Question 47
Let $x, y, z \gt 0$ such that $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} \lt \frac{1}{xyz}$. Show that
$$
\frac{2x}{\sqrt{1+x^2}} + \frac{2y}{\sqrt{1+y^2}} + \frac{2z}{\sqrt{1+z^2}} \lt 3.
$$
$$
\frac{2x}{\sqrt{1+x^2}} + \frac{2y}{\sqrt{1+y^2}} + \frac{2z}{\sqrt{1+z^2}} \lt 3.
$$