The AM-GM Inequality

The Arithmetic-Geometric Mean Inequality is simple, but quite useful for majorization. For completeness, we give the statement and proof here.

Theorem: If $x\geq 0$ and $y\geq 0$ then $\sqrt{xy}\leq\frac{1}{2}(x+y),$ with equality if and only if $x=y.$

Proof: Expand $(\sqrt{x}-\sqrt{y})^2\geq 0,$ and collect terms. QED

Corollary: $\mid xy\mid\leq\frac{1}{2}(x^2+y^2)$

Proof: Just a simple rewrite of the theorem. QED