Slice holomorphic functions in the unit ball: boundedness of $L$-index in a direction and related properties - A. I. Bandura; T. M. Salo; O. B. Skaskiv (Математичні Студії)
Let $b\in\mathbb{C}^n\setminus\{0\}$ 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 $\{z^0+tb: t\in\mathbb{C}\}$ with the unit ball $\mathbb{B}^n=\{z\in\mathbb{C}^: \ |z|:=\sqrt{|z|_1^2+…+|z_n|^2}<1\}$ for any $z^0\in\mathbb{B}^n$. For this class of functions we consider the concept of boundedness of $L$-index in the direction $b,$ where $L: \mathbb{B}^n→\mathbb{R}_+$ is a positive continuous function such that $L(z)>\frac{β|b|}{1-|z|}$ and $β>1$ is some constant. For functions from this class we deduce analog of Hayman's Theorem. It is criterion useful in applications to differential equations. We introduce a concept of function having bounded value $L$-distribution in direction for the slice holomorphic functions in the unit ball. It is proved that slice holomorphic function in the unit ball has bounded value $L$-distribution in a direction if and only if its directional derivative has bounded $L$-index in the same direction. Other propositions concern existence theorems. We show that for any slice holomorphic function $F$ with bounded multiplicities of zeros on any slice in the fixed direction there exists such a positive continuous function $L$ that the function $F$ has bounded $L$-index in the direction.