  A consistent BPS formalism to study the existence of topological axially
symmetric vortices in generalized versions of the Born-Infeld-Higgs
electrodynamics is implemented. Such a generalization modifies the field
dynamics via introduction of three non-negative functions depending only in the
Higgs field, namely, $G(|\phi|)$, $w(|\phi|) $ and $V(|\phi|)$. A set of
first-order differential equations is attained when these functions satisfy a
constraint related to the Ampere law. Such a constraint allows to minimize the
system energy in such way that it becomes proportional to the magnetic flux.
Our results provides an enhancement of topological vortex solutions in
Born-Infeld-Higgs electrodynamics. Finally, we analyze a set of models such
that a generalized version of Maxwell-Higgs electrodynamics is recovered in a
certain limit of the theory.
