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Math Help - Double-integral inequalities for monotone functions

  1. #1
    Senior Member bkarpuz's Avatar
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    Exclamation 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
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  2. #2
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    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?
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  3. #3
    Senior Member bkarpuz's Avatar
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    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.
    Last edited by bkarpuz; June 18th 2013 at 09:56 PM.
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    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.
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    Senior Member bkarpuz's Avatar
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    Re: Double-integral inequalities for monotone functions

    Quote Originally Posted by chiro View Post
    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.
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  6. #6
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    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.
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