Some properties of analytic in a ball functions of bounded $\mathbf{L}$-index in joint variables

TL;DR

This paper generalizes the concept of boundedness of the L-index in joint variables for analytic functions in a ball, providing criteria, growth estimates, and applications to partial differential equations.

Contribution

It introduces new boundedness criteria for analytic functions in a ball, extending previous work on L-index in joint variables and applying results to PDE systems.

Findings

Established criteria for boundedness of L-index in a ball

Derived growth estimates for analytic functions of bounded L-index

Applied results to analyze solutions of PDE systems

Abstract

A concept of boundedness of the $\mathbf{L}$-index in joint variables (see in Bandura A. I., Bordulyak M. T., Skaskiv O. B. "Sufficient conditions of boundedness of L-index in joint variables", Mat. Stud. 45 (2016), 12--26. dx.doi.org/10.15330/ms.45.1.12-26) is generalized for analytic in a ball function. There are proved criteria of boundedness of the $\mathbf{L}$-index in joint variables which describe local behavior of partial derivatives and maximum modudus on a skeleton of a polydisc, properties of power series expansion. Also we obtained analog of Hayman's Theorem. As a result, they are applied to study linear higher-order systems of partial differential equations and to deduce sufficient conditions of boundedness of the $\mathbf{L}$-index in joint variables for their analytic solutions and to estimate it growth. We used an exhaustion of ball in $\mathbb{C}^n$ by polydiscs. Also growth estimates of analytic in ball functions of bounded $\mathbf{L}$-index in joint variables are obtained. Note that this paper and a paper "Analytic functions in a bidisc of bounded L-index in joint variables" (arXiv:1609.04190) do not overlap because analytic in a ball and analytic in a bidisc functions are different classes of holomorphic functions. Some results have similar formulations but a bidisc and a ball are particular geometric objects.