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Math Help - inequality

  1. #1
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    inequality

    a,b,c>0 and a^4+b^4+c^4 = 3 . Prove the :
    sigma\frac{1}{4-ab} =< 1
    Last edited by math; October 16th 2005 at 12:01 AM.
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  2. #2
    hpe
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    Quote Originally Posted by math
    a,b,c>0 and a^4+b^4+c^4 = 3 . Prove the :
    sigma\frac{1}{4-ab} =< 1
    Do you mean this?
    If a, \, b, \, c > 0 and a^4 + b^4 + c^4=3, then \sum \frac{1}{4-ab} \le 1
    What kind of sum? There is only one term. Or do you mean
    \frac{1}{4-ab} + \frac{1}{4-bc} + \frac{1}{4-ac}\le 1?

    This inequality appears to be correct. It can probably be proved using Lagrange multipliers: Set  F(a,b,c) = a^4 + b^4 + c^4 and G(a,b,c) = \frac{1}{4-ab} + \frac{1}{4-bc} + \frac{1}{4-ac}. You want to find the maximum of G subject to F(a,b,c) = 3. The Lagrange equations are
      \frac{b}{(4-ab)^2} + \frac{c}{(4-ac)^2} = 4 \lambda a^3
      \frac{a}{(4-ab)^2} + \frac{c}{(4-bc)^2} = 4 \lambda b^3
      \frac{b}{(4-bc)^2} + \frac{a}{(4-ac)^2} = 4 \lambda c^3
    Good luck
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