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Math Help - proove number inequality

  1. #1
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    proove number inequality

    a,b,c,d\in\mathbb{R^{+}}\;\;,a+b+c+d=1.

    Then prove that

    \left( a+\dfrac{1}{b}\right).\left(b+\dfrac{1}{c}\right).  \left(c+\dfrac{1}{a}\right)\geq \left(\dfrac{10}{3}\right)^3

    Anyone an idea on how to start with this exercise?
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  2. #2
    MHF Contributor alexmahone's Avatar
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    Re: proove number inequality

    Quote Originally Posted by wnvl View Post
    a,b,c,d\in\mathbb{R^{+}}\;\;,a+b+c+d=1.

    Then prove that

    \left( a+\dfrac{1}{b}\right).\left(b+\dfrac{1}{c}\right).  \left(c+\dfrac{1}{a}\right)\geq \left(\dfrac{10}{3}\right)^3

    Anyone an idea on how to start with this exercise?
    What's the point of defining d if it doesn't appear in the inequality?
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  3. #3
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    Re: proove number inequality

    Quote Originally Posted by alexmahone View Post
    What's the point of defining d if it doesn't appear in the inequality?
    a,b,c,d\in\mathbb{R^{+}}\;\;,a+b+c+d=1

    This is a way of stating that 0<a+b+c<1
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