Boundary behavior of derivatives of analytic functions
type: Item
instance of: scholarly article
title: Boundary behavior of derivatives of analytic functions
zbMATH Open document ID: 0769.30020
author: Sheldon Axler
author: Kehe Zhu
published in: Michigan Mathematical Journal
publication date: 1992-09-27T00:00:00Z
review text: Denote by \(D\) the complex open unit disk and \(U\) the closed subalgebra of \(L^ \infty(D)\) generated by the bounded complex harmonic functions on \(D\). Define \(P\) to be the linear operator on \(L^ 1(D)\) \[ P(g,z):= {1\over\pi} \int_ D {{g(w)} \over {(1-z\overline{w})^ 2}}dw. \] The authors show that an analytic function \(f\) on \(D\) is in \(P(U)\) if and only if \((1-| z|^ 2)^ n f^{(n)}(z)\in U\) for some positive integer \(n\). Further they give several results that prove there is a closed connection between the behaviour of functions of the form \((1- | z|^ 2)^ n f^{(n)}(z)\), the Gleason parts, the Bloch space and the operator \(P\). Complete proofs are given.
Mathematics Subject Classification ID: 30D45
zbMATH DE Number: 63017
zbMATH Keywords: Gleason parts
zbMATH Keywords: Bloch space
reviewed by: Miguel A. Jiménez
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029004460
OpenAlex ID: W2002549613
DOI: 10.1307/MMJ/1029004460
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description: scientific article; zbMATH DE number 63017