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.
 A: Remember that for any 3 positive numbers we have $${a+b+c\over 3} \geq \sqrt[3]{abc}$$
this is inequality between arithmetic mean and geometic mean.
Use it twice. First $a= (xy)^2$, $b=....$
and second $a= x^2$,... and you get:
$$(\frac{xy}{z}+\frac{zx}{y}+\frac{yz}{x})(\frac{x}{yz}+\frac{y}{xz}+\frac{z}{xy})\geq 9$$
A: By Cauchy-Schwarz inequality https://math.stackexchange.com/tags/cauchy-schwarz-inequality/info
$$\left(\frac{xy}{z}+\frac{zx}{y}+\frac{yz}{x}\right)\left(\frac{x}{yz}+\frac{y}{xz}+\frac{z}{xy}\right)=\sum_{cyc}\frac{xy}{z}\sum_{cyc}\frac{z}{xy}\geq\left(\sum_{cyc}\sqrt{\frac{xy}{z}\cdot\frac{z}{xy}}\right)^2=9.$$
The equality occurs for $x=y=z$, which says that we gat a minimal value.
Also, we can use AM-GM:
$$\sum_{cyc}\frac{xy}{z}\sum_{cyc}\frac{z}{xy}\geq3\sqrt[3]{\prod_{cyc}\frac{xy}{z}}\cdot3\sqrt[3]{\prod_{cyc}\frac{z}{xy}}=9.$$
A: 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 $$A>0$$
