# 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].