Theorems of Morera type for domains with weak cone condition
type: Item
instance of: scholarly article
title: Theorems of Morera type for domains with weak cone condition
zbMATH Open document ID: 0846.30001
published in: Russian Mathematics
publication date: 1995-12-14T00:00:00Z
review text: The author gives two theorems of Morera type in the complex plane and three multidimensional theorems of Morera type. We consider two typical results. If \(G\) is an open set in the complex plane \(\mathbb{C}\), we write \(G\in A(\alpha, H)\) if for any \(z\in G\) there exists \(\theta\in [0, 2\pi)\) such that \(z+ T(\alpha, H) e^{i\theta}\in G\), where \[ T(\alpha, H):= \{z\in \mathbb{C}: 0\leq \text{Re } z\leq H, |\arg z|\leq \alpha\}. \] Theorem 1: Let \(\alpha\leq \pi/3\), \(G\in A(\alpha, H)\), \(f\in C(G)\), and \(\int_{\partial S} f(z) dz= 0\) for any circular sector \(S\subset G\) with a fixed opening \(\varphi\leq \alpha\) and a radius \(r< H\sin \alpha/2(1+ \sin \alpha)\). Then \(f\) is holomorphic in \(G\). Let \(\omega_\nu(z):= dz\wedge dz[\nu]\), \(1\leq \nu\leq n\), \(z= (z_1,\dots, z_n)\in \mathbb{C}^n\) and \(B^n_R\subset \mathbb{C}^n\) be the open ball \(|z|< R\). Theorem 4: Assume \(0< \varepsilon< R\), \(f\in C(B^n_R)\), and \(\int_{\partial D} f(z) \omega_\nu(z)= 0\) for \(1\leq \nu\leq n\) and for any ball \(D\subset B^n_R\) with a center in \(B^n_\varepsilon\). Then \(f\) is holomorphic in \(B^n_R\).
Mathematics Subject Classification ID: 30A99
Mathematics Subject Classification ID: 32A10
zbMATH DE Number: 820003
zbMATH Keywords: theorem of Morera type
zbMATH Keywords: multidimensional theorems
author: Valeriĭ V. Volchkov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 820003