# Thread: Double-integral inequalities for monotone functions

1. ## Double-integral inequalities for monotone functions

Dear MHF members,

I met somewhere an integral inequality as follows.
$\int_{\tau(a)}^{\tau(b)}f(x)\int_{x}^{b}g(y) \mathrm{d}y \mathrm{d}x\geq\int_{a}^{b}g(y)\int_{\tau(a)}^{ \tau(y)}f(x) \mathrm{d}x \mathrm{d}y,$
where $f,g$ are nonnegative continuous functions
and $\tau$ is a nondecreasing function with $\tau(x)\leq x$.
Have any of you met before with something like this
or can help me how to deduce this?

Thank you very much.
bkarpuz

2. ## 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?

3. ## 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 $\tau$ is differentiable, I have the following proof.

Proof. Let
$\varphi(t):=\int_{\tau(a)}^{\tau(t)}f(x)\int_{x}^{ t}g(y) \mathrm{d}y \mathrm{d}x-\int_{a}^{t}g(y)\int_{\tau(a)}^{ \tau(y)}f(x) \mathrm{d}x \mathrm{d}y$,
which yields
$\varphi^{\prime}(t)=\bigg(\int_{\tau(t)}^{t}g(y) \mathrm{d}y\bigg)f(\tau(t))\tau^{\prime}(t)\geq0.$
Hence, for $b\geq a$, we get
$\varphi(b)\geq\varphi(a)=0$,
which completes the proof. $\rule{0.2cm}{0.2cm}$

But what if $\tau$ is not differentiable?
Thank you.

4. ## 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.

5. ## Re: Double-integral inequalities for monotone functions

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.

6. ## 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.