  We study the validity of Bekenstein's entropy bound for a charged black hole
in the context of nonlinear electrodynamics. Bekenstein's inequalities are
commonly understood as universal relations between the entropy, the charge, the
momentum, and the energy of a physical system but independent of its dynamics.
In particular, we consider the Born-Infeld electrodynamics coupled to gravity
as described by General Relativity. Following the steps that lead to these
inequalities, we study the absorption of a charged test particle by the black
hole and verify that the entropy bound is violated. We find a modified upper
bound for the entropy that depends on the maximum field parameter of the
Born-Infeld theory.
