SOME CRITERIA OF BOUNDEDNESS OF L-INDEX
IN A DIRECTION FOR SLICE HOLOMORPHIC
FUNCTIONS IN THE UNIT BALL
Andriy Bandura1, Lyubov Shegda2,
Oleh Skaskiv2, Liana Smolovyk4
1,2,4Department of Advanced Mathematics
Ivano-Frankivsk National Technical University
of Oil and Gas
15 Karpatska str., Ivano-Frankivsk - 76019, UKRAINE
3Department of Mechanics and Mathematics
Ivan Franko National University of Lviv
1 Universytetska str., Lviv - 79000, UKRAINE

Abstract

Let $\mathbf{b}\in\mathbb{C}^n\setminus\{\mathbf{0}\}$ be a fixed direction and $\mathbf{L}: \mathbb{B}^n\to\mathbb{R}_+$ be a positive continuous function such that $L(z)>\frac{\beta\vert\mathbf{b}\vert}{1-\vert z\vert},$ where $\beta>1$ is some constant. We consider slice holomorphic functions of several complex variables in the unit ball, i.e. we study functions which are analytic in intersection of every slice $\{z^0+t\mathbf{b}: t\in\mathbb{C}\}$ with the unit ball $\mathbb{B}^n=\{z\in\mathbb{C}^n: \ \vert z\vert:=\sqrt{\vert z\vert _1^2+\ldots+\vert z_n\vert^2}<1\}$ for any $z^0\in\mathbb{B}^n$. For functions from this class we prove some criteria of boundedness of $L$-index in the direction describing local behavior of maximum modulus, minimum modulus of the slice holomorphic function and providing estimates of logarithmic derivative and distribution of zeros. Moreover, we obtain an analog of logarithmic criterion. Note that the hypothesis on holomorphy in one direction together with the hypothesis on joint continuity do not imply holomorphy in whole $n$-dimensional unit ball.

Citation details of the article



Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 34
Issue: 4
Year: 2021

DOI: 10.12732/ijam.v34i4.13

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