If-x-y-z-gt-0-and-xyz-1-Prove-that-2-x-2-1-xz-1-xy-2-y-2-1-yz-1-xy-2-z-2-1-xz-1-yz-3-2-

Question Number 205770 by hardmath last updated on 30/Mar/24

If x,y,z>0 and xyz = 1 Prove that: ((((√2)x)^2 )/((1+xz)(1+xy))) + ((((√2)y)^2 )/((1+yz)(1+xy))) + ((((√2)z)^2 )/((1+xz)(1+yz))) ≥ (3/2)

$$\mathrm{If}\:\:\mathrm{x},\mathrm{y},\mathrm{z}>\mathrm{0}\:\:\mathrm{and}\:\:\mathrm{xyz}\:=\:\mathrm{1} \\ $$$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\frac{\left(\sqrt{\mathrm{2}}\mathrm{x}\right)^{\mathrm{2}} }{\left(\mathrm{1}+\mathrm{xz}\right)\left(\mathrm{1}+\mathrm{xy}\right)}\:+\:\frac{\left(\sqrt{\mathrm{2}}\mathrm{y}\right)^{\mathrm{2}} }{\left(\mathrm{1}+\mathrm{yz}\right)\left(\mathrm{1}+\mathrm{xy}\right)}\:+\:\frac{\left(\sqrt{\mathrm{2}}\mathrm{z}\right)^{\mathrm{2}} }{\left(\mathrm{1}+\mathrm{xz}\right)\left(\mathrm{1}+\mathrm{yz}\right)}\:\geqslant\:\frac{\mathrm{3}}{\mathrm{2}} \\ $$

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If-x-y-z-gt-0-and-xyz-1-Prove-that-2-x-2-1-xz-1-xy-2-y-2-1-yz-1-xy-2-z-2-1-xz-1-yz-3-2-

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