Solution of an Inequality Show that $(ab + bc + ca)(ab^{-1}+ bc^{-1} + ca^{-1}) \geq (a + b + c)^{2}.$
We, at first, change the given expression $(ab + bc + ca)(ab^{-1} + bc^{-1} +ca^{-1})$ into $(ab^2 + bc^2 + ca^2 )(1/a + 1/b + 1/c)$ . Now we will show that this term $\geq (a + b+ c)^{2}.$ Again by Chebyshev's inequality we showed $(1/a + 1/b + 1/c) \geq \frac{9}{a + b + c}.$ But after that we face some problem. Please help or give some hint so that we can solve this problem.

It follows from the Cauchy-Schwarz inequality:
$$(ab + bc + ca)(ab^{-1} + bc^{-1} + ca^{-1}) \ge \left(\sqrt{ab}\cdot \sqrt{ab^{-1}} + \sqrt{bc}\cdot \sqrt{bc^{-1}} + \sqrt{ca}\cdot \sqrt{ca^{-1}}\right)^2 = \cdots$$
Now finish the rest.
