New mean value theorems for polyanalytic functions
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.
scientific article; zbMATH DE number 830722
scientific article