Nevanlinna’s coefficients and Douglas algebras

Abstract

is the Blaschke product corresponding to {zn}. While the funcions P,Q,R and S arose from classical function theory, it turns out that they are also connected with more recent developments. It is part of Nevanlinna's theory that the functions P/R, Q/R, S/R and 1/R belong to U and are linked with π in many ways. (See Lemma 1.) Suppose (NP) has a solution / 0 satisfying sup{|/0(z)|, z E D} < 1. Our main result is that then P/R, Q/R, S/R and 1/R all belong to a certain subalgebra of H°° depending only on π which we shall denote by CDAπ. This algebra is part of the theory of Douglas algebras through the work of S.Y. Chang and D.E. Marshall ([1], [2?]). Our results in particular answer