In this note, we continue the investigations of [3], proving another analogue of the maximum modulus theorem, this time for the sequence space bv, and we investigate maximal functions for such theorems. As in [3], we use the notation f∈MM if f is analytic in the disk |z| <1, continuous for |z| ≤ 1 and satisfies |f(z)| ≤ 1 on |z| = 1. We also write f∈SL if f∈MM and f(0) = 0. Whenever x={xk} is a sequence of complex numbers, we write f(x) = {f(xk)}.

In [3], we proved analogues of the maximum modulus theorem for the sequence spaces 5, m and c, and analogues of the Schwarz Lemma for the sequence spaces c0, lp and bv0. We begin this note with the sequence space bv.