# 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].

#### References

 [a1] R. Ahlswede, D.E. Daykin, "An inequality for the weights of two families, their unions and intersections" Z. Wahrscheinlichkeitsth. verw. Gebiete , 43 (1978) pp. 183–185 [a2] B. Bollobás, "Combinatorics" , Cambridge Univ. Press (1986) [a3] P.C. Fishburn, "A correlational inequality for linear extensions of a poset" Order , 1 (1984) pp. 127–137 [a4] P.C. Fishburn, "Correlation in partially ordered sets" Discrete Appl. Math. , 39 (1992) pp. 173–191 [a5] P.C. Fishburn, P.G. Doyle, L.A. Shepp, "The match set of a random permutation has the FKG property" Ann. of Probab. , 16 (1988) pp. 1194–1214 [a6] C.M. Fortuin, P.N. Kasteleyn, J. Ginibre, "Correlation inequalities for some partially ordered sets" Comm. Math. Phys. , 22 (1971) pp. 89–103 [a7] R.L. Graham, "Applications of the FKG inequality and its relatives" , Proc. 12th Internat. Symp. Math. Programming , Springer (1983) pp. 115–131 [a8] L.A. Shepp, "The XYZ conjecture and the FKG inequality" Ann. of Probab. , 10 (1982) pp. 824–827
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