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 :

\frac{{(\sqrt{2} x)}^{2}}{(1 + xz) (1 + xy)} + \frac{{(\sqrt{2} y)}^{2}}{(1 + yz) (1 + xy)} + \frac{{(\sqrt{2} z)}^{2}}{(1 + xz) (1 + yz)} ⩾ \frac{3}{2}