Inequalities for Ising models and field theories which obey the Lee-Yang Theorem

A series of inequalities for partition, correlation, and Ursell functions are derived as consequences of the Lee-Yang Theorem. In particular, the n-point Schwinger functions of even φ⁴ models are bounded in terms of the 2-point function as strongly as is the case for Gaussian fields; this strengthens recent results of Glimm and Jaffe and shows that renormalizability of the 2-point function by fourth degree counter-terms implies existence of a φ⁴ field theory with a moment generating function which is entire of exponential order at most two. It is also noted that if any (even) truncated Schwinger function vanishes identically, the resulting field theory is a generalized free field.