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Question Number 34031 by naka3546 last updated on 29/Apr/18

Prove  that  for  every  positive  real  numbers  x, y, z  and   xyz  =  1,  hold  (x + y + z)^2 ((1/x^2 ) + (1/y^2 ) + (1/z^2 ))  ≥  9 + 2(x^3  + y^3  + z^3 ) + 4((1/x^3 ) + (1/y^3 ) + (1/z^3 ))

$${Prove}\:\:{that}\:\:{for}\:\:{every}\:\:{positive}\:\:{real}\:\:{numbers}\:\:{x},\:{y},\:{z}\:\:{and}\:\:\:{xyz}\:\:=\:\:\mathrm{1},\:\:{hold} \\ $$$$\left({x}\:+\:{y}\:+\:{z}\right)^{\mathrm{2}} \left(\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\:+\:\frac{\mathrm{1}}{{y}^{\mathrm{2}} }\:+\:\frac{\mathrm{1}}{{z}^{\mathrm{2}} }\right)\:\:\geqslant\:\:\mathrm{9}\:+\:\mathrm{2}\left({x}^{\mathrm{3}} \:+\:{y}^{\mathrm{3}} \:+\:{z}^{\mathrm{3}} \right)\:+\:\mathrm{4}\left(\frac{\mathrm{1}}{{x}^{\mathrm{3}} }\:+\:\frac{\mathrm{1}}{{y}^{\mathrm{3}} }\:+\:\frac{\mathrm{1}}{{z}^{\mathrm{3}} }\right)\: \\ $$

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