An extremum Problem related to Morera's theorem
type: Item
instance of: scholarly article
title: An extremum Problem related to Morera's theorem
zbMATH Open document ID: 0899.30002
DOI: 10.1007/BF02305150
published in: Mathematical Notes
publication date: 1998-11-15T00:00:00Z
review text: The following extension of Morera's theorem has been proved by \textit{C. Berenstein} and \textit{R. Gay} [J. Anal. Math. 52 133-166 (1989; Zbl 0668.30037)]: Let \(T \subseteq B_r= \{z\in\mathbb{C}:| z| <r\}\) be a given triangle, \(f\) be a continuous function in \(B_R\), and \(2r\leq R\). Then \(f\) is holomorphic in \(B_R\) iff \[ \int_{\partial (\sigma T)} f(z)dz=0, \quad \forall \sigma\in M: \sigma T\subseteq B_R. \tag{1} \] Here \(M\) is the group of Euclidean motions of the complex plane \(\mathbb{C}\). The present author has studied the problem of finding the minimum value \(R=R(T)\) such that for any function \(f\in C(B_R)\) condition (1) implies that \(f\) is holomorphic and has found the value for squares and half-disks. In the article under review the author proves that for regular triangles \(T\) of side \(a\) \(R(T)= a\sqrt 3/2\).
Mathematics Subject Classification ID: 30A05
zbMATH DE Number: 1158963
zbMATH Keywords: description of holomorphic functions
zbMATH Keywords: Morera's theorem
author: Valeriĭ V. Volchkov
reviewed by: Alexander I. Kheyfits
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf02305150
OpenAlex ID: W4231660833
description: scientific article; zbMATH DE number 1158963