Question 118
Show that
$$
\frac{27}{4}(x+y)(y+z)(z+x) \geq (\sqrt{x+y}+\sqrt{y+z}+\sqrt{z+x})^2 \geq 6\sqrt{3}
$$
for any positive real numbers $x$, $y$, $z$ with $xy + yz + zx = 1$.
$$
\frac{27}{4}(x+y)(y+z)(z+x) \geq (\sqrt{x+y}+\sqrt{y+z}+\sqrt{z+x})^2 \geq 6\sqrt{3}
$$
for any positive real numbers $x$, $y$, $z$ with $xy + yz + zx = 1$.