Double-integral inequalities for monotone functions

Dear **MHF** members,

I met somewhere an integral inequality as follows.

where are nonnegative continuous functions

and is a nondecreasing function with .

Have any of you met before with something like this

or can help me how to deduce this?

Thank you very much.

**bkarpuz**

Re: Double-integral inequalities for monotone functions

Hey bkarpuz.

This looks like a change of order: can you try doing a change of order first and then seeing if you can use your constraint (non-decreasing function) to prove the result?

Re: Double-integral inequalities for monotone functions

My friend **chiro**, you are always doing the same.

Please give me somthing more than I have. :)

Okay, when is differentiable, I have the following proof.

**Proof**. Let

,

which yields

Hence, for , we get

,

which completes the proof.

But what if is not differentiable?

Thank you.

Re: Double-integral inequalities for monotone functions

If it is not differentiable you will need to resort to the appropriate Measure Theory results and use the appropriate measure.

Re: Double-integral inequalities for monotone functions

Quote:

Originally Posted by

**chiro** If it is not differentiable you will need to resort to the appropriate Measure Theory results and use the appropriate measure.

What kind of appropriate measure are you talking about **chiro**?

This is usual integration the functions are continuous.

I will be very glad if you do **not** spam my posts again.

Re: Double-integral inequalities for monotone functions

If something is not differentiable then you use a different measure to integrate.

If something has an analytic smooth anti-derivative you use the Riemann integral and measure.

Other-wise you use another measure that allows you to integrate the function.

An example of a measure that is used for continuous but not differentiable integration is the Brownian Motion stochastic integral.

Take a look at the Lebesgue integral and the appropriate measure theory.