# Complex inequality having $3$ variables [duplicate]

If $$x,y,z>0.$$ Then prove that $$\displaystyle \frac{yz}{x}+\frac{zx}{y}+\frac{xy}{z}\geq x+y+z$$

what i try

put $$\displaystyle x=a/b,y=b/c,z=c/a$$ and $$xyz=1$$

Then $$\displaystyle \frac{b^2}{a^2}+\frac{c^2}{b^2}+\frac{a^2}{c^2}\geq \frac{(a+b+c)^2}{a^2+b^2+c^2}$$ (Titu,s lima)

How do i solve it Help me please

## marked as duplicate by Martin R, Community♦Mar 27 at 19:15

Hint : $$yz/x + xz/y \ge 2z$$
It's $$\sum_{cyc}(x^2y^2-x^2yz)\geq0$$ or $$\sum_{cyc}z^2(x-y)^2\geq0.$$