Boundedness of the L-index in a direction of the sum and product of slice holomorphic functions in the unit ball

Abstract

Let $\mathbf{b}\in\mathbb{C}^n\setminus\{\mathbf{0}\}$ be a fixed direction. 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
$\mathbb{B}^n=\{z\in\mathbb{C}^{n}: \ |z|:=\sqrt{|z|_1^2+\ldots+|z_n|^2}<1\}$ for any
$z^0\in\mathbb{B}^n$. For this class of functions
there is considered the concept of boundedness of $L$-index in the direction $\mathbf{b},$ where
${L}: \mathbb{B}^n\to\mathbb{R}_+$ is a positive continuous function such that
$L(z)>\frac{\beta|\mathbf{b}|}{1-|z|}$ and $\beta>1$ is some constant.
There are presented sufficient conditions that the sum of slice holomorphic functions of bounded $L$-index in direction
belong this class. This class of slice holomorphic functions is closed under the operation of multiplication.