In this paper we modify the maximum principal of (Galloway, 2000) for totally geodesic null hypersurfaces ?by proving a geometric maximum principle which obeys mean curvature inequalities of a family of totally umbilical null hypersurfaces of a spacetime manifold (Theorem~6). As a physical interpretation we show that, in particular, for a prescribed class of spacetimes the geometric inequality of the Theorem~6 for totally umbilical null hypersurfaces ?is valid as well as it establishes a link with Galloway's vanishing mean curvature totally geodesic null hypersurfaces that arise most naturally in general relativity, such as black hole event horizons (Theorem 7).
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