A lower bound of the Trautman-Bondi energy

We obtain an energy inequality on null surfaces $u=const$ in the Bondi-Sachs formalism. We show that for a sufficiently regular event horizon $H$ there is an affine radial coordinate which is constant on $H$. Then the energy inequality can be prolongated to the horizon giving an estimation which is closely related to the Penrose inequality. We test it for the Kerr solution written in the Fletcher-Lun coordinates.