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Math Help - need help to prove inequalities

  1. #1
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    need help to prove inequalities

    Hello..

    I need help to prove this inequalities
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  2. #2
    Super Member Aryth's Avatar
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    Re: need help to prove inequalities

    I don't know how good you are at inequalities but it basically goes like this:

    \frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} = \frac{a^2cd}{abcd} + \frac{b^2ad}{abcd} + \frac{c^2ab}{abcd} + \frac{d^2bc}{abcd}


    = \frac{a^2cd + b^2ad + c^2ab + d^2bc}{abcd} \leq \frac{a^2 + b^2 + c^2 + d^2}{abcd} \leq \frac{a + b + c + d}{abcd} \leq \frac{4}{abcd}

    So, \frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} \leq \frac{4}{abcd}
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  3. #3
    MHF Contributor alexmahone's Avatar
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    Re: need help to prove inequalities

    Quote Originally Posted by Aryth View Post
    = \frac{a^2cd + b^2ad + c^2ab + d^2bc}{abcd} \leq \frac{a^2 + b^2 + c^2 + d^2}{abcd} \leq \frac{a + b + c + d}{abcd}
    Care to explain the first inequality?

    The second inequality is false. Take a = 3, b = 0.3, c = 0.3, d = 0.4 for a counterexample.
    Last edited by alexmahone; September 4th 2011 at 01:07 PM.
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