Prove inequality $ (a+b+c)(a^7+b^7+c^7)\ge(a^5+b^5+c^5)(a^3+b^3+c^3)$ for $a,b,c\ge 0$ Prove that:
$$ (a+b+c)(a^7+b^7+c^7)\ge(a^5+b^5+c^5)(a^3+b^3+c^3)$$
I already know that this can be proven using Cauchy Schwarz, but I don't really see how to apply it here. I'm looking for hints. 

I do not see a direct application of Cauchy-Schwarz. Anyway after the multiplication, we obtain
$$\sum_{sym}a^7b\geq \sum_{sym}a^5b^3$$
which holds by Muirhead's inequality.
