  We bound integral means of the Bergman projection of a function in terms of
integral means of the original function. As an application of these results, we
bound certain weighted Bergman space norms of derivatives of Bergman
projections in terms of weighted $L^p$ norms of certain derivatives of the
original function in the $\theta$ direction. These results easily imply the
well known result that the Bergman projection is bounded from the Sobolev space
$W^{k,p}$ into itself for $1 < p < \infty$. We also apply our results to derive
certain regularity results involving extremal problems in Bergman spaces.
Lastly, we construct a function that approaches $0$ uniformly at the boundary
of the unit disc but whose Bergman projection is not in $H^2$.
