<s>[INST] How prove this ineuality$\sqrt{x+1}+\sqrt{y+1}+\sqrt{z+1}\le\sqrt{xy+yz+zx+9}$ let $$x,y,z\in(-1,1), x+y+z=-xyz$$
show that
$$\sqrt{x+1}+\sqrt{y+1}+\sqrt{z+1}\le\sqrt{xy+yz+zx+9}$$
This problem is my frends ask me,I remenber this is old inequality,But Now I can't it
$$x+y+z+3+2\sum_{cyc}\sqrt{xy+x+y+1}\le xy+yz+xz+9$$
 [/INST] Do as you said and we only have to prove
$$2\sum_{cyc}\sqrt{xy+x+y+1}\le xy+yz+xz+6-x-y-z$$
This is right, since $\displaystyle x+y+z=-xyz$ and $x,y,z,xy,yz,xz\in(-1;1)$, we have:
$2\sqrt{xy+x+y+1}
=2\sqrt{xy+x+y+1-x-y-z-xyz}
=2\sqrt{xy-xyz+1-z}
=2\sqrt{(1-z)(1+xy)}
\le 1-z+1+xy$
Do this with the other two and sum up, we have the inequality.
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