Analytic functions in the unit ball of bounded Lindex: asymptotic and local properties 

Author 
andriykopanytsia@gmail.com, olskask@gmail.com
Department of Advanced Mathematics, IvanoFrankivsk National Technical University of Oil and Gas, IvanoFrankivsk, 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 vectorfunction, $\mathbb{B}^n$ 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 higherorder 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 higherorder systems of PDE

DOI 
doi:10.15330/ms.48.1.3773

Reference 
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Pages 
3773

Volume 
48

Issue 
1

Year 
2017

Journal 
Matematychni Studii

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