Holley inequality

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