  We study the effect of competing interactions on ensemble inequivalence. We
consider a one-dimensional Ising model with ferromagnetic mean-field
interactions and short-range couplings which can be either ferromagnetic or
antiferromagnetic. Despite the relative simplicity of the model, our
calculations in the microcanonical ensemble reveal a rich phase diagram. The
comparison with the corresponding phase diagram in the canonical ensemble shows
the presence of phase transition points and lines which are different in the
two ensembles. As an example, in a region of the phase diagram where the
canonical ensemble shows a critical point and a critical end point, the
microcanonical ensemble has an additional critical point and also a triple
point. The regions of ensemble inequivalence typically occur at lower
temperatures and at larger absolute values of the competing couplings. The
presence of two free parameters in the model allows us to obtain a fourth-order
critical point, which can be fully characterized by deriving its Landau normal
form.
