Confusions in Holder's Inequality Holder's Inequality states that for nonnegative real numbers $a_1,...,a_n$ and $b_1,...,b_n$ we have $$\left(\sum_{i=1}^na_i\right)^p\left(\sum_{i=1}^nb_i\right)^q\ge \left(\sum_{i=1}\sqrt[p+q]{a_i^pb_i^q}\right)^{p+q}$$
Where $p$ and $q$ are positive real numbers.
Here is my problem : $a,b,c$ are positive reals, prove that $$(a^5-a^2+3)(b^5-b^2+3)(c^5-c^2+3)\ge (a+b+c)^3$$
But for $x>0$ we have $x^5-x^2+3\ge x^3+2$ so we only need to prove $$(a^3+2)(b^3+2)(c^3+2)\ge (a+b+c)^3$$
But when I read the solution, it says ''From Holder's Inequality, it follows ''  $$(a^3+2)(b^3+2)(c^3+2)\ge (a+b+c)^3$$
But I don't see that.

I would think Holder's inequality implies
$$(a^3+2)(b^3+2)(c^3+2)\geq \left((a^3b^3c^3)^{1/3}+ (2^3)^{1/3} \right)^3=(abc+2)^3$$
So it would be good to know how your last inequality follows from Holder's.
