Prove that $ \frac{x}{yz} + \frac{y}{xz} + \frac{z}{xy} \geq \frac{1}{x} + \frac{1}{y} + \frac{1}{z} $ Prove that $ \frac{x}{yz} + \frac{y}{xz} + \frac{z}{xy}  \geq  \frac{1}{x} + \frac{1}{y} + \frac{1}{z} $ for $x,y,z \in \Bbb{R}$ and $xyz > 0 $.
I know that i can use the axioms of the real numbers, but i can't finde an usefull equivalent transformation 

multiplying your given inequality by $$xyz>0$$ we get
$$x^2+y^2+z^2\geq xy+xz+zy$$
and this is equivalent to
$$(x-y)^2+(y-z)^2+(z-x)^2\geq  0$$ which is true.
