S = 1 + 2\left( \frac{1 + b}{a} + \frac{1}{b} - \frac{1}{a + b} \right)

February 22, 2026

$S = 1 + 2\left( \frac{1 + b}{a} + \frac{1}{b} - \frac{1}{a + b} \right)$

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= 1 + 2z\left( \frac{1}{a} + \frac{1}{b} - \frac{1}{a + b} \right) + 2\frac{b}{a}
This still depends on \( z \). But since the original expression is homogeneous (scaling \( x, y, z \) by a constant does not change the value), we can set \( z = 1 \) without loss of generality.
Let \( z = 1 \), \( y = 1 + b \), \( x = 1 + b + a \), with \( a, b > 0 \). Then:
Now define:
f(a, b) = \frac{1 + b}{a} + \frac{1}{b} - \frac{1}{a + b}
We seek to minimize \( S = 1 + 2f(a, b) \), so minimize \( f(a, b) \).

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