<s>[INST] 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.
 [/INST] Hints (assuming $\,a,b,c\,$ all have the same sign):

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


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