(Theorem): Consider where and either or . Let and let be either convex or monotonic, then
Proof
There are two cases:
+ If then the above problem is alway true.
+ If then we can write as:
Now we need to prove that:
Without loss of generality, suppose that then:
Let
then we have
From van khea’s inequality with convex funtion we have
Therefore the proof is completed.
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