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Math Help - bound on sum of logarithms

  1. #1
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    bound on sum of logarithms

    I have that

     \prod_{i,j} (1-y_{ij}) \geq l , where 1> y_{ij}\geq 0

    I need to bound the following using  l

     \prod_{i,j} (1-a_{ij}y_{ij}) \geq ? , where 1> y_{ij}\geq 0 , and 0.5 \geq a_{ij}\geq 0

    Such that ? has to be a function of  l ,  f(l)

    Can I use something like Holder's inequality?

    I started by taking the logarithm of both sides

     \sum_{i,j} \log (1-y_{ij}) \geq \log(l)
     \sum_{i,j} \log (1-a_{ij}y_{ij}) \geq ?

    And I believe that the solution is

     \sum_{i,j} \log (1-a_{ij}y_{ij}) \geq \log (\frac{1+l}{2})

    but I don't know how to prove it. Any pointers?
    Last edited by robustor; November 29th 2011 at 12:28 PM.
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