Question 74
Let $k$ be a real number with $k\gt 2$.
a) Show that for all positive numbers $x, y$ and $z$ the following inequality holds:
$$
\sqrt{x+y}+\sqrt{y+z}+\sqrt{z+x}\gt 2 \sqrt{\frac{(x+y)(y+z)(z+x)}{x y+y z+z x}}
$$
b) Prove that there exist positive numbers $x, y$ and $z$ for which
$$
\sqrt{x+y}+\sqrt{y+z}+\sqrt{z+x}\lt k \sqrt{\frac{(x+y)(y+z)(z+x)}{x y+y z+z x}}
$$
a) Show that for all positive numbers $x, y$ and $z$ the following inequality holds:
$$
\sqrt{x+y}+\sqrt{y+z}+\sqrt{z+x}\gt 2 \sqrt{\frac{(x+y)(y+z)(z+x)}{x y+y z+z x}}
$$
b) Prove that there exist positive numbers $x, y$ and $z$ for which
$$
\sqrt{x+y}+\sqrt{y+z}+\sqrt{z+x}\lt k \sqrt{\frac{(x+y)(y+z)(z+x)}{x y+y z+z x}}
$$