I will just give the solution for 2. (a) since the remaining case is very similar.

Clearly, if is increasing then we know that for all .

Hence, we have

which completes the first part of the inequality.

For the remaining part, we consider that is concave down (convex).

This indicates that holds for all (draw a graph for the increasing convex function and the line passing at the points and ).

Integrating both sides of this inequality, we get

which completes the proof.