  Let $(M,g)$ be a complete connected $n$-dimensional Riemannian spin manifold
without boundary such that the scalar curvature satisfies $R_g\geq -n(n-1)$ and
$\mathcal{E}\subset M$ be an asymptotically hyperbolic end, we prove that the
mass functional of the end $\mathcal{E}$ is timelike future-directed or zero.
Moreover, it vanishes if and only if $(M,g)$ is isometric to the hyperbolic
space. We also consider the mass of an asymptotically hyperbolic manifold with
compact boundary, we prove the mass is timelike future-directed if the mean
curvature of the boundary is bounded from below by a function defined using
distance estimates. As an application, the mass is timelike future-directed if
the mean curvature of the boundary is bounded from below by $-(n-1)$ or the
scalar curvature satisfies $R_g\geq (-1+\kappa)n(n-1)$ for any positive
constant $\kappa$ less than one.
