Composition, product and sum of analytic functions of bounded Lindex in direction in the unit ball 

Author 
andriykopanytsia@gmail.com
IvanoFrankivsk National Technical University of Oil and Gas, IvanoFrankivsk, 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 onedimensional case these results are new for functions analytic in the unit disc.

Keywords 
bounded index; bounded Lindex in direction; analytic function; unit ball; composite function;
bounded lindex; sum; existence theorem

DOI 
doi:10.15330/ms.50.2.115134

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Pages 
115134

Volume 
50

Issue 
2

Year 
2018

Journal 
Matematychni Studii

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