On a theorem of Korenblum (Q1329666)

\(A^{-\alpha}\) \((\alpha> 0)\) is the set of functions \(f\) analytic in the unit disk \(U= \{z: | z|< 1\}\) satisfying \[ \sup_{z\in U} (1- | z|)^ \alpha | f(z)|< \infty. \] For a given sequence \(Z\) from \(U\) we denote by \(\rho(Z)\) the infimum of those \(\alpha\) such that \(Z\) is the sequence of zeros of a function in \(A^{-\alpha}\). It is proved that \(\rho(Z)\) can be expressed in terms of a certain density of \(Z\). This notion of density was introduced by \textit{B. Korenblum} [Acta Math. 135, 187-219 (1975; Zbl 0323.30030)], who proved a less precise result. Some fine estimates of \textit{E. J. Specht} [see Trans. Am. Math. Soc. 71, 183-196 (1951; Zbl 0043.302)] on the mapping function and its derivative in conformal mapping of so called nearly circular regions play a crucial role in the proof.