Find the minimum value of an expression with three variables How can I find the minimum of the following expression:
$(\frac{xy}{z}+\frac{zx}{y}+\frac{yz}{x})(\frac{x}{yz}+\frac{y}{xz}+\frac{z}{xy})$?
($x, y, z$ are non-zero real numbers)
The expression can be simplified to $\frac{((xy)^2+(zx)^2+(yz)^2)(x^2+y^2+z^2)}{(xyz)^2}$, but I am not sure that it will help anyhow.

expanding gives
$$3+\frac{x^2}{y^2}+\frac{y^2}{x^2}+\frac{x^2}{z^2}+\frac{z^2}{x^2}+\frac{y^2}{z^2}+\frac{z^2}{y^2}$$ and now use that $$A+\frac{1}{A}\geq 2$$ for $$A>0$$
