Summary
It is shown that if (X, ℱ μ) is a product of totally ordered measure spaces andf j (j=1,2,3,4) are measurable non-negative functions onX satisfyingf 1(x)f2(y)≦f3(x∨y)f4(x∧y), where (∨, ∧) are the lattice operations onX, then (∫f 1 dμ)(∫f 2 dμ)≦(∫f 3 dμ)(∫f 4 dμ). This generalises results of Ahlswede and Daykin (for counting measure on finite sets) and Preston (for special choices off j).
Article PDF
Similar content being viewed by others
References
Ahlswede, R., Daykin, D.E.: An inequality for the weights of two families of sets, their unions and intersections, Z. Wahrscheinlichkeitstheorie verw. Gebiete43, 183–185 (1978)
Ahlswede, R., Daykin, D.E.: Inequalities for a pair of mapsS×S→S withS a finite set. Math. Z.165, 267–289 (1979)
Batty, C.J.K.: An extension of an inequality of R. Holley. Quart. J. Math. (Oxford) (2)27, 457–461 (1976)
Birkhoff, G.: Lattice theory. 3rd ed. Providence: Amer. Math. Soc. 1967
Daykin, D.E.: A lattice is distributive iff ¦A¦¦B¦≦¦A∨B¦¦A∧B¦. Nanta Math.10, no. 1, 58–60 (1977)
Daykin, D.E.: Inequalities among the subsets of a finite set. [To appear in Nanta Math.]
Dunford, N., Schwartz, J.T.: Linear operators I. New York: Interscience 1958
Edwards, D.A.: On the Holley-Preston inequalities. Proc. Royal Soc. Edinburgh78 A, 265–272 (1978)
Fortuin, C.M., Kastelyn, P.W., Ginibre, J.: Correlation inequalities on some partially ordered sets. Commun. Math. Phys.22, 89–103 (1971)
Holley, R.: Remarks on the FKG inequalities. Commun. Math. Phys.36, 227–231 (1974)
Kemperman, J.H.B.: On the FKG-inequality for measures on a partially ordered space. Proc. Nederl. Akad. Wet. Ser. A80, 313–331 (1977)
Kleitman, D.J.: Families of non-disjoint subsets. J. Combinatorial Theory1, 153–155 (1966)
Preston, C.J.: A generalisation of the FKG inequalities. Commun. Math. Phys.36, 233–241 (1974)
Seymour, P.D., Welsh, D.J.A.: Combinatorial applications of an inequality from statistical mechanics. Math. Proc. Cambridge Philos. Soc.77, 485–495 (1975)
Welsh, D.J.A.: In Problèmes combinatoires et théorie des graphes (Paris 1976). Colloques internationaux du C.N.R.S. no. 260, C.N.R.S., Paris, 1978
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Batty, C.J.K., Bollmann, H.W. Generalised Holley-Preston inequalities on measure spaces and their products. Z. Wahrscheinlichkeitstheorie verw Gebiete 53, 157–173 (1980). https://doi.org/10.1007/BF01013313
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01013313