Area extremal problems for non-vanishing functions

Abstract:

We consider the problem of finding the extremal function $f$which minimises the Bergman space $A^2$norm for the class of non-vanishing functions whose first $n+1$ Taylor coefficients are given.$$We define an analytic function $K$ in terms of $f$ and show that the functions $K$ and $f$ satisfy a certain differential equation. This equation yields a set of relationships between the area moments and the circle moments of $|f|^2$, which in particular shows that the outer part of $f$is a polynomial of degree at most $n$.