How can I prove that $4(x^3 + y^3 + z^3 +3) \geq 3(x+1)(y+1)(z+1)$ for $x, y, z > 0$ and $x, y, z \in \mathbb{R}$?

Question

I have to find proof for $4(x^3 + y^3 + z^3 +3) \geq 3(x+1)(y+1)(z+1)$ if $x, y, z > 0$ and $x, y, z \in \mathbb{R}$. I tried it using the generalized mean inequality, but couldn't find proof that way.

Answer 1

use Holder inequality and AM-GM inequality $$LHS=\sum_{cyc}(1^3+1^3)(1^3+1^3)(x^3+1^3)\ge \sum_{cyc}(x+1)^3\ge 3(x+1)(y+1)(z+1)$$

Answer 2

$$\sum_{cyc} (x^3+1) \geq 3 \prod_{cyc}(x^3+1)^{1/3} \geq \frac{3}{4} \prod_{cyc} (x+1),$$

where the second step uses Hölder's inequality -- $(x^3+1^3)^{1/3} (1^{3/2}+1^{3/2})^{2/3} \geq (x+1)$.