# Analytic functions in the unit ball of bounded L-index: asymptotic and local properties

Author
Department of Advanced Mathematics, Ivano-Frankivsk National Technical University of Oil and Gas, Ivano-Frankivsk, Ukraine; Department of Mechinics and Mathematics, Ivan Franko National University of Lviv, Lviv, Ukraine
Abstract
We have generalized some criteria of boundedness of $\mathbf{L}$-index in joint variables for analytic functions in the unit ball, where $\mathbf{L}\colon \mathbb{B}^n\to \mathbb{R}^n_+$ is a continuous vector-function, $\mathbb{B}^n$ is the unit ball in $\mathbb{C}^n.$ One of propositions gives an estimate of the coefficients of power series expansions by a dominating homogeneous polynomial for analytic functions in the unit ball. Also we provide growth estimates of these functions. They describe the behavior of maximum modulus of analytic function on a skeleton in a polydisc by behavior of the function $\mathbf{L}.$ Most of our results are based on polydisc exhaustion of the unit ball. Nevertheless, we have generalized criteria of boundedness of $\mathbf{L}$-index in joint variables which describe local behavior of partial derivatives on sphere in $\mathbb{C}^n.$ The proposition uses a ball exhaustion. An analog of Hayman's theorem is applied to investigation of boundedness of $\mathbf{L}$-index in joint variables for analytic solutions in the unit ball of some linear higher-order systems of PDE's. There were found sufficient conditions providing the boundedness. Growth estimates of analytic solutions in the unit ball are also obtained.
Keywords
analytic function in a ball; bounded index in joint variables; maximum modulus; partial derivative; Cauchys integral formula; geometric exhaustion; growth estimates; linear higher-order systems of PDE
DOI
doi:10.15330/ms.48.1.37-73
Reference
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Pages
37-73
Volume
48
Issue
1
Year
2017
Journal
Matematychni Studii
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