  We establish gradient estimates for solutions to the Dirichlet problem for
the constant mean curvature equation in hyperbolic space. We obtain these
estimates on bounded strictly convex domains by using the maximum principles
theory of $\Phi$-functions of Payne and Philippin. These estimates are then
employed to solve the Dirichlet problem when the mean curvature $H$ satisfies
$H<1$ under suitable boundary conditions.
