Composition, product and sum of analytic functions of bounded L-index in direction in the unit ball

Author
A. I. Bandura
Ivano-Frankivsk National Technical University of Oil and Gas, Ivano-Frankivsk, Ukraine
Abstract
In this paper, we investigate a composition of entire function of one variable and analytic function in the unit ball. There are obtained conditions which provide equivalence of bounded\-ness of $L$-index in a direction for such a composition and boundedness of $l$-index of initial function of one variable, where the continuous function $L\colon \mathbb{B}^n\to \mathbb{R}_+$ is constructed by the continuous function $l\colon \mathbb{C}\to \mathbb{R}_+.$ We present sufficient conditions for boundedness of $L$-index in the direction for sum and for product of functions analytic in the unit ball. The class of analytic functions in the unit ball having bounded $L$-index in direction is very wide because it contains all analytic functions with bounded multiplicities of zeros on every complex line $\{z^0+t\mathbf{b}\colon t\in\mathbb{C}\}$. It is a statement of proved existence theorem. In the one-dimensional case these results are new for functions analytic in the unit disc.
Keywords
bounded index; bounded L-index in direction; analytic function; unit ball; composite function; bounded l-index; sum; existence theorem
DOI
doi:10.15330/ms.50.2.115-134
Reference
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Pages
115-134
Volume
50
Issue
2
Year
2018
Journal
Matematychni Studii
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