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