A proof (but with technical restrictions) was obtained in [@Acena:2010ws] [@Clement:2011kz]. The first general and pure quasi-local result was proven in [@Dain:2011pi], where the relevant role of the stability condition for minimal surfaces was pointed out. The generalization to trapped surfaces and non-vacuum has been proved in [@Jaramillo:2011pg]. The electromagnetic charge was included in [@Clement:2011np] and [@Clement:2012vb]. This inequality has been extended to higher dimensions in [@Hollands:2011sy] and [@Paetz:2013rka]. In [@Yazadjiev:2012bx] [@Yazadjiev:2013hk] and [@Fajman:2013ffa] it has been also extended to Einstein-Maxwell dilaton gravity. In [@Reiris:2013jaa] related inequalities that involve the shape of the black hole were proved.

![Axially symmetric two-surface. The axial Killing vector $\eta$ is tangent to the surface. The null vectors $\ell^a$ and $k^a$ are normal to ${\cal S}$[]{data-label="fig:axial-2s"}](axial-two-surface.pdf){width="20.00000%"}

To describe the concept of stable trapped surface (this condition was first introduced in [@andersson08]) used in theorem \[t:2\] let us consider an axially symmetric closed two-surface ${\Sigma}$ with the topology of a two-sphere. The surface ${\Sigma}$ is embedded in the spacetime. Let $\ell^a$ and $k^a$ be null vectors spanning the normal plane to ${\Sigma}$ and normalized as $\ell^a k_a =
-1$, see figure \[fig:axial-2s\]. The expansion is defined by $\theta^{(\ell)}= \nabla_a\ell^a$, where $\nabla$ is the spacetime connection. The 