  Bekenstein and Mayo proposed a generalised bound for the entropy, which
implies some inequalities between the charge, energy, angular momentum, and the
size of the macroscopic system. Dain has shown that Maxwell's electrodynamics
satisfies all three inequalities. We investigate the validity of these
relations in the context of nonlinear electrodynamics and show that Born-Infeld
electrodynamics satisfies all of them. However, contrary to the linear theory,
there is no rigidity statement in Born-Infeld. We study the physical meaning
and the relationship between these inequalities and, in particular, we analyse
the connection between the energy-angular momentum inequality and causality.
