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Math Help - inequalities

  1. #1
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    inequalities

    Prove that

    a^4 + b^4 + c^4 > (a^2 * bc) + (b^2 * ac) + (c^2 * ab)

    holds for all real numbers a, b, c.
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  2. #2
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    Quote Originally Posted by the undertaker View Post
    Prove that

    a^4 + b^4 + c^4 > a^2  bc + b^2  ac + c^2  ab

    holds for all real numbers a, b, c.
    First note that 0\leqslant (b^2-c^2)^2 + (c^2-a^2)^2 + (a^2-b^2)^2 = 2(a^4 + b^4 + c^4) - 2(b^2c^2+c^2a^2+a^2b^2), from which b^2c^2+c^2a^2+a^2b^2 \leqslant a^4 + b^4 + c^4.

    Then by the Cauchy–Schwarz inequality a^2  bc + b^2  ac + c^2  ab \leqslant (a^4 + b^4 + c^4)^{1/2}(b^2c^2+c^2a^2+a^2b^2)^{1/2} \leqslant a^4 + b^4 + c^4.
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