Ball exhaustion of the Reinhardt domain: properties of analytic functions of bounded $\mathbf{L}$-index in joint variables

This article is a continuation of the research on the concept of an analytic function of a bounded $\mathbf{L}$-index, conducted during 2019-2025 in the articlesA. Bandura, T. Salo and O. Skaskiv, in which the construction of a complete and exhaustive theory of analytic functions of a bounded $\mathbf{L}$-index in joint variables in an arbitrary complete Reinhardt domain was successfully started. In this article, the criterion for the boundedness of the $\mathbf{L}$-index of a function analytic in a Reinhardt domain in terms of its maximum modulus on balls with an arbitrary center and an arbitrary radius is proved.