Ahlswede–Daykin inequality

The Ahlswede–Daykin inequality, also known as the four functions theorem (or inequality), is a correlation-type inequality for four functions on a finite distributive lattice. It is a fundamental tool in statistical mechanics and probabilistic combinatorics (especially random graphs and the probabilistic method).

The inequality states that if $$f_1,f_2,f_3,f_4$$ are nonnegative functions on a finite distributive lattice such that


 * $$f_1(x)f_2(y)\le f_3(x\vee y)f_4(x\wedge y)$$

for all x, y in the lattice, then


 * $$f_1(X)f_2(Y)\le f_3(X\vee Y)f_4(X\wedge Y)$$

for all subsets X, Y of the lattice, where


 * $$f(X) = \sum_{x\in X}f(x)$$

and


 * $$X\vee Y = \{x\vee y\mid x\in X, y\in Y\}$$
 * $$X\wedge Y = \{x\wedge y\mid x\in X, y\in Y\}.$$

The Ahlswede–Daykin inequality can be used to provide a short proof of both the Holley inequality and the FKG inequality. It also implies the XYZ inequality.

For a proof, see the original article or.

Generalizations
The "four functions theorem" was independently generalized to 2k functions in and.