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].

See also Correlation inequalities; Holley inequality.

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