S \geq 1.

February 22, 2026

$S \geq 1.$

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S = \frac{x^2}{y} + \frac{y^2}{z} + \frac{z^2}{x},
given that $ x + y + z = 1 $ and $ x, y, z > 0 $, we apply the **Cauchy-Schwarz Inequality** in the following form:
Since $ x + y + z = 1 $, we substitute into the inequality:
To check if this bound is achievable, consider the equality condition in Cauchy-Schwarz: equality occurs when
Let this common value be $ k $. Then,
\frac{x^2}{y} = \frac{y^2}{z} \Rightarrow x^2 z = y^3, \quad \frac{y^2}{z} = \frac{z^2}{x} \Rightarrow y^2 x = z^3.

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