New correlation inequalities for the Ising model and $P\left(\phi \right)$ theories

The reflection principle combined with an extension of Ginibre's discussion of the Griffiths inequalities leads to new correlation inequalities for the Ising model and $P\left(\phi \right)$ theories. In particular the Ising correlation $〈{\sigma }_0{\sigma }_{\left(j_1\cdots j_d\right)}〉$ in $d$ dimensions in an arbitrary external magnetic field is shown to be nonincreasing in each coordinate variable $\left|j_l\right|$.