View source for Holley inequality

An inequality for a finite [[Distributive lattice|distributive lattice]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110230/h1102301.png" />, saying that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110230/h1102302.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110230/h1102303.png" /> map <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110230/h1102304.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110230/h1102305.png" /> and satisfy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110230/h1102306.png" /> and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110230/h1102307.png" /></td> </tr></table>

then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110230/h1102308.png" /></td> </tr></table>

for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110230/h1102309.png" /> that is non-decreasing in the sense that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110230/h11023010.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110230/h11023011.png" />. It is due to R. Holley [[#References|[a4]]] and was motivated by the related [[FKG inequality|FKG inequality]] [[#References|[a3]]]. It is an easy corollary [[#References|[a2]]] of the more powerful [[Ahlswede–Daykin inequality|Ahlswede–Daykin inequality]] [[#References|[a1]]].

See also [[Correlation inequalities|Correlation inequalities]]; [[Fishburn–Shepp inequality|Fishburn–Shepp inequality]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.C. Fishburn,   "Correlation in partially ordered sets"  ''Discrete Appl. Math.'' , '''39'''  (1992)  pp. 173–191</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  C.M. Fortuin,   P.N. Kasteleyn,   J. Ginibre,   "Correlation inequalities for some partially ordered sets"  ''Comm. Math. Phys.'' , '''22'''  (1971)  pp. 89–103</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>