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Square root inequality from Gohberg’s book

This is a problem from the book « Abstract Inequalities » of Gohberg

Let $x,y,z$ be nonnegative. Prove that:
$$x^{\frac12}+y^{\frac12}+z^{\frac12}\le\left(xy+yz+xz\right)^{\frac13}+\left(xy+yz+xz\right)^{\frac13}+\left(xy+yz+xz\right)^{\frac13}$$

If $x^{\frac12},y^{\frac12},z^{\frac12}$ are positives, the solution
is immediate. I need to prove it for the remaining cases.
$(x,y,z)\in\{(0,0,0),(0,1,1),(1,0,1),(1,1,0)\}$ can be proved using AM-GM inequality.
Now, for $(x,y,z)\in\{(0,0,0),(1,0,1)\}$, there exist $a,b,c>0$ such that $x=a^2, y=b^2$ and $z=c^2$. Then
\$\begin{aligned}x+y+z&=3a^2+b^2+c^2\\&\le(3a^2b^2+3b^2c^2+3a^2c^2)^{\frac13}+3(a^2b^2+b^2c^2+a^2c^2)^{\frac13}+3(a^2b^2+b^2c^2+a^2c^2)^{\frac13}\\&=(3a^2b+3b^2c+3a^2c)^{\frac13}+(a^2b+b^2c)^{\frac13}+(a^2b+b^2c)^{\frac13}\\&\le(a^2b+b^2c+a^2c)^{\frac13

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