  Let $L$ be a finite distributive lattice and $\mu : L \to {\mathbb R}^{+}$ a
log-supermodular function. For functions $k: L \to {\mathbb R}^{+}$ let
$$E_{\mu} (k; q) \defeq \sum_{x\in L} k(x) \mu (x) q^{{\mathrm rank}(x)} \in
{\mathbb R}^{+}[q].$$ We prove for any pair $g,h: L\to {\mathbb R}^{+}$ of
monotonely increasing functions, that $$E_{\mu} (g; q)\cdot E_{\mu} (h; q) \ll
  E_{\mu} (1; q)\cdot E_{\mu} (gh; q), $$ where ``$ \ll $'' denotes
coefficientwise inequality of real polynomials. The FKG inequality of Fortuin,
Kasteleyn and Ginibre (1971) is the real number inequality obtained by
specializing to $q=1$.
  The polynomial FKG inequality has applications to $f$-vectors of joins and
intersections of simplicial complexes, to Betti numbers of intersections of
certain Schubert varieties, and to the following kind of correlation inequality
for power series weighted by Young tableaux. Let $Y$ be the set of all integer
partitions. Given functions $k, \mu: Y \rarr \R^+$, and parameters $0\le s\le
t$, define the formal power series $$F_{\mu}(k ; z) \defeq \sum_{\la\in Y}
k(\la) \mu(\la) (f_{\la})^t \frac{z^{|\la|}}{(|\la| !)^s} \in \R^+ [[z]], $$
%\sum_{\la\in Y} k(\la) \mu(\la) (f_{\la})^t \frac{z^{|\la|}}{|\la| !} \in \R^+
[[z]],$$ where $f_{\la}$ is the number of standard Young tableaux of shape
$\la$. Assume that $\mu: Y\rarr \R^+$ is log-supermodular, and that $g, h: Y
\rarr \R^+$ are monotonely increasing with respect to containment order of
partition shapes. Then $$F_{\mu}(g;z) \cdot F_{\mu}(h;z) \ll F_{\mu}(1;z) \cdot
F_{\mu}(gh;z). $$
