I know that is concave in . So there's a unique maximum point of on .

Since the function is strictly increasing, and have the same maximum point.

Can I now conclude that has also a unique maximum point?

Also is concave in and is strictly increasing.

Then the maximum point of on is unique. Am I right?

If the maximum points of and are unique I think it follows from the uniform convergence that it holds

.

Am I right?

Sorry, I made some mistakes in the definitions. There should be a instead of the . So, the corrected version: