Growth and zero sets of analytic families of Cauchy-Stieltjes integrals
Let \(X\) be the boundary of the unit disk \(D\). The authors continue their study of functions \(f(z)\), \(z\in D\) which can be represented as \[ f(z)=\int_ X (1-xz)^{-\alpha} d\mu(x), \] where \(\mu\) is a probability measure and \(\alpha>0\). (A slightly different formula works for \(\alpha=0\).) The functions so representable form \({\mathcal F}_ \alpha\). For functions in a given \({\mathcal F}_ \alpha\) the authors determine the corresponding Hardy and Bergman spaces to which they belong. The extremal function \(f(z)= (1-z)^{-\alpha}\) shows what to expect. When \(\alpha\leq 1\), the zeros satisfy a Blaschke condition while if \(\alpha>1\) there is a weaker condition: \(\prod^ n(1/| a_ k|= o(n^{\alpha-1})\), but neither is sufficient. However, the Blaschke condition remains sharp when the zeros are confined to a single ray. (This is related to earlier work of \textit{F. Bagemihl}, \textit{P. Erdős} and \textit{W. Seidel} [Ann. Sci. École Norm. Super., III. Sér. 70, 133- 147 (1953; Zbl 0053.238)].) An extensive study is made of radial growth as \(r\to 1\). This is of significance only when \(\alpha>1\). An interesting device (p.244) is a representation of \(f\in{\mathcal F}_ \alpha\) in terms of a function \(g\in{\mathcal F}_ 1\), valid on any concentric disk.
scientific article; zbMATH DE number 528675
scientific article