Suppose are any positive real numbers such that Prove that:
Hence, we set , therefore it suffices to find .
Considering the symmetricity, and it suffices to examine the partial derivative of with respect to .
It follows that the critical points are and .
On the other hand, let
We find that the critical point for is , and , which is the max value of (at ) at the same time because the other critical point makes attain .
Therefore, the given inequality holds.
Not. I know this is a long solution and not so nice. :S