**Problem**. Let be a continuously differentiable function and suppose that Show that

**Solution**. Let

Then and so integration by parts with together with the fact that give

Thus, by Cauchy-Schwarz,

On the other hand, by Cauchy-Schwarz again,

and hence

The result now follows from

**Exercise 1**. Explain why, in the solution of the above problem, the result follows from

*Hint*. If then for all

**Exercise 2**. Let be a continuously differentiable function and suppose that Show that

*Hint*. If then

**Exercise 3**. This exercise generalizes the above problem. Let be a continuously differentiable function. Show that

*Hint*. Apply the result given in the problem to the function

**Exercise 4**. Prove the inequality given in Exercise 3 for i.e. by evaluating each integral, show that