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. 

Using Holder's inequality, we have that
\begin{align*}
a^5+b^5+c^5 &= a^{\frac{1}{3}}a^{\frac{14}{3}} + b^{\frac{1}{3}}b^{\frac{14}{3}}+c^{\frac{1}{3}}c^{\frac{14}{3}}\\
&\le (a+b+c)^{\frac{1}{3}}(a^7+b^7+c^7)^{\frac{2}{3}},
\end{align*}
and
\begin{align*}
a^3+b^3+c^3 &= a^{\frac{2}{3}}a^{\frac{7}{3}} + b^{\frac{2}{3}}a^{\frac{7}{3}}+c^{\frac{2}{3}}a^{\frac{7}{3}}\\
&\le (a+b+c)^{\frac{2}{3}}(a^7+b^7+c^7)^{\frac{1}{3}}.
\end{align*}
Multiplying them together, we are done.
