<s>[INST] Prove that $3(a^3+b^3+c^3) > (a+b+c)(a^2+b^2+c^2)$ Let $a,b,c$ be positive, not equal. Prove that $3(a^3+b^3+c^3) > (a+b+c)(a^2+b^2+c^2)$.
I know the proof by subtracting LHS by RHS and then doing some arrangement.
But isn't there any inequality which can be used in $(1+1+1)(a^3+b^3+c^3) >(a+b+c)(a^2+b^2+c^2)$, or an easy proof?
 [/INST] As is often the case, we can use Cauchy-Schwarz.   
Since the $L^3$ norm is larger then the $L^4$ norm, we have that $$\left(a^{3}+b^{3}+c^{3}\right)^{\frac{1}{3}}\geq\left(a^{4}+b^{4}+c^{4}\right)^{\frac{1}{4}}$$ with equality if and only if $a=b=c.$ Using Cauchy Schwarz, we have that $$\sqrt{3\left(a^{4}+b^{4}+c^{4}\right)}\geq\left(a^{2}+b^{2}+c^{2}\right)$$ and $$\sqrt{3\left(a^{2}+b^{2}+c^{2}\right)}\geq\left(a+b+c\right).$$  Multiplying these two inequalities, we get $$3\left(a^{4}+b^{4}+c^{4}\right)^{\frac{3}{4}}\geq\left(a+b+c\right)\left(a^{2}+b^{2}+c^{2}\right),$$ which implies the result. 
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