Difference between revisions of "Karamata's Inequality"
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Revision as of 16:48, 29 December 2021
Karamata's Inequality states that if majorizes
and
is a convex function, then
Proof
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:
.
Therefore,
Thus we have proven Karamata`s Theorem This article is a stub. Help us out by expanding it.