Difference between revisions of "Karamata's Inequality"
(Proof of the Inequality) |
Robot7620 2 (talk | contribs) m (Changed ending to be grammatically correct) |
||
Line 25: | Line 25: | ||
<cmath>\sum_{i=1}^{n}f(a_i) \geq \sum_{i=1}^{n}f(b_i)</cmath> | <cmath>\sum_{i=1}^{n}f(a_i) \geq \sum_{i=1}^{n}f(b_i)</cmath> | ||
− | + | Thus we have proven Karamata`s Theorem | |
{{stub}} | {{stub}} | ||
Revision as of 00:30, 24 November 2018
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.