Characteristics of an IFR pdf

Hi,

I am looking for help in solving a problem that crept up in my doctoral thesis.

Let f be a pdf, twice differentiable and F its cdf. Both are defined over .

I wish to prove that

(1).

I wish to prove it for all functions f which are IFR in the sense developed by Richard E. Barlow and Franck Proschan in their book Mathematical theory of reliability - Google Book Search.

When a pdf is IFR, then is weakly increasing. This, of course, leads to .

So that we can write

(2)

Now (1) can also be written . (3)

(3) is true if f is IFR and if .

What happens when ?

If , f'(x) is always positive because f is a pdf. So f is increasing over .

If there exists a such that , then f exhibits a maximum at that point. but f being a pdf, that means that , which contradicts the hypothesis unless f is the null function, which we discard. So . Hence f is striclty increasing. But by definition of F, .

My hunch is that this last equality contradicts the fact that f is strictly increasing and positive.

Is this the case and can we then say that if f is IFR then and so (1) is true when f is IFR?

If that is so, then I have have won my day.

Any help on this problem would be very much appreciated!

Thank you.