How to prove fraction inequality proof

How can I prove the following inequality?

$$\frac{a^3}{b^3} + \frac{b^3}{c^3} + \frac{c^3}{a^3} \geq \frac{a}{b} + \frac{b}{c} + \frac{c}{a}$$

I've tried the get common denominators but that doesn't really seem to help since although the denominators are the same, the numerators are hard to compare after doing so.

Solution 1:

Hints (assuming $\,a,b,c\,$ all have the same sign):

• $\displaystyle\frac{a^3}{b^3} + \frac{b^3}{c^3} + \frac{c^3}{a^3} \geq \frac{1}{9}\left(\frac{a}{b} + \frac{b}{c} + \frac{c}{a}\right)^3\,$ by the generalized means inequality;

• $\displaystyle \frac{a}{b} + \frac{b}{c} + \frac{c}{a} \ge 3\,$ by AM-GM.

Solution 2:

Hint: Assume $a, b, c > 0$, apply AM-GM inequality: $\dfrac{a^3}{b^3} + 1 + 1 \ge \dfrac{3a}{b}$.