Namespaces
Variants
Actions

Difference between revisions of "Holley inequality"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Tex done)
(MSC 05A20)
 
Line 1: Line 1:
 +
{{TEX|done}}{{MSC|05A20}}
 +
 
An inequality for a finite [[distributive lattice]] $(\Gamma,{\prec})$, saying that if $\mu_1$ and $\mu_2$ map $\Gamma$ into $(0,\infty)$ and satisfy $\sum_\Gamma \mu_1(a) = \sum_\Gamma \mu_2(a)$ and
 
An inequality for a finite [[distributive lattice]] $(\Gamma,{\prec})$, saying that if $\mu_1$ and $\mu_2$ map $\Gamma$ into $(0,\infty)$ and satisfy $\sum_\Gamma \mu_1(a) = \sum_\Gamma \mu_2(a)$ and
 
$$
 
$$
Line 18: Line 20:
 
<TR><TD valign="top">[a4]</TD> <TD valign="top">  R. Holley,  "Remarks on the FKG inequalities"  ''Comm. Math. Phys.'' , '''36'''  (1974)  pp. 227–231</TD></TR>
 
<TR><TD valign="top">[a4]</TD> <TD valign="top">  R. Holley,  "Remarks on the FKG inequalities"  ''Comm. Math. Phys.'' , '''36'''  (1974)  pp. 227–231</TD></TR>
 
</table>
 
</table>
 
{{TEX|done}}
 

Latest revision as of 18:02, 16 December 2016

2020 Mathematics Subject Classification: Primary: 05A20 [MSN][ZBL]

An inequality for a finite distributive lattice $(\Gamma,{\prec})$, saying that if $\mu_1$ and $\mu_2$ map $\Gamma$ into $(0,\infty)$ and satisfy $\sum_\Gamma \mu_1(a) = \sum_\Gamma \mu_2(a)$ and $$ \mu_1(a) \mu_2(b) \le \mu_1(a \vee v) \mu_2(a \wedge b) $$ for all $a,b \in \Gamma$, then $$ \sum_\Gamma f(a) \mu_1(a) \ge \sum_\Gamma f(a) \mu_2(a) $$ for every $f : \Gamma \rightarrow \mathbf{R}$ that is non-decreasing in the sense that $a \prec b$ implies $f(a) \le f(b)$. It is due to R. Holley [a4] and was motivated by the related FKG inequality [a3]. It is an easy corollary [a2] of the more powerful Ahlswede–Daykin inequality [a1].

See also Correlation inequalities; Fishburn–Shepp 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] P.C. Fishburn, "Correlation in partially ordered sets" Discrete Appl. Math. , 39 (1992) pp. 173–191
[a3] C.M. Fortuin, P.N. Kasteleyn, J. Ginibre, "Correlation inequalities for some partially ordered sets" Comm. Math. Phys. , 22 (1971) pp. 89–103
[a4] R. Holley, "Remarks on the FKG inequalities" Comm. Math. Phys. , 36 (1974) pp. 227–231
How to Cite This Entry:
Holley inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Holley_inequality&oldid=40021
This article was adapted from an original article by P.C. Fishburn (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article