Difference between revisions of "Karamata's Inequality"
(Proof of the Inequality) |
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<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}} | ||
==See also== | ==See also== | ||
− | [[Category: | + | |
− | [[Category: | + | [[Category:Algebra]] |
+ | [[Category:Inequalities]] |
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.