Hey Juju.

One suggestion I would have is to look at the parts that are increasing and decreasing and use the fact that if f() is monotonic increasing and E[X] < E[Y], then E[f(X)] < E[f(Y)]. Similarly if f is monotonic decreasing, you will have E[f(X)] < E[f(Y)].

However if there are turning points, then one needs to look at a mixture of this above results. If for example f(x) is always greater than 0 for all x, then you should get the above result that E[f(X)] < E[f(y)] for E[X] < E[Y].