in the spirit of [@Booth:2007wu], although restricted only to axial symmetry. In the article [@Dain:2007pk] it has been conjectured that, within axially symmetry, to prove the stability of a nearly extreme black hole is perhaps simpler than a Schwarzschild black hole. It is possible that this quasi-local extremality criteria will have relevant applications in this context. Note also that the inequality allows to define, at least formally, the positive temperature of a dynamical black hole $\kappa$ by the formula (\[eq:7\]) (see Refs. [@Ashtekar03] [@Ashtekar02] for a related discussion of the first law in dynamical horizons). If inequality holds, then ${m_{bh}}$ defines a non-trivial quantity that increase monotonically with time, like the black hole area $A$.

It is important to emphasize that the physical arguments presented above in support of are certainly weaker in comparison with the ones behind the Penrose inequalities that support the inequalities (\[eq:pen\]) and (\[eq:mj\]) that we will discuss bellow. A counter example of any of these inequality will prove that the standard picture of the gravitational collapse is wrong. On the other hand, a counter example of will just prove that the quasi-local mass is not appropriate to describe the evolution of a non-stationary black hole. 