Question 48
Let $\alpha$ be an arbitrary positive real number. Determine for this number $\alpha$ the greatest real number $C$ such that the inequality
$$
\left(1 + \frac{\alpha}{x^2}\right) \left(1 + \frac{\alpha}{y^2}\right) \left(1 + \frac{\alpha}{z^2}\right) \geq C \cdot \left(\frac{x}{z} + \frac{z}{x} + 2\right)
$$
is valid for all positive real numbers $x, y$ and $z$ satisfying $xy + yz + zx = \alpha$. When does equality occur?
$$
\left(1 + \frac{\alpha}{x^2}\right) \left(1 + \frac{\alpha}{y^2}\right) \left(1 + \frac{\alpha}{z^2}\right) \geq C \cdot \left(\frac{x}{z} + \frac{z}{x} + 2\right)
$$
is valid for all positive real numbers $x, y$ and $z$ satisfying $xy + yz + zx = \alpha$. When does equality occur?