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?
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.
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.
Hence, for , we get
which completes the proof.
But what if is not differentiable?
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.