Slice holomorphic functions in several variables with bounded \(L\)-index in direction (Q2306618)

Summary: In this paper, for a given direction \(\mathbf{b} \in \mathbb{C}^n \backslash \{0 \}\) we investigate slice entire functions of several complex variables, i.e., we consider functions which are entire on a complex line \(\{z^0 + t \mathbf{b} : t \in \mathbb{C} \}\) for any \(z^0 \in \mathbb{C}^n\). Unlike to quaternionic analysis, we fix the direction \(\mathbf{b} \). The usage of the term slice entire function is wider than in quaternionic analysis. It does not imply joint holomorphy. For example, it allows consideration of functions which are holomorphic in variable \(z_1\) and continuous in variable \(z_2\). For this class of functions there is introduced a concept of boundedness of \(L\)-index in the direction \(\mathbf{b}\) where \(\mathbf{L} : \mathbb{C}^n \rightarrow \mathbb{R}_+\) is a positive continuous function. We present necessary and sufficient conditions of boundedness of \(L\)-index in the direction. In this paper, there are considered local behavior of directional derivatives and maximum modulus on a circle for functions from this class. Also, we show that every slice holomorphic and joint continuous function has bounded \(L\)-index in direction in any bounded domain and for any continuous function \(L : \mathbb{C}^n \rightarrow \mathbb{R}_+\).