  Smooth solutions to the axially symmetric Navier-Stokes equations obey the
following maximum
principle:$\|ru_\theta(r,z,t)\|_{L^\infty}\leq\|ru_\theta(r,z,0)\|_{L^\infty}.$
We first prove the global regularity of solutions if
$\|ru_\theta(r,z,0)\|_{L^\infty}$ or $ \|ru_\theta(r,z,t)\|_{L^\infty(r\leq
r_0)}$ is small compared with certain dimensionless quantity of the initial
data. This result improves the one in Zhen Lei and Qi S. Zhang \cite{1}. As a
corollary, we also prove the global regularity under the assumption that
$|ru_\theta(r,z,t)|\leq\ |\ln r|^{-3/2},\ \ \forall\
0<r\leq\delta_0\in(0,1/2).$
