Ahlswede-Daykin inequality

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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 be a finite distributive lattice (see also FKG inequality), such as the power set of a finite set ordered by proper inclusion. For subsets , of , define and . If or is empty, . Given , let .

The Ahlswede–Daykin inequality says that if , , , and map into such that


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.


[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
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