I was trying to prove an inequality with 3 variables, and after simplification, it boiled down to trying to prove

$$x^2 y+xz^2 +y^2 z \geq x+y+z$$, where $xyz=1$, and $x,y,z$ are all positive real numbers.

Once, I prove this I would be able to prove the overall inequality, however I am stuck at this part. The original question is supposed to be an olympiad style question, so i hope that i can prove this lemma without Lagrange Multipliers, etc.

Sincere thanks for any help!

(I have tried AM-GM, but that would lead to $x^2 y+xz^2 +y^2 z \geq 3\sqrt[3]{x^3y^3z^3}=3$ which is not sharp enough)

  • $\begingroup$ can you post the original problem please? $\endgroup$ May 26 '15 at 17:38

Hint: By AM-GM, $$x^2y+x^2y+xz^2\ge 3\left(x^5y^2z^2\right)^{\frac{1}{3}}=3x.$$

  • $\begingroup$ i think you got the question incorrectly, the 2nd term in the LHS is not what the OP has given $\endgroup$ Dec 17 '12 at 17:05
  • 1
    $\begingroup$ @dineshdileep: It is just a hint rather than a full answer. $\endgroup$
    – 23rd
    Dec 17 '12 at 17:08

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