I could not obtain proof for either one of the following inequalities:
I tried to prove it by using their tangents, but it seems not to be so simple.
There's probably a nicer proof, but here is one anyway:
Let . The derivative of the function is , which is equal to 0 at , and respectively negative and positive at the left and right of this point. As a consequence, for every , which is exactly what we want.
Now, I see that I always had mistakes in the calculation.
I again checked my way for the last time and saw that it works.
Let me explain it below.
Set , we clearly know that for every since is convex.
What I need to find is that
Solving (1), we get
Clearly, (3) satisfies (2), and thus the solution is complete.
I really still don't know where I had mistakes all the time.