Growth estimates for analytic functions in the unit polydisc with bounded $L$-index in direction

Abstract

This paper investigates the growth properties of analytic functions defined in the unit polydisc of $\mathbb{C}^n$ having bounded $L$-index in a direction. Special attention is devoted to the behavior of such functions under composition and to the relationships between their growth characteristics and the corresponding properties of their components. We establish new estimates for the maximum modulus and propose sufficient conditions under which the composition preserves given growth classes. The obtained results extend several known one-dimensional theorems to the multidimensional setting and refine existing bounds in the theory of entire and analytic functions. Furthermore, we analyze the interplay between the geometry of the polydisc and the growth indicators, revealing specific features that arise in higher dimensions. At the end, we consider one equation from the electromagnetism's theory written by directional derivative and study its analytic solutions.