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A fundamental tool in [[statistical mechanics]] and [[Combinatorics#Probabilistic_combinatorics|probabilistic combinatorics]] (especially [[random graph]]s and the [[probabilistic method]]), the '''Ahlswede–Daykin inequality''' {{harv|Ahlswede|Daykin|1978}}, also known as the '''four functions theorem''' (or '''inequality'''),
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is a [[correlation]]-type inequality for four functions on a finite [[distributive lattice]].
 
It states that if <math>f_1,f_2,f_3,f_4</math> are nonnegative functions on a finite distributive lattice such that
 
:<math>f_1(x)f_2(y)\le f_3(x\vee y)f_4(x\wedge y)</math>
 
for all ''x'', ''y'' in the lattice, then  
 
:<math>f_1(X)f_2(Y)\le f_3(X\vee Y)f_4(X\wedge Y)</math>
 
for all subsets ''X'', ''Y'' of the lattice, where
 
:<math>f(X) = \sum_{x\in X}f(x)</math>
 
and
 
:<math>X\vee Y = \{x\vee y\mid x\in X, y\in Y\}</math>
:<math>X\wedge Y = \{x\wedge y\mid x\in X, y\in Y\}.</math>
 
The Ahlswede–Daykin inequality can be used to provide a short proof of both the [[Holley inequality]] and the [[FKG inequality]]. It also implies the [[Fishburn–Shepp inequality]].
 
For a proof, see the original article {{harv|Ahlswede|Daykin|1978}} or {{harv|Alon|Spencer|2000}}.
 
==Generalizations==
 
The "four functions theorem" was independently generalized to 2''k'' functions in {{harv|Aharoni|Keich|1996}} and {{harv|Rinott|Saks|1991}}.
 
==References==
*{{Citation | last1=Ahlswede | first1=Rudolf | last2=Daykin | first2=David E. | title=An inequality for the weights of two families of sets, their unions and intersections | doi=10.1007/BF00536201 | mr=0491189 | year=1978 | journal=Probability Theory and Related Fields | issn=0178-8051 | volume=43 | issue=3 | pages=183–185}}
*{{Citation|last1=Alon|first1=N.|last2=Spencer|first2=J. H.|title=The probabilistic method. Second edition. With an appendix on the life and work of Paul Erdős.|publisher=Wiley-Interscience, New York|year=2000|isbn=0-471-37046-0 |mr=1885388}}
*{{springer|id=a/a110440|authorlink=Peter Fishburn|first=P.C. |last=Fishburn}}
*{{Citation | last1=Aharoni | first1=Ron | last2=Keich | first2=Uri | title= A Generalization of the Ahlswede Daykin Inequality | doi=10.1016/0012-365X(94)00294-S | year=1996| journal=Discrete Mathematics | volume=152 | pages=1–12}}
*{{Citation | last1=Rinott| first1=Yosef | last2=Saks | first2=Michael | title= Correlation inequalities and a conjecture for permanents | doi=10.1007/BF01202353 | year=1991| journal=COMBINATORICA | volume=13 | issue=3 | pages=269–277}}
 
{{DEFAULTSORT:Ahlswede-Daykin inequality}}
[[Category:Inequalities]]
[[Category:Statistical mechanics]]

Latest revision as of 21:08, 16 December 2014

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