$(y+\sqrt{zx}+z)^2 \leq (y+2z)(x+y+z)$
It remains to prove that,
${\sum_{cyc} \frac{2x^2+xy}{y+2z}\geq \sum_{cyc} x}$
$\leftrightarrow \sum_{cyc} \frac{2x^2+2xy+2xz}{y+2z} \geq \sum_{cyc} x$
$\leftrightarrow \sum_{cyc} \frac{x}{y+2z}\geq 1$
It is true by Cauchy
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