Let $a,b,c$ be positive numbers such that $abc=1$. Prove that $\frac{a-1}{b}+\frac{b-1}{c}+\frac{c-1}{a} \geq 0$

Question

Question

Let $a,b,c$ be positive numbers such that $abc=1$. Prove that $$\frac{a-1}{b}+\frac{b-1}{c}+\frac{c-1}{a} \geq 0$$

My try

I have simplified this to the equivalent inequality $$ab^2+bc^2+ca^2 \geq ab+bc+ca$$

Now, I have tried AM-GM, but it does not work.

Any hint will be greatly appreciated.

Answer 1

By AM-GM : $ab^2+2bc^2\ge 3bc, ...$