# Math Help - Proof Inequality

1. ## Proof Inequality

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?

2. I would start by reducing each side.

$a^3b^3+b^3c^3+c^3a^3 \geq a^4bc+b^4ac+c^4ab$

$abc(\frac{a^2b^2}{c}+\frac{b^2c^2}{a}+\frac{a^2c^2 }{b}) \geq abc(a^3+b^3+c^3)$

$\frac{a^2b^2}{c}+\frac{b^2c^2}{a}+\frac{a^2c^2}{b} \geq a^3+b^3+c^3$

Do you have any other further information?

i.e is a>b>c>0? or are a,b & c integers?

3. Originally Posted by AirRafer
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?
it's not always true. for example it doesn't hold for a = b = 1 and c = 2. my guess is that we need the condition $0 \leq a,b,c \leq 1.$

4. Oh,yes.it doesn't always true.
There isn't a good way to proof it even the 0<a,b,c<1