A ratio-sum inequality

AM-GM.
$$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\ge3\sqrt[3]{\frac{a}{b}\cdot\frac{b}{c}\cdot\frac{c}{a}}=3\sqrt[3]{1}=3.$$

Equality case. $\frac ab=\frac bc=\frac ca$, which implies $a=b=c$.

Conclusion. The inequality holds, with equality iff $a=b=c$.