  In this paper we prove an extrinsic one-sided curvature estimate for disks
embedded in $\mathbb{R}^3$ with constant mean curvature which is independent of
the value of the constant mean curvature. We apply this extrinsic one-sided
curvature estimate in [24] to prove to prove a weak chord arc type result for
these disks. In Section 4 we apply this weak chord arc result to obtain an
intrinsic version of the one-sided curvature estimate for disks embedded in
$\mathbb{R}^3$ with constant mean curvature. In a natural sense, these
one-sided curvature estimates generalize respectively, the extrinsic and
intrinsic one-sided curvature estimates for minimal disks embedded in
$\mathbb{R}^3$ given by Colding and Minicozzi in Theorem 0.2 of [8] and in
Corollary 0.8 of [9].
