  Given a constant mean curvature surface that bounds a compact manifold with
nonnegative scalar curvature, we obtain intrinsic conditions on the surface
that guarantee the positivity of its Hawking mass. We also obtain estimates of
the Bartnik mass of such surfaces, without assumptions on the integral of the
squared mean curvature. If the ambient manifold has negative scalar curvature,
our method also applies and yields estimates on the hyperbolic Bartnik mass of
these surfaces.
