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
.
Thus,