Question 58
Positive real numbers $x$, $y$, $z$ satisfy $xyz + xy + yz + zx = x + y + z + 1$. Prove that
$$
\begin{aligned}&\frac{1}{3} \left( \sqrt{\frac{1+x^2}{1+x}} + \sqrt{\frac{1+y^2}{1+y}} + \sqrt{\frac{1+z^2}{1+z}} \right)\\
&\le \left( \frac{x+y+z}{3} \right)^{5/8}.\end{aligned}
$$
$$
\begin{aligned}&\frac{1}{3} \left( \sqrt{\frac{1+x^2}{1+x}} + \sqrt{\frac{1+y^2}{1+y}} + \sqrt{\frac{1+z^2}{1+z}} \right)\\
&\le \left( \frac{x+y+z}{3} \right)^{5/8}.\end{aligned}
$$