  In this paper we study asymptotically hyperbolic manifolds given as graphs of
asymptotically constant functions over hyperbolic space $\bH^n$. The graphs are
considered as subsets of $\bH^{n+1}$ and carry the induced metric. For such
manifolds the scalar curvature appears in the divergence of a 1-form involving
the integrand for the asymptotically hyperbolic mass. Integrating this
divergence we estimate the mass by an integral over an inner boundary. In case
the inner boundary satisfies a convexity condition this can in turn be
estimated in terms of the area of the inner boundary. The resulting estimates
are similar to the conjectured Penrose inequality for asymptotically hyperbolic
manifolds. The work presented here is inspired by Lam's article concerning the
asymptotically Euclidean case.
