An extremal problem in Banach algebras

We study asymptotics of a class of extremal problems r(n) (A, epsilon) rela ted to norm controlled inversion in Banach algebras. In a general setting w e prove estimates that can be considered as quantitative refinements of a t heorem of Jan-Erik Bjork [1]. In the last section we specialize further and consider a class of analytic Beurling algebras. In particular, a question raised by Jan-Erik Bjork in [1] is answered in the negative.