MathLabs

Problem 4

Let x,y,zx,y,z be positive numbers such that 1x+1y+1z=1\frac1x+\frac1y+\frac1z=1. Show that x+yz+y+zx+z+xy≥xyz+x+y+z\sqrt{x+yz}+\sqrt{y+zx}+\sqrt{z+xy}\ge\sqrt{xyz}+\sqrt{x}+\sqrt{y}+\sqrt{z}.
Step 5 of 8: Prove the pairwise estimate
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(x+yz)(y+zx)−xy(1+z)2=z(x−y)2≥0.(x+yz)(y+zx)-xy(1+z)^2=z(x-y)^2\ge0.
Detailed analysis

Expanding the difference gives the displayed square, so positivity yields the corresponding square-root inequality.