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Exponential function sum problem

  1. Feb 2, 2012 #1
    1. The problem statement, all variables and given/known data
    I need to prove that
    (1+[itex]x_{1})[/itex]·...·(1+[itex]x_{n}[/itex])≥(1-[itex]Ʃ^{n}_{i=1}x_{i}^2[/itex])[itex]e^{Ʃ^{n}_{i=1}x_{i}}[/itex]
    with all 0≤[itex]x_{i}[/itex]≤1

    I've already proven that

    (1+[itex]x_{1}[/itex])·...·(1+[itex]x_{n}[/itex])≤[itex]e^{Ʃ^{n}_{i=1}x_{i}}[/itex]
    with all 0≤[itex]x_{i}[/itex]≤1

    and (1-[itex]x_{1}[/itex])·...·(1-[itex]x_{i}[/itex])≥1-Ʃ[itex]^{n}_{i=1}x_{i}[/itex] with all 0≤[itex]x_{i}[/itex]≤1 ,

    but can't figure out what to do with the main problem :D
     
  2. jcsd
  3. Feb 2, 2012 #2
    Take logarithm on both sides, i.e.,
    Ʃ log(1+x_i)≥log(1-Ʃ x_i^2)+Ʃ x_i
    then realize that x(1-x)≤log(1+x)≤x for 0<x<1, substitute in and you'll see
     
  4. Feb 7, 2012 #3
    you cant take logarithm because you can't know if 1-sum(x_i)^2 is negative or not.
     
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