\quad (=\text{Kerr black
      hole}). \label{eq:pendj}
  \end{aligned}$$ The inequality is quasi-local and the inequality is global. The global inequality implies the following two inequalities $$\begin{aligned}
  \sqrt{\frac{A}{16\pi}} &\leq m \quad (=\text{Schwarzschild}),\label{eq:penddd}\\
\sqrt{|J|} &\leq m. \quad (=\text{extreme Kerr black hole}).\label{eq:mjdd}\end{aligned}$$ That is:

> *The three geometrical inequalities (\[eq:pen\]), (\[eq:mj\]) and (\[eq:JA\]) valid for the Kerr black holes are expected to hold also for axially symmetric, dynamical black holes.*

The Penrose inequality is valid also without the axial symmetry assumption. It is important to emphasize that all the quantities involved in the geometrical inequalities above can be calculated on the initial surface. For simplicity, we have avoided the distinction between event horizon and apparent horizons (defined in terms of trapped surfaces) to calculate the area $A$. This point is important for the Penrose inequality (see the discussion in [@Mars:2009cj]) but not for the other inequalities which are the main subject of this review. In particular the horizon area $A$ in (\[eq:JAd\]) is the area of an appropriated defined trapped surface.

A counter example of the global inequality (\[eq:pendj\]) will imply that cosmic censorship is not true. Conversely a proof of it gives indirect evidence of the validity of censorship, since it is very hard 