Bergman space extremal problems for non-vanishing functions

We consider the problem of minimising the Bergman Space A ² norm of functions analytic and non-vanishing in the unit disc, which satisfy a finite number of constraints of the form l _i (f) = c _i , where each l _i (f) is a finite linear combination of Taylor coefficients of f evaluated at certain points of the disc. We show that when the class of functions satisfying the constraints is nonempty, an extremal function exists and that every extremal function has rational outer part of a specific form.