Question 135
Prove that the inequality
$$ \frac{x}{\sqrt{x^2 + y + z}} + \frac{y}{\sqrt{x + y^2 + z}} + \frac{z}{\sqrt{x + y + z^2}} \le \sqrt{3} $$
holds for any nonnegative real numbers $x, y, z$ satisfying $x^2 + y^2 + z^2 = 3$.
$$ \frac{x}{\sqrt{x^2 + y + z}} + \frac{y}{\sqrt{x + y^2 + z}} + \frac{z}{\sqrt{x + y + z^2}} \le \sqrt{3} $$
holds for any nonnegative real numbers $x, y, z$ satisfying $x^2 + y^2 + z^2 = 3$.