  Countable projective limits of countable inductive limits, called PLB-spaces,
of weighted Banach spaces of continuous functions have recently been
investigated by Agethen, Bierstedt and Bonet. In a previous article, the author
extended their investigation to the case of holomorphic functions and
characterized when spaces over the unit disc w.r.t. weights whose decay,
roughly speaking, is neither faster nor slower than that of a polynomial are
ultrabornological or barrelled. In this note, we prove a similar
characterization for the case of weights which tend to zero logarithmically.
