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Math Help - Problem from Analisis 3 exam.

  1. #1
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    Problem from Analisis 3 exam.

    Could there be function:

    0,\infty) \rightarrow C" alt="f0,\infty) \rightarrow C" /> so true: \frac{1}{f} \in L^1(0,\infty) and \overline{lim}_{t\rightarrow 0+} \frac{\int_0^t |f(x)|dx}{t^2} ?

    Do I need to use some of inequalities or in any other way?

    Thank you.
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  2. #2
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    Quote Originally Posted by veljko View Post
    Could there be function:

    0,\infty) \rightarrow C" alt="f0,\infty) \rightarrow C" /> so true: \frac{1}{f} \in L^1(0,\infty) and \overline{lim}_{t\rightarrow 0+} \frac{\int_0^t |f(x)|dx}{t^2} ?

    Do I need to use some of inequalities or in any other way?

    Thank you.
    You have likely gotten no responses since this is incomprehensible. Try rewriting it. What is L^1(0,\infty) Do you mean the space with the \ell_1 norm \|(x_1,\cdots,x_n)\|=\sum_{j=1}^{n}|x_j|?
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  3. #3
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    Update:

    Is there a function:

    0,\infty) \rightarrow C" alt="f0,\infty) \rightarrow C" /> so true: \frac{1}{f} \in L^1(0,\infty) and \overline{lim}_{t\rightarrow 0+} \frac{\int_0^t |f(x)|dx}{t^2} < \infty?

    p.s. = \{f: \int_{(0,\infty)}|f|dx < + \infty  \}
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  4. #4
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    edit:nevermind.
    edit2: there's probably a one-liner involving maximal functions.
    Last edited by maddas; April 13th 2010 at 07:46 AM.
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  5. #5
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    Quote Originally Posted by veljko View Post
    Update:

    Is there a function:

    0,\infty) \rightarrow C" alt="f0,\infty) \rightarrow C" /> so true: \frac{1}{f} \in L^1(0,\infty) and \overline{lim}_{t\rightarrow 0+} \frac{\int_0^t |f(x)|dx}{t^2} < \infty?

    p.s. = \{f: \int_{(0,\infty)}|f|dx < + \infty  \}
    How about f(t)=e^t.
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  6. #6
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    Doesn't \frac1{t^2}\int_0^te^x = \frac{e^t-1}{t^2} which goes to infinity?....
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