*Problem:*

Prove that for any real numbers the following inequality holds:

.

*Solution:*

Observe that

.

But can be considered as a constant. So,

.

Notice now that

.

So, the inequality reduces to

.

Now, using Cauchy-Schwartz inequality for integrals, we get that

.

But, we must now observe that

,

which comes to the conclusion, *Q.E.D.*