New mean value theorems for polyanalytic functions
type: Item
instance of: scholarly article
title: New mean value theorems for polyanalytic functions
zbMATH Open document ID: 0841.30036
DOI: 10.1007/BF02362407
published in: Mathematical Notes
publication date: 1996-01-08T00:00:00Z
review text: In the paper under review the functions obeyed to the mean value formula for the polyanalytic functions on the circles of some fixed radii are investigated. Namely, the author considers the next problem: let \(n\) be a fixed natural number, \(f \in C (\mathbb{C})\) and let, for a fixed \(r > 0\) and all \(z \in \mathbb{C}\), the following equality be valid: \[ \iint_{K (z,r)} (\zeta - z)^n f(\zeta) d \xi d \eta = 0 \quad (\zeta = \xi + i \eta) \] where \(K(z,r) = \{\zeta \in \mathbb{C} : |\zeta - z |\leq r\}\). It is true that \(f\) is \(n\)-analytic (i.e., \((\partial/ \partial \zeta)^n f = 0)\)? In the paper some essentially stronger results than the ``two radii'' Zalcman theorem [[\textit{L. Zalcman}, Arch. Rat. Mech. Analysis. 47, 237-254 (1972; Zbl 0251.30047)]) and also its analogy for the bounded domain are obtained. Also some special cases are considered, when the equation for the mean values is approximately valid but nevertheless, it allows to make conclusion of the function polyanalyticity under certain conditions.
reviewed by: I.E.Tralle
Mathematics Subject Classification ID: 30G20
zbMATH DE Number: 830722
zbMATH Keywords: Bessel inequality
zbMATH Keywords: Bessel functions
zbMATH Keywords: polyanalytic functions
zbMATH Keywords: Zalcman theorem
author: Valeriĭ V. Volchkov
MaRDI profile type: Publication
cites work: A theorem of Fédoroff
cites work: Analyticity and the Pompeiu problem
cites work: Offbeat Integral Geometry
cites work: Q3353339
cites work: Q5629972
cites work: Q3892486
cites work: Automorphic systems of scalar fields
cites work: Uniqueness theorems for multiple lacunary trigonometric series
full work available at URL: https://doi.org/10.1007/bf02362407
OpenAlex ID: W2031871524
description: scientific article; zbMATH DE number 830722