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. 

If all of $a$, $b$ and $c$ are zero, then the inequality holds trivially, and so we can assume that at least one of $a$, $b$, and $c$ are positive.
The Cauchy-Schwarz inequality gives us that for any natural number $n$,
$$
 \left(a^n + b^n + c^n\right)\left(a^{n+2} + b^{n+2} + c^{n+2} \right)
 \geq \left(a^{n+1} + b^{n+1} + c^{n+1} \right)^2
$$
and so
$$
  \frac{a^n + b^n + c^n}{a^{n+1} + b^{n+1} + c^{n+1}} \geq \frac{a^{n+1} + b^{n+1} + c^{n+1}}{a^{n+2} + b^{n+2} + c^{n+2}}.
$$
We thus have that
$$
  \frac{a + b + c}{a^2 + b^2 + c^2} \geq \frac{a^2 + b^2 + c^2}{a^3 + b^3 + c^3} \geq \frac{a^3 + b^3 + c^3}{a^4 + b^4 + c^4} \geq \frac{a^4 + b^4 + c^4}{a^5 + b^5 + c^5} \geq \frac{a^5 + b^5 + c^5}{a^6 + b^6 + c^6},
$$
and in a similar way, it follows that
$$
  \frac{a^2 + b^2 + c^2}{a^3 + b^3 + c^3} \geq \frac{a^6 + b^6 + c^6}{a^7 + b^7 + c^7}.
$$
We thus have that
$$
  \frac{a + b + c}{a^3 + b^3 + c^3} = \frac{a + b + c}{a^2 + b^2 + c^2} \cdot \frac{a^2 + b^2 + c^2}{a^3 + b^3 + c^3} \geq \frac{a^5 + b^5 + c^5}{a^6 + b^6 + c^6} \cdot \frac{a^6 + b^6 + c^6}{a^7 + b^7 + c^7} = \frac{a^5 + b^5 + c^5}{a^7 + b^7 + c^7},
$$
which implies that
$$
  \left(a + b + c\right) \left(a^7 + b^7 + c^7 \right) \geq \left(a^5 + b^5 + c^5\right) \left(a^3 + b^3 + c^3 \right),
$$
as required.
