Slice Holomorphic Functions in the Unit Ball Having a Bounded L-Index in Direction

Let $\mathbf{b}\in {\mathbb{C}}^n\backslash \left\{\mathbf{0}\right\}$ be a fixed direction. We consider slice holomorphic functions of several complex variables in the unit ball, i.e., we study functions that are analytic in the intersection of every slice $\{z^0+t\mathbf{b}:t\in \mathbb{C}\}$ with the unit ball ${\mathbb{B}}^n=\{z\in {\mathbb{C}}^{:}\left|z\right|:=\sqrt{{\left|z\right|}_1^2+\dots +{\left|z_n\right|}^2}<1\}$ for any $z^0\in {\mathbb{B}}^n$. For this class of functions, there is introduced a concept of boundedness of L-index in the direction $\mathbf{b}$, where $\mathbf{L}:{\mathbb{B}}^n\to {\mathbb{R}}_{+}$ is a positive continuous function such that $L\left(z\right)>\frac{\beta \left|\mathbf{b}\right|}{1-\left|z\right|},$ where $\beta >1$ is some constant. For functions from this class, we describe a local behavior of modulus of directional derivatives on every ’circle’ $\{z+t\mathbf{b}:|t|=r/L(z\left)\right\}$ with $r\in (0;\beta ],$$t\in \mathbb{C},$$z\in {\mathbb{C}}^n$. It is estimated by the value of the function at the center of the circle. Other propositions concern a connection between the boundedness of L-index in the direction $\mathbf{b}$ of the slice holomorphic function F and the boundedness of $l_z$-index of the slice function $g_z\left(t\right)=F(z+t\mathbf{b})$ with $l_z\left(t\right)=L(z+t\mathbf{b}).$ In addition, we show that every slice holomorphic and joint continuous function in the unit ball has a bounded L-index in direction in any domain compactly embedded in the unit ball and for any continuous function $L:{\mathbb{B}}^n\to {\mathbb{R}}_{+}.$ View Full-Text