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, ββc, h∼0 (βc 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*(β) and the susceptibility χ (β,h): e.g., m*(β) is upper semicontinuous, and a discontinuity in m*(β) at β0 implies that the susceptibility cannot be bounded (near h=0) by an integrable function of h as β→β0 from the left. We also find some inequalities among the critical indices.