  We study the nonlinear stability of a large class of inhomogeneous steady
state solutions to the Hamiltonian Mean Field (HMF) model. Under a simple
criterion, we prove the nonlinear stability of steady states which are
decreasing functions of the microscopic energy. To achieve this task, we extend
to this context the strategy based on generalized rearrangement techniques
which was developed recently for the gravitational Vlasov-Poisson equation.
Explicit stability inequalities are established and our analysis is able to
treat non compactly supported steady states to HMF, which are physically
relevant in this context but induces additional difficulties, compared to the
Vlasov-Poisson system.
