Why is $\left(\frac{1}{y} + \frac{1}{z}\right)\left(\frac{1}{z} + \frac{1}{x}\right)\left(\frac{1}{x} + \frac{1}{y}\right) \geq \frac{8}{xyz}$?

Question

The solution of a question in my book uses the property that $$\left(\frac{1}{y} + \frac{1}{z}\right)\left(\frac{1}{z} + \frac{1}{x}\right)\left(\frac{1}{x} + \frac{1}{y}\right) \geq \frac{8}{xyz}$$ Where did they get it from? Is this using $AM >= GM >= HM$? In that case of what is the arithmetic/geometric/harmonic mean? Or is this some random mathematical property?

Answer 1

Hint: if your variables assumed to be positive, you can use that $$\frac{1}{x}+\frac{1}{y}\geq \frac{2}{\sqrt{xy}}$$ for each factor.

Answer 2

It comes indeed from the AM-GM inequality. If we assume that $x, y, z > 0$, then $$ \frac{1}{y} + \frac{1}{z} \geq \frac{2}{\sqrt{yz}} $$ and similarly for the other pairs. Multiplying these three equations gives the desired inequality.