  In a statistical mechanics model with unbounded spins, we prove uniqueness of
the Gibbs measure under various assumptions on finite volume functional
inequalities. We follow the approach of G. Royer (1999) and obtain uniqueness
by showing convergence properties of a Glauber-Langevin dynamics. The result
was known when the measures on the box $[-n,n]^d$ (with free boundary
conditions) satisfied the same logarithmic Sobolev inequality. We generalize
this in two directions: either the constants may be allowed to grow
sub-linearly in the diameter, or we may suppose a weaker inequality than
log-Sobolev, but stronger than Poincar\'e. We conclude by giving a heuristic
argument showing that this could be the right inequalities to look at.
