+ C(z)p(z) + D(z) I < M,

Using condition (ii) in this last inequality leads to the required result (4.1-17).

As in the previous case for A(z) = 0, instead of prescribing the constant N in Theorem 4.1d, in some cases we can use (ii) to determine an appropriate N = N(M, n, A, B, C, D) so that (4.1-8) implies jp(z)l < N . This can be accomplished by solving (ii) for N and by taking the supremum of the resulting function over U. If this supremum is finite, we have the following version of Theorem 4.1d.