Composition of entire and analytic functions in the unit ball

Abstract

In this paper, we investigate a composition of entire function of several complex variables and analytic function in the unit ball. We modified early known results with conditions providing equivalence of boundedness of $L$-index in a direction for such a composition and boundedness of $l$-index of initial function of one variable, where the continuous function $L:\mathbb{B}^n\to \mathbb{R}_+$ is constructed by the continuous function $l: \mathbb{C}^m\to \mathbb{R}_+.$ Taking into account new ideas from recent results on composition of entire functions, we remove a condition that a directional derivative of the inner function $\Phi$ in the composition does not equal to zero. Instead of the condition we construct a greater function $L(z)$ for which $F(z)=f(\underbrace{\Phi(z),\ldots,\Phi(z)}_{m\text{ times}})$ has bounded $L$-index in a direction, where $f\colon \mathbb{C}^m\to \mathbb{C}$ is an entire function of bounded $l$-index in the direction $(1,\ldots,1)$, $\Phi\colon \mathbb{B}^n\to \mathbb{C}$ is an analytic function in the unit ball.

We weaken the condition $|\partial_{\mathbf{b}}^k\Phi(z)|\le K|\partial_{\mathbf{b}}\Phi(z)|^k$ for all $z\in\mathbb{B}^n$, where $K\geq 1$ is a constant, $\mathbf{b}\in\mathbb{C}^n\setminus\{0\}$ is a given direction and $${\partial_{\mathbf{b}} F(z)}:=\sum\limits_{j=1}^{n}\!\frac{\partial F(z)}{\partial z_{j}}{b_{j}}, \ \partial_{\mathbf{b}}^k F(z):=\partial_{\mathbf{b}}\big(\partial_{\mathbf{b}}^{k-1} F(z)\big).$$ It is replaced by the condition $|\partial_{\mathbf{b}}^k\Phi(z)|\le K(l(\Phi(z)))^{1/(N_{\mathbf{1}}(f,l)+1)}|\partial_{\mathbf{b}}\Phi(z)|^k$, where $N_{\mathbf{1}}(f,l)$ is the $l$-index of the function $f$ in the direction $\mathbf{1}=(1,\ldots,1).$ The described result is an improvement of previous one. It is also a new result for the one-dimensional case $n=1,$ $m=1$, i.e. for an analytic function $\Phi$ in the unit disc and for an entire function $f: \mathbb{C}\to\mathbb{C}$ of bounded $l$-index.