Question 43
Let $x$, $y$ and $z$ be any positive real numbers. Prove the inequality
$$
(x + y + z) \left( \frac{1}{x} + \frac{1}{y} + \frac{1}{z} \right) \le m^2, \quad \text{where } m = \min \left( \frac{x}{y} + \frac{y}{z} + \frac{z}{x}, \frac{y}{x} + \frac{z}{y} + \frac{x}{z} \right).
$$
Find when the equality holds.
$$
(x + y + z) \left( \frac{1}{x} + \frac{1}{y} + \frac{1}{z} \right) \le m^2, \quad \text{where } m = \min \left( \frac{x}{y} + \frac{y}{z} + \frac{z}{x}, \frac{y}{x} + \frac{z}{y} + \frac{x}{z} \right).
$$
Find when the equality holds.