COMPOSITION OF SLICE ENTIRE FUNCTIONS AND BOUNDED L-INDEX IN DIRECTION

Abstract

We study the following question: "Let $f: \mathbb{C}\to \mathbb{C}$ be an entire function of bounded $l$-index, $\Phi: \mathbb{C}^n\to \mathbb{C}$ be a slice entire function, $n\geq2,$ $l:\mathbb{C}\to \mathbb{R}_+$ be a continuous function.We study the following question: "Let $f: \mathbb{C}\to \mathbb{C}$ be an entire function of bounded $l$-index, $\Phi: \mathbb{C}^n\to \mathbb{C}$ be a slice entire function, $n\geq2,$ $l:\mathbb{C}\to \mathbb{R}_+$ be a continuous function.What is a  positive continuous function $L:\mathbb{C}^n\to \mathbb{R}_+$ and a direction $\mathbf{b}\in\mathbb{C}^n\setminus\{\mathbf{0}\}$  such that the composite function $f(\Phi(z))$ has bounded $L$-index in the direction~$\mathbf{b}$?". In the present paper, early known results on boundedness of $L$-index in direction for the composition of entire functions$f(\Phi(z))$ are generalized to the case where  $\Phi: \mathbb{C}^n\to \mathbb{C}$ is a slice entire function, i.e.it is an entire function on a complex line $\{z^0+t\mathbf{b}: t\in\mathbb{C}\}$ for any $z^0\in\mathbb{C}^n$ andfor a given direction $\mathbf{b}\in\mathbb{C}^n\setminus\{\mathbf{0}\}$.These slice entire functions are not joint holomorphic in the general case. For~example, it allows consideration of functions which are holomorphic in variable $z_1$ and  continuous in variable $z_2.$