  For general ferromagnetic Ising models whose coupling matrix has bounded
spectral radius, we show that the log-Sobolev constant satisfies a simple bound
expressed only in terms of the susceptibility of the model. This bound implies
very generally that the log-Sobolev constant is uniform in the system size up
to the critical point (including on lattices), without using any mixing
conditions. Moreover, if the susceptibility satisfies the mean-field bound as
the critical point is approached, our bound implies that the log-Sobolev
constant depends polynomially on the distance to the critical point and on the
volume. In particular, this applies to the Ising model on subsets of
$\mathbb{Z}^d$ when $d>4$.
  The proof uses a general criterion for the log-Sobolev inequality in terms of
the Polchinski (renormalisation group) equation, a recently proved remarkable
correlation inequality for Ising models with general external fields, the
Perron--Frobenius theorem, and the log-Sobolev inequality for product Bernoulli
measures.
