<s>[INST] Prove that $\sqrt{x^{3} + y^{3}+1} + \sqrt{z^{3} + y^{3}+1} + \sqrt{x^{3} + z^{3}+1} \ge 2 + \sqrt{2(x^{3} + y^{3}+z^{3})+1}$ Given $x, y,z \ge 0$. Prove that 
$$\sqrt{x^{3} + y^{3}+1} + \sqrt{z^{3} + y^{3}+1}  + \sqrt{x^{3} + z^{3}+1} \ge 2 + \sqrt{2(x^{3} + y^{3}+z^{3})+1} $$

Attempt
Notice that $$(x^{3} + y^{3}+1) + (z^{3} + y^{3}+1)  + (x^{3} + z^{3}+1) = 2 + 2(x^{3} + y^{3}+z^{3})+1  $$
So the sum of squares must be between two forms as below:
$$2 + \sqrt{2(x^{3} + y^{3}+z^{3})+1} \le \sqrt{x^{3} + y^{3}+1} + \sqrt{z^{3} + y^{3}+1}  + \sqrt{x^{3} + z^{3}+1}  \le 2 + 2(x^{3} + y^{3}+z^{3})+1$$
If I work backwards I will get, by letting the form in each root be $a,b,c$ respectively:
$$\sqrt{a} + \sqrt{b}  + \sqrt{c} \ge 2 + \sqrt{a+b+c-2} $$
$$ a + b  +c + 2(\sqrt{ab} + \sqrt{bc}  + \sqrt{ca}) \ge 4 + (a+b+c-2) + 4\sqrt{a+b+c-2}$$
$$  (\sqrt{ab} + \sqrt{bc}  + \sqrt{ca}) \ge 1 + 2\sqrt{a+b+c-2}$$
$$  ((\sqrt{ab} + \sqrt{bc}  + \sqrt{ca}) -1))^{2} \ge  4(a+b+c-2)$$
$$ (ab + bc  + ca + 2a^{2}\sqrt{bc} +2b^{2}\sqrt{ac} +2c^{2}\sqrt{ab}) - 2(\sqrt{ab} + \sqrt{bc}  + \sqrt{ca}) + 1 \ge  4(a+b+c-2)$$
How to continue? 

Apart from the attempt above, I have tried: 


*

*AM-GM for the sum of the 3 squares

*AM-GM for the sum of the 3 squares, then using Holder's inequality from the GM

*AM-GM for each square
 [/INST] It should be 
$$ (ab + bc  + ca + 2a\sqrt{bc} +2b\sqrt{ac} +2c\sqrt{ab}) - 2(\sqrt{ab} + \sqrt{bc}  + \sqrt{ca}) + 1 \ge  4(a+b+c-2),$$
where $a\geq1$, $b\geq1$ and $c\geq1$, for which it's not so easy to find a proof.
By the way, the Karamata's inequality  helps.
Indeed, let $f(x)=\sqrt{x+1},$ $x^3+y^3=c$, $x^3+z^3=b$ and $y^3+z^3=a$.
Thus, $f$ is a concave function and we need to prove that
$$\sum_{cyc}\sqrt{a+1}\geq2+\sqrt{a+b+c+1},$$ which is just Karamata because if $a\geq b\geq c$ so
$$(a+b+c,0,0)\succ(a,b,c)$$ and
$$f(a+b+c)+f(0)+f(0)\leq f(a)+f(a)+f(b)$$ and we are done!
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