  Estimates for the norm of the second fundamental form, $|A|$, play a crucial
role in studying the geometry of surfaces. In fact, when $|A|$ is bounded the
surface cannot bend too sharply. In this paper we prove that for an embedded
geodesic disk with bounded $L^2$ norm of $|A|$, $|A|$ is bounded at interior
points, provided that the $W^{1,p}$ norm of its mean curvature is sufficiently
small, $p>2$. In doing this we generalize some renowned estimates on $|A|$ for
minimal surfaces.
