  We prove a version the Penrose inequality for black hole space-times which
are perturbations of the Schwarzschild exterior in a slab around a null
hypersurface $\underline{\mathcal{N}}_0$. $\underline{\mathcal{N}}_0$
terminates at past null infinity $\mathcal{I}^-$ and
$\mathcal{S}_0:=\partial\underline{\mathcal{N}}_0$ is chosen to be a marginally
outer trapped sphere. We show that the area of $\mathcal{S}_0$ yields a lower
bound for the Bondi energy of sections of past null infinity, thus also for the
total ADM energy. Our argument is perturbative, and rests on suitably deforming
the initial null hypersurface $\underline{\mathcal{N}}_0$ to one for which the
natural "luminosity" foliation originally introduced by Hawking yields a
monotonically increasing Hawking mass, and for which the leaves of this
foliation become asymptotically round. It is to ensure the latter (essential)
property that we perform the deformation of the initial nullhypersurface
$\underline{\mathcal{N}}_0$.
