Question 93
Prove the inequality
$$
4 \left( \sqrt[3]{\frac{a}{b}} + \sqrt[3]{\frac{b}{c}} + \sqrt[3]{\frac{c}{a}} \right) \le 3 \left( 2 + a + b + c + \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right)^{2/3}
$$
for positive real numbers $a$, $b$ and $c$ with $abc = 1$.
$$
4 \left( \sqrt[3]{\frac{a}{b}} + \sqrt[3]{\frac{b}{c}} + \sqrt[3]{\frac{c}{a}} \right) \le 3 \left( 2 + a + b + c + \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right)^{2/3}
$$
for positive real numbers $a$, $b$ and $c$ with $abc = 1$.