Английская Википедия:Ahlswede–Daykin inequality

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Шаблон:Short description The Ahlswede–Daykin inequality Шаблон:Harv, also known as the four functions theorem (or inequality), is a correlation-type inequality for four functions on a finite distributive lattice. It is a fundamental tool in statistical mechanics and probabilistic combinatorics (especially random graphs and the probabilistic method).

The inequality 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 XYZ inequality.

For a proof, see the original article Шаблон:Harv or Шаблон:Harv.

Generalizations

The "four functions theorem" was independently generalized to 2k functions in Шаблон:Harv and Шаблон:Harv.

References

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