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This inequality often appears in Analysis: , for , and . It does seem quite tricky to prove, and using Binomial Theorem leads to a mess and doesn’t work!

It turns out that the key is to use convexity, and we can even prove a stronger version of the above, namely .

Proof: Consider which is convex on . Let . By convexity, we have for .

Choose . Then we have , which implies .

Thus,

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