Write the second integral as , and make the substitution . Then it becomes . Integrate that by parts, to get .
Now suppose that b = f(a). Then the previous equation says that . Thus equality holds for the desired inequality in that case.
Next, suppose that b > f(a). By the result in the "equality" case, we know that . But is an increasing function, taking its minimum value a at the left end of the interval [f(a),b]. So . Hence
A similar argument shows that the strict inequality also holds when b < f(a).
[But I don't see how this has anything to do with Young's inequality (except that they both say that ab is less than something).]