  We prove a Lieb-Thirring type inequality for a complex perturbation of a
d-dimensional massive Dirac operator $D_m, m\geq 0$ whose spectrum is $]-\infty
, -m]\cup[m , +\infty[$. The difficulty of the study is that the unperturbed
operator is not bounded from below in this case, and, to overcome it, we use
the methods of complex function theory. The methods of the article also give
similar results for complex perturbations of the Klein-Gordon operator.
