Sharp estimates of uniform harmonic majorants in the plane
type: Item
instance of: scholarly article
title: Sharp estimates of uniform harmonic majorants in the plane
zbMATH Open document ID: 0626.30022
DOI: 10.1007/BF02384434
author: Matts Essén
published in: Arkiv för Matematik
publication date: 1987-01-01T00:00:00Z
review text: S-Y. A. Chang and D. Marshall have proved that an analytic function f in the unit disk with Dirichlet integral \(\leq \pi\) satisfies \[ \int^{\pi}_{-\pi}\exp (| f(e^{i\theta})-f(0)|^ 2)d\theta \leq C<\infty. \] This is the critical endpoint improvement of a theorem from Beurling's thesis. The present author considerably generalizes the Chang-Marshall theorem by proving that if \(\Omega\) is a plane domain containing the origin with area \(\leq \pi\) then \(\exp | w|^ 2\) has a harmonic majorant in \(\Omega\). In fact, this result is just a special case of his general theorem, which states that certain functions \(\Phi\) (\(| w|)\) have harmonic majorants in \(\Omega\) when \(\Omega\) is not too large. For example, if \(0\in \Omega\) and \[ \int^{1}_{0}r\theta (r)dr+\lambda^ 2\int^{\infty}_{1}t^{2\lambda -1}\theta (t)dt=\pi, \] where \(\lambda >0\) and \(\theta\) (r) is the angular measure of \(\Omega \cap (| w| =r)\), then \(\exp (| w|^{2\lambda})\) has a harmonic majorant in \(\Omega\). The proof is technically quite interesting. It involves Moser's integrability theorem, symmetrization, Ahlfors' distortion theorem, and the reduction of harmonic measure estimates in the possibly multiply connected \(\Omega\) to some in simply connected domains.
Mathematics Subject Classification ID: 30C85
zbMATH DE Number: 4017273
zbMATH Keywords: harmonic majorants
zbMATH Keywords: harmonic measure
reviewed by: Albert Baernstein II
MaRDI profile type: Publication
cites work: Q5179675
cites work: Integral means, univalent functions and circular symmetrization
cites work: On a Sharp Inequality Concerning the Dirichlet Integral
cites work: The cos \(\pi\lambda\) theorem. With a paper by Christopher Borell
cites work: Harmonic Majorization and Classical Analysis
cites work: Q5614914
cites work: Q5594005
full work available at URL: https://doi.org/10.1007/bf02384434
OpenAlex ID: W1978822113
description: scientific article; zbMATH DE number 4017273