Local Ward identities and the decay of correlations in ferromagnets

Using local Ward identities we prove a number of correlation inequalities for N-component, isotropically coupled, pair interacting ferromagnets; some for all N≧2 and some for N=2, 3, 4. These are used to prove a mass gap above the mean field temperature, for all N≧2. For N=2, 3, 4 we prove an upper bound on a critical exponent, and a lower bound on the susceptability which diverges as m→0.