Integrals of subharmonic functions along two curves
type: Item
instance of: scholarly article
title: Integrals of subharmonic functions along two curves
zbMATH Open document ID: 0794.31003
DOI: 10.1016/0019-3577(93)90015-Q
published in: Indagationes Mathematicae. New Series
publication date: 1994-09-08T00:00:00Z
review text: The problem of the constant \(K\) determination which satisfies the inequality \[ \int_{C_ 0} u(z)| dz|\leq K \int_ C u(z)| dz| \] is considered in the paper. \(C_ 0\) is a circle, \(C\) a rectifiable curve surrounding \(C_ 0\), \(u(z)\) a complex function defined in \(\overline{D}\), the adherence of the interior \(D\) of \(C\). If \({\mathcal F}\) is the class of the functions \(u(z)\) defined in \(\overline{D}\), the author denotes with \(K({\mathcal F},C)\) the least constant satisfying the above inequality and he considers the unit circle \(| z|=1\) as \(C_ 0\). The main results of the paper are contained in four theorems with the following contents: (i) If \(C\) is convex and \({\mathcal F}\) is the class \(S^ +\) of the subharmonic positive functions in \(D\) and upper semicontinuous in \(\overline{D}\), \(K<2\); (ii) If \(C\) is a circle, then \(K({\mathcal A},C)\leq 1\), where \({\mathcal A}= \{v\mid v=e^ u\), \(u\in S\}\); \[ K(S^ +,C)^{-1}= \inf_{0\leq\theta\leq2\pi} |(\partial/\partial \theta)\phi(e^{i\theta})|, \tag{iii} \] where \(w=\phi(z)\) maps \(| w|<1\) conformally onto \(D\); (iv) \(K(S^ +,C)+ K({\mathcal H}^ +,C)\) and \(K({\mathcal A},C)= K({\mathcal P},C)\), where \({\mathcal H}^ +\) is the class of positive, harmonic functions in \(D\) and continuous in \(\overline{D}\), \({\mathcal P}= \{u\mid u(z)= | P(z)|\}\), \(P(z)\), a polynomial.
Mathematics Subject Classification ID: 31A05
Mathematics Subject Classification ID: 30A10
zbMATH DE Number: 549283
zbMATH Keywords: integral inequality for subharmonic functions in \(\mathbb{R}^ 2\)
author: Walter K. Hayman
reviewed by: Vasile Ionescu
MaRDI profile type: Publication
cites work: Harmonic measure and arclength
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description: scientific article; zbMATH DE number 549283