Max–min inequality

In mathematics, the max–min inequality is as follows:

For any function ${\displaystyle \ f:Z\times W\to \mathbb {R} \ ,}$

${\displaystyle \sup _{z\in Z}\inf _{w\in W}f(z,w)\leq \inf _{w\in W}\sup _{z\in Z}f(z,w)\ .}$

When equality holds one says that f, W, and Z satisfies a strong max–min property (or a saddle-point property). The example function ${\displaystyle \ f(z,w)=\sin(z+w)\ }$ illustrates that the equality does not hold for every function.

A theorem giving conditions on f, W, and Z which guarantee the saddle point property is called a minimax theorem.