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-

Tinku Tara June 4, 2023 None 0 Comments

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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