More Inequalities for Ising Ferromagnets

We consider an Ising spin system with ferromagnetic pair interactions. Combining the old Griffiths inequalities with the recent Fortuin-Kasteleyn-Ginibre inequalities we obtain some new bounds on the correlations and thermodynamic functions. These bounds appear to be of interest in the neighborhood of the two-phase region of these systems, $\beta \gtrsim {\beta }_c$, $h\sim 0$ (${\beta }_c$ is the reciprocal of the critical temperature, and $h$ is the external magnetic field), where they yield relations between singularities in the spontanuous magnetization $m^{*}\left(\beta \right)$ and the susceptibility $\chi (\beta ,h)$: e.g., $m^{*}\left(\beta \right)$ is upper semicontinuous, and a discontinuity in $m^{*}\left(\beta \right)$ at ${\beta }_0$ implies that the susceptibility cannot be bounded (near $h=0\mathrm{}$) by an integrable function of $h$ as $\beta \to {\beta }_0$ from the left. We also find some inequalities among the critical indices.