TITLE: Is it true that $\left(\frac{a^2+b^2+c^2}{a+b+c}\right)^{a+b+c}≥a^ab^bc^c$?
QUESTION [5 upvotes]: Let $a,b,c\in\mathbb{R}_{>0}$. Is it true that:
$$
\left(\frac{a^2+b^2+c^2}{a+b+c}\right)^{a+b+c}≥a^ab^bc^c
$$
I remarked that the inequality is (a bit weirdly) homogeneous, but couldn't use it. Also directly taking the logarithm doesn't seem to help; how to decide wether it's true?

REPLY [4 votes]: Since $a,b,c>0$, 
\begin{align}
\frac{a \log a + b \log b + c \log c}{a+b+c} \le \log \left(\sum_{cyc}a\times \frac{a}{a+ b+ c}\right)
\end{align}
by Jensen's inequality on the $\log$. Taking exponent gives the required result.