  Let (L,\preccurlyeq) be a finite distributive lattice, and suppose that the
functions f_1,f_2:L\to R are monotone increasing with respect to the partial
order \preccurlyeq. Given \mu a probability measure on L, denote by E(f_i) the
average of f_i over L with respect to \mu, i=1,2. Then the
  FKG inequality provides a condition on the measure \mu under which the
covariance, Cov(f_1,f_2):=E(f_1f_2)-E(f_1)E(f_2), is nonnegative. In this paper
we derive a ``third-order'' generalization of the FKG inequality.
  We also establish fourth- and fifth-order generalizations of the FKG
inequality and formulate a conjecture for a general mth-order generalization.
For functions and measures on R^n we establish these inequalities by extending
the method of diffusion processes. We provide several applications of the
third-order inequality, generalizing earlier applications of the FKG
inequality.
  Finally, we remark on some connections between the theory of total positivity
and the existence of inequalities of FKG-type within the context of Riemannian
manifolds.
