Composition of entire function and analytic functions in the unit ball with a vanished gradient - A. I. Bandura; T. M. Salo; O. B. Skaskiv (Математичні Студії)
The composition $H(z)=f(Φ(z))$ is studied, where $f$ is an entire function of a single complex variable and $Φ$ is an analytic function in the $n$-dimensional unit ball with a vanished gradient. We found conditions by the function $Φ$ providing boundedness of the $L$-index in joint variables for the function $H$, if the function $f$ has bounded $l$-index for some positive continuous function $l$ and $L(z)= l(Φ(z))(\max\{1,|Φ_{z_1}'(z)|\},…, \max\{1,|Φ_{z_n}'(z)|\}),$ $z\in\mathbb{B}^n.$ Such a constructed function $L$ allows us to consider a function $Φ$ with a nonempty zero set for its gradient. The obtained results complement earlier published results with $\mathop{grad}Φ(z)=(\frac{\partial Φ(z)}{\partial z_1}, …, \frac{\partial Φ(z)}{\partial z_j},…,\frac{\partial Φ(z)}{\partial z_n})\ne 0.$ Also, we study a more general composition $H(w)=G(Φ(w))$, where $G: \mathbb{C}^n→ \mathbb{C}$ is an entire function of the bounded $L$-index in joint variables, $Φ: \mathbb{B}^m→ \mathbb{C}^n$ is a vector-valued analytic function, and $L: \mathbb{C}^n→\mathbb{R}^n_+$ is a continuous function. If the $L$-index of the function $G$ equals zero, then we construct a function $\widetilde{L}: \mathbb{B}^m→\mathbb{R}^m_+$ such that the function $H$ has bounded $\widetilde{L}$-index in the joint variables $w_1,$ $…,$ $w_m$. These results are also new in one-dimensional case, i.e. for functions analytic in the unit disc.