Abstract
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 $Re \int _\Omega f(z) \overline {k(z)}\, \nu (z) \, dA(z)$ 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.
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