  We study ferromagnetic Ising models on finite graphs with an inhomogeneous
external field, where a subset of vertices is designated as the boundary. We
show that the influence of boundary conditions on any given spin is maximised
when the external field is identically $0$. One corollary is that spin-spin
correlations are maximised when the external field vanishes and the boundary
condition is free, which proves a conjecture of Shlosman. In particular, the
random field Ising model on ${\mathbb Z}^d$, $d\geq 3$, exhibits exponential
decay of correlations in the entire high temperature regime of the pure Ising
model. Another corollary is that the pure Ising model in $d\geq 3$ satisfies
the conjectured strong spatial mixing property in the entire high temperature
regime.
