Namely $$\begin{aligned}
  \label{eq:11}
  m^2 &= \sqrt{m^4-J^2+J^2},\\
& \leq |J| +\sqrt{m^4-J^2}, \label{eq:11b} \\
& \leq \frac{A}{8\pi}+ \sqrt{m^4-J^2},\label{eq:11c}\end{aligned}$$ where in the line (\[eq:11b\]) we have used (\[eq:mj\]) and in line (\[eq:11c\]) we have used (\[eq:JA\]). In that sense, the inequalities (\[eq:pen\]), (\[eq:mj\]) and (\[eq:JA\]) are more fundamental than (\[eq:10\]). However, the inequality (\[eq:10\]) is important by itself since it related with the Penrose inequality with angular momentum, see [@Khuri:2013wha] [@Dain:2013qia].

We have seen that for stationary black holes the inequalities (\[eq:pen\]), (\[eq:mj\]) and (\[eq:JA\]) are straightforward consequences of the area formula (\[eq:3\]).

![Schematic representation of an initial data for a non-stationary black hole. The black ring represents a trapped surface. Outside and inside the trapped surface the gravitational field is highly dynamical.[]{data-label="fig:3"}](dynamical-bh-p.pdf){width="6cm"}

However, black holes are in general non stationary, see figure \[fig:3\]. Astrophysical phenomena like the formation of a black hole by gravitational collapse or a binary black hole collision are highly dynamical. For such systems, the black hole can not be characterized by few parameters as in the stationary case. In fact, even stationary but non-vacuum black holes have a complicated structure (for example black holes surrounded by a rotating ring of matter, see the numerical studies in [@Ansorg05]). Remarkably, 