Correlation inequalities and a conjecture for permanents

Abstract

This paper presents conditions on nonnegative real valued functionsf ₁,f ₂,...,f _m andg ₁,g ₂,...g _m implying an inequality of the type

$$\mathop \Pi \limits_{i = 1}^m \int {f_i (x)d\mu } (x) \leqslant \mathop \Pi \limits_{i = 1}^m \int {g_i (x)d\mu } (x).$$

This “2m-function” theorem generalizes the “4-function” theorem of [2], which in turn generalizes a “2-function” theorem ([8]) and the celebrated FKG inequality. It also contains (and was partly inspired by) an “m against 2” inequality that was deduced in [5] from a general product theorem.