  We discuss the regularity of extremal functions in certain weighted Bergman
and Fock type spaces. Given an appropriate analytic function $k$, the
corresponding extremal function is the function with unit norm maximizing
$\text{Re} \int_\Omega f(z) \overline{k(z)}\, \nu(z) \, dA(z)$ over all
functions $f$ of unit norm, where $\nu$ is the weight function and $\Omega$ is
the domain of the functions in the space. We consider the case where $\nu(z)$
is a decreasing radial function satisfying some additional assumptions, and
where $\Omega$ is either a disc centered at the origin or the entire complex
plane. We show that if $k$ grows slowly in a certain sense, then $f$ must grow
slowly in a related sense. We also discuss a relation between the integrability
and growth of certain log-convex functions, and apply the result to obtain
information about the growth of integral means of extremal functions in Fock
type spaces.
