  We study estimates for Hardy space norms of analytic projections. We first
find a sufficient condition for the Bergman projection of a function in the
unit disc to belong to the Hardy space $H^p$ for $1 < p < \infty$. We apply the
result to prove a converse to an extension of Ryabykh's theorem about Hardy
space regularity for Bergman space extremal functions. We also prove that the
$H^q$ norm of the Szeg\"{o} projection of $F^{p/2} \overline{F}^{(p/2)-1}$
cannot be too small if $F$ is analytic, for certain values of $p$ and $q$. We
apply this to show that the best analytic approximation in $L^p$ of a function
in both $L^p$ and $L^q$ will also lie in $L^q$, for certain values of $p$ and
$q$.
