a,b,c > 0
a^3 b^3 + b^3 c^3 + c^3 a^3 ≥ a^4 bc + b^4 ac + c^4 ab
How to proof the inequality?
I would start by reducing each side.
$\displaystyle a^3b^3+b^3c^3+c^3a^3 \geq a^4bc+b^4ac+c^4ab$
$\displaystyle abc(\frac{a^2b^2}{c}+\frac{b^2c^2}{a}+\frac{a^2c^2 }{b}) \geq abc(a^3+b^3+c^3)$
$\displaystyle \frac{a^2b^2}{c}+\frac{b^2c^2}{a}+\frac{a^2c^2}{b} \geq a^3+b^3+c^3$
Do you have any other further information?
i.e is a>b>c>0? or are a,b & c integers?