Counting functions and majorization for Jensen measures (Q1059186)

We establish for uniform algebras a generalization of the classical identities of Hardy and Stein. We use this and an estimate based on the isoperimetric inequality to give a proof of H. Alexander's spectral area theorem. We use similar methods to prove a slight generalization of the following result of \textit{S. Axler} and \textit{J. H. Shapiro} [Math. Ann. 271, 161-183 (1985; Zbl 0541.30041)]: if f is a bounded holomorphic function on the unit ball in \({\mathbb{C}}^ n\) and the cluster set at each point in the boundary of the ball has zero area then f belongs to VMOA.