Inequality: Let x,yz∈(0,∞) be defined so that 3xyz=1...

[gavrist]

Hello guys! It's my first time posting on this forum, and obvioulsy I'm looking for help with an exercice. 3xyz=1, proove the inequality.

I have used Titu's Andreescu's inequality, and now i am blocked in this point:

Thanks!


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[stapel]

Do I understand correctly the exercise to be as follows?

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\(\displaystyle \mbox{Let }\, x,\, y\, z\, \in\, (0, \infty)\, \mbox{ be defined so that }\, 3xyz\, =\, 1.\)

\(\displaystyle \mbox{Then prove the following inequality:}\)

. . .\(\displaystyle \dfrac{3xy^3}{3x^4\, +\, y\, +\, z}\, +\, \dfrac{3yz^3}{3y^4\, +\, z\, +\, x}\, +\, \dfrac{3zx^3}{3z^4\, +\, x\, +\, y}\, \ge\, 1\)

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By "Titu's Andreescu's inequality", do you mean the Cauchy-Schwarz inequality? Or his Lemma, which he used in a proof of the Cauchy-Schwarz inequality? (here) Or another of his Lemma's, which follows from the Cauchy-Schwarz inequality? (here)

Please reply with the specific result that you're using, along with a clear listing of the steps by which you arrived that the point where you're getting stuck. Thank you!

 

[gavrist]

Sorry for not being understood. Above is the Inequality that i've used.

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