# Ahlswede-Daykin inequality

four-functions inequality

An inequality in which an inequality for four functions on a finite distributive lattice applies also to additive extensions of the functions on lattice subsets. Let $( \Gamma, \prec )$ be a finite distributive lattice (see also FKG inequality), such as the power set of a finite set ordered by proper inclusion. For subsets $A$, $B$ of $\Gamma$, define $A \lor B = \{ {a \lor b } : {a \in A, b \in B } \}$ and $A \wedge B = \{ {a \wedge b } : {a \in A, b \in B } \}$. If $A$ or $B$ is empty, $A \lor B = A \wedge B = \emptyset$. Given $f : \Gamma \rightarrow \mathbf R$, let $f ( A ) = \sum _ {a \in A } f ( a )$.

The Ahlswede–Daykin inequality says that if $f _ {1}$, $f _ {2}$, $f _ {3}$, and $f _ {4}$ map $\Gamma$ into $[ 0, \infty )$ such that

$$f _ {1} ( a ) f _ {2} ( b ) \leq f _ {3} ( a \lor b ) f _ {4} ( a \wedge b ) \textrm{ for all } a, b \in \Gamma,$$

then

$$f _ {1} ( A ) f _ {2} ( B ) \leq f _ {3} ( A \lor B ) f _ {4} ( A \wedge B ) \textrm{ for all } A, B \subseteq \Gamma .$$

See [a1] or [a2], [a4], [a7] for a proof.

The inequality is very basic and is used in proofs of other inequalities (cf. [a2], [a3], [a4], [a5], [a7]), including the FKG inequality [a6] and the Fishburn–Shepp inequality [a3], [a8].

How to Cite This Entry:
Ahlswede-Daykin inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ahlswede-Daykin_inequality&oldid=45140
This article was adapted from an original article by P.C. Fishburn (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article