<s>[INST] Prove that if $a$,$b$,$c$ are non-negative real numbers such that $a+b+c =3$, then $abc(a^2 + b^2 + c^2)\leq 3$ Prove that if $ a,b,c $ are non-negative real numbers such that $a+b+c = 3$, then $$ abc(a^2 + b^2 + c^2) \le 3 $$
My attempt : 
I tried AM-GM inequality, tried to convert it to $a+b+c$, but I think I cannot get the manipulation of $abc$.
 [/INST] Let:
$$ M_p = \left(\frac{a^p+b^p+c^p}{3}\right)^\frac{1}{p} $$
for $p>0$ and $M_0 = \sqrt[3]{abc}$. Since $M_p$ is a log-concave and increasing function,
$$ M_0^3 M_2^2 \leq M_{\frac{4}{5}}^5 \leq M_1^{5} = 1 $$
and the claim follows.
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