  Hawking's black hole area theorem was proven using the null energy condition
(NEC), a pointwise condition violated by quantum fields. The violation of the
NEC is usually cited as the reason that black hole evaporation is allowed in
the context of semiclassical gravity. Here we provide two generalizations of
the classical black hole area theorem: First, a proof of the original theorem
with an averaged condition, the weakest possible energy condition to prove the
theorem using focusing of null geodesics. Second, a proof of an area-type
result that allows for the shrinking of the black hole horizon but provides a
bound on it. This bound can be translated to a bound on the black hole
evaporation rate using a condition inspired from quantum energy inequalities.
Finally, we show how our bound can be applied to two cases that violate
classical energy conditions.
