On some classes of subharmonic functions which have nonnegative harmonic majorants in a half-plane
type: Item
instance of: scholarly article
title: On some classes of subharmonic functions which have nonnegative harmonic majorants in a half-plane
zbMATH Open document ID: 0834.31003
author: A. M. Dzhrbashyan
published in: Journal of Contemporary Mathematical Analysis. Armenian Academy of Sciences
publication date: 1995-10-18T00:00:00Z
review text: The author studies functions \(u(z)\) subharmonic in the upper half-plane \(H\) which have nonnegative harmonic majorants there. The theory is less clean than with functions in the unit disk, because the natural exhaustion of \(H\) includes at each stage part of \(\partial H= \mathbb{R}\). The object is to make minimal assumptions on the boundary values of \(u\). The author introduces a function \(\Omega (x)\), \(-\infty< x< \infty\), with \(\Omega (x)> C(1+ x^2)^{-1}\) and the class \(S(\Omega)\) of subharmonic \(u\) in \(H\) with the standard condition \[ \liminf_{R\to \infty} R^{-1} \int_0^\pi u^+ (Re^{i \theta}) \sin (\theta) d\theta< \infty \tag \(*\) \] augmented by a somewhat intricate condition in terms of \(\Omega\) about the boundary values of \(u\) on \(\mathbb{R}\). He then shows that a representation analogous to that of Nevanlinna holds for functions of \(S(\Omega)\): \(u\) is a Green potential \(+\) the Poisson integral of a (signed) measure \(\mu\); \(\Omega\) plays a role in the exact form of \(\mu\). \{Reviewer's remarks. At the very end of \S 2, replace (1.20) by (1.21). The region \(G_{P,p}^{ (+)}\) in \S 2 is not a `segment', but the intersection of a disk with a half-plane\}.
Mathematics Subject Classification ID: 31A10
Mathematics Subject Classification ID: 30D50
Mathematics Subject Classification ID: 30C15
Mathematics Subject Classification ID: 30E20
zbMATH DE Number: 795462
zbMATH Keywords: Nevanlinna factorization
zbMATH Keywords: subharmonic functions
zbMATH Keywords: harmonic majorants
reviewed by: David Drasin
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 795462