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Math Help - Proof Inequality

  1. #1
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    Proof Inequality

    a,b,c > 0
    a^3 b^3 + b^3 c^3 + c^3 a^3 ≥ a^4 bc + b^4 ac + c^4 ab

    How to proof the inequality?
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  2. #2
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    I would start by reducing each side.

    a^3b^3+b^3c^3+c^3a^3 \geq a^4bc+b^4ac+c^4ab

    abc(\frac{a^2b^2}{c}+\frac{b^2c^2}{a}+\frac{a^2c^2  }{b}) \geq abc(a^3+b^3+c^3)

    \frac{a^2b^2}{c}+\frac{b^2c^2}{a}+\frac{a^2c^2}{b} \geq a^3+b^3+c^3

    Do you have any other further information?

    i.e is a>b>c>0? or are a,b & c integers?
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  3. #3
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    Quote Originally Posted by AirRafer View Post
    a,b,c > 0
    a^3 b^3 + b^3 c^3 + c^3 a^3 ≥ a^4 bc + b^4 ac + c^4 ab

    How to proof the inequality?
    it's not always true. for example it doesn't hold for a = b = 1 and c = 2. my guess is that we need the condition 0 \leq a,b,c \leq 1.
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  4. #4
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    Red face

    Oh,yes.it doesn't always true.
    There isn't a good way to proof it even the 0<a,b,c<1
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