Monotonicity of Ursell functions in the Ising model

In this paper, we consider Ising models with ferromagnetic pair interactions. We prove that the Ursell functions $u_{2k}$ satisfy: $(-1{)}^{k-1}{u}_{2k}$ is increasing in each interaction. As an application, we prove a 1983 conjecture by Nishimori and Griffiths about the partition function of the Ising model with complex external field $h$: its closest zero to the origin (in the variable $h$) moves towards the origin as an arbitrary interaction increases.