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Estimate integral. (Lp Spaces, Holder)

  1. Apr 11, 2013 #1
    1. The problem statement, all variables and given/known data
    Show that: [itex]\left(\int^{0}_{1}\frac{x^{\frac{1}{2}}dx}{(1-x)^{\frac{1}{3}}}\right)^{3}\leq\frac{8}{5}[/itex]

    2. Relevant equations
    Holder inequality.

    3. The attempt at a solution
    First, I took the cube root of each side. This let me just deal with the 1-norm on the left. Then I broke the function into 2 parts: f=numerator. g=1/denominator. The 2-norm of each function is integrable. |f|2=(1/2)^(1/2). |g|2=3^(1/2). |f|2*|g|2=(3/2)^(1/2)>(8/5)^(1/3), which is counter the claim. So I obviously missed something.

    Where is the flaw in this approach? Are there better approaches to this problem?
     
  2. jcsd
  3. Apr 11, 2013 #2

    micromass

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    You have used the inequality

    [tex]\int_0^1 |fg|\leq \left(\int_0^1 |f|^p\right)^{1/p}\left(\int_0^1 |g|^q\right)^{1/q}[/tex]

    with the functions ##f(x)=x^{1/2}## and ##g(x) = (1-x)^{-1/3}##. This is correct.

    However, you have used ##p=q=2##. This gives you an inequality, but not the correct one. Try to use other constants.
     
  4. Apr 11, 2013 #3
    I was hoping that I was missing a more strategic approach. (I had already tried working through it with q=3 but ran into unboundedness of g. Working with 2 solved the unboundedness problem, but missed the mark.)

    Problem does solve with p=3 and q=3/2. Thanks for the help.
     
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