Composition of entire and analytic functions in the unit ball

In this paper, we investigate a composition of entire function several complex variables and analytic in the unit ball. We modified early known results with conditions providing equivalence boundedness $L$-index direction for such $l$-index initial one variable, where continuous $L:\mathbb{B}^n\to \mathbb{R}_+$ is constructed by $l: \mathbb{C}^m\to \mathbb{R}_+.$ Taking into account new ideas from recent on functions, remove condition that directional derivative inner $\Phi$ does not equal to zero. Instead construct greater $L(z)$ which $F(z)=f(\underbrace{\Phi(z),\ldots,\Phi(z)}_{m\text{ times}})$ has bounded direction, $f\colon \mathbb{C}$ an $(1,\ldots,1)$, $\Phi\colon \mathbb{B}^n\to ball.
 weaken $|\partial_{\mathbf{b}}^k\Phi(z)|\le K|\partial_{\mathbf{b}}\Phi(z)|^k$ all $z\in\mathbb{B}^n$, $K\geq 1$ constant, $\mathbf{b}\in\mathbb{C}^n\setminus\{0\}$ given $${\partial_{\mathbf{b}} F(z)}:=\sum\limits_{j=1}^{n}\!\frac{\partial F(z)}{\partial z_{j}}{b_{j}}, \ \partial_{\mathbf{b}}^k F(z):=\partial_{\mathbf{b}}\big(\partial_{\mathbf{b}}^{k-1} F(z)\big).$$ It replaced K(l(\Phi(z)))^{1/(N_{\mathbf{1}}(f,l)+1)}|\partial_{\mathbf{b}}\Phi(z)|^k$, $N_{\mathbf{1}}(f,l)$ $f$ $\mathbf{1}=(1,\ldots,1).$ The described result improvement previous one. also one-dimensional case $n=1,$ $m=1$, i.e. disc $f: \mathbb{C}\to\mathbb{C}$ $l$-index.