Ahlswede–Daykin inequality
A fundamental tool in statistical mechanics and probabilistic combinatorics, the Ahlswede–Daykin inequality , also known as the four functions theorem,
is a correlation-type inequality for four functions on a finite distributive lattice.
It states that if are nonnegative functions on a finite distributive lattice such that
for all x, y in the lattice, then
for all subsets X, Y of the lattice, where
and
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 Fishburn–Shepp inequality.
For a proof, see the original article or.Generalizations
The "four functions theorem" was independently generalized to 2k functions in and.
