Analytic functions in a bidisc of bounded $\mathbf{L}$-index in joint variables

TL;DR

This paper generalizes the concept of bounded L-index in joint variables to analytic functions in a bidisc, providing criteria for boundedness, local behavior descriptions, and improved conditions for boundedness.

Contribution

It extends the bounded L-index concept to bidisc functions and offers new criteria and improvements over existing conditions for boundedness.

Findings

Criteria for boundedness of L-index in bidisc functions

Descriptions of local behavior of partial derivatives

Improved sufficient conditions for boundedness

Abstract

A concept of boundedness of L-index in joint variables (see in Bordulyak M.T. The space of entire in $\mathbb{C}^n$ functions of bounded L-index, Mat. Stud., 4 (1995), 53--58. (in Ukrainian)) is generalised for analytic in a bidisc function. We proved criteria of boundedness of $\mathbf{L}$-index in joint variables which describe local behaviour of partial derivative and give an estimate of maximum modulus on a skeleton of polydisc. Some improvements of known sufficient conditions of boundednees of $\mathbf{L}$-index in joint variables are obtained.