  We generalize a well-known sufficient condition for interpolating sequences
for the Hilbert Bergman spaces to other Bergman spaces with normal weights (as
defined by Shields and Williams) and obtain new results regarding the
membership of the derivative of a Blashke product or a general inner function
in such spaces. We also apply duality techniques to obtain further results of
this type and obtain new results about interpolating Blaschke products.
