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\displaystyle 4x^{4}(x^{2}+y^{2}+z^{2}+3(yz+zx+xy))^{4}-27x^{4}(x+y+2z)^{4}(x+y)^2(yz+zx+xy)= (x^2+y^2-2z^2)^2(z^2+3z(x+y)+x^2+3xy+y^2)(z^2+9z(x+y)+4x^2+9xy+4y^2) +27(x+y)^2(x+y+2z)^2(z^2-xy)^2 \ge 0 \end{aligned}
\begin{aligned} \displaystyle \sum_{cyc} x\frac{x}{\sqrt{x+y}} \le \sqrt{\frac{3}{2}(x^{2}+y^{2}+z^{2})}
\begin{aligned} \displaystyle \sum_{cyc} x\frac{\sqrt{x}}{\sqrt{x+y}} \le \sqrt{\frac{(xy+yz+zx+x^{2}+y^{2}+z^{2})(x^{2}y+y^{2}x+x^{2}z+z^{2}x+z^{2}y+y^{2}z)}{(x+y)(y+z)(z+x)}}
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