Boundedness of the $L$-index in the direction of the composition of slice entire functions and slice holomorphic functions in the unit ball

Abstract

UDC 517.55

We study the composition $F(z):=f(\underbrace{\Phi(z),\ldots,\Phi(z)}_{m\text{ times}})\colon\mathbb{C}^n\to\mathbb{C}$, $n\geq 1, m\geq 1,$ where $\Phi\colon\mathbb{B}^n\to\mathbb{C}$ is a slice holomoprphic function in the unit ball and  $f\colon\mathbb{C}^m\to\mathbb{C}$ is a slice holomoprphic function in the whole $m$-dimensional complex space $\mathbb{C}^m$, i.e., a slice function $g_z(\tau)=f(z+\mathbf{b}\tau)$ is an entire function of $\tau\in\mathbb{C}$ for any fixed $z\in\mathbb{C}^m$ and a given direction $\mathbf{b}\in\mathbb{C}^m$.  Theslice holomorphicity in a unit ball $\mathbb{B}^n$ means that, for a fixed direction $\mathbf{b}\in\mathbb{C}^n\setminus\{\mathbf{0}\}$ and any point $z^0\in \mathbb{B}^n$  of the unit ball, the function is holomorphic on its restriction to the slice $\{z^0+t\mathbf{b}\colon t\in\mathbb{C}\}\cap\mathbb{B}^n.$  An additional assumption about equicontinuity  of these functions allows us to construct an analog of the theory of entire functions with bounded index. This analog can be applied to the investigation of the properties of slice-holomorphic solutions of directional differential equations that describe local behaviors and  the distribution of values. We establish conditions sufficient for the boundedness of the $L$-index in the direction $\mathbf{b}$ for the function $F(z).$ Some of the obtained results are also new in one-dimensional case, i.e., for $n=1$ and $m=1.$ The indicated conditions are obtained by using two different approaches: an analog of Hayman's theorem and analog of the logarithmic criterion.  We also present examples of functions whose composition satisfies all conditions of only one of the obtained theorems.