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\frac{x + y}{x - y} + \frac{y + z}{y - z} + \frac{z + x}{z - x}
given that \( x > y > z \).
We are to minimize the expression
S = \frac{x + y}{x - y} + \frac{y + z}{y - z} + \frac{z + x}{z - x}
under the condition \( x > y > z > 0 \). Note that all denominators are positive since \( x > y > z \), so each fraction is positive and unbounded as any variable approaches another.
We analyze the structure of each term. Consider the identity:
\frac{x + y}{x - y} = \frac{(x - y) + 2y}{x - y} = 1 + \frac{2y}{x - y}
\frac{y + z}{y - z} = 1 + \frac{2z}{y - z}, \quad \frac{z + x}{z - x} = \frac{z + x}{-(x - z)} = -\frac{z + x}{x - z} = -\left( \frac{(x - z) + 2z}{x - z} \right) = -1 - \frac{2z}{x - z}
Adding the three:
S = \left(1 + \frac{2y}{x - y}\right) + \left(1 + \frac{2z}{y - z}\right) + \left(-1 - \frac{2z}{x - z}\right)