Is it true that $\left(\frac{a^2+b^2+c^2}{a+b+c}\right)^{a+b+c}≥a^ab^bc^c$?

Question

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?

Answer 1

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.