Try out the Personality Quiz!
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 .