Entire functions of bounded index in frame

Abstract

We introduce a concept of entire functions having bounded index in a variable direction, i.e. in a frame.
An entire function $F\colon\ \mathbb{C}^n\to \mathbb{C}$ is called a function of bounded frame index in a frame $\mathbf{b}(z)$,
if~there exists $m_{0} \in\mathbb{Z}_{+}$ such that for every $m \in\mathbb{Z}_{+}$ and for all $z\in \mathbb{C}^{n}$
one has
$\displaystyle
\frac{|{\partial^{m}_{\mathbf{b}(z)}F(z)}|}{m!}
\leq\max_{0\leq k \leq m_{0}} \frac{|{\partial^{k}_{\mathbf{b}(z)}F(z)}|}{k!},
$
where $\partial^{0}_{\mathbf{b}(z)}F(z)=F(z),$
$\partial^{1}_{\mathbf{b}(z)}F(z)=\sum_{j=1}^n \frac{\partial F}{\partial z_j}(z)\cdot b_j(z),$ 
$\partial^{k}_{\mathbf{b}(z)}F(z)=\partial_{\mathbf{b}(z)}(\partial^{k-1}_{\mathbf{b}(z)}F(z))$ for $k\ge 2$
and $\mathbf{b}\colon\ \mathbb{C}^n\to\mathbb{C}^n$ is a entire vector-valued function.
There are investigated properties of these functions. We established analogs of propositions known for entire functions
of bounded index in direction. The main idea of proof is usage the slice $\{z+t\mathbf{b}(z)\colon\ t\in\mathbb{C}\}$ for given $z\in\mathbb{C}^n.$
We proved the following criterion (Theorem 1) describing local behavior of modulus $\partial_{\mathbf{b}(z)}^kF(z+t\mathbf{b}(z))$ on the circle $|t|=\eta$: 

An entire~function
$F\colon\ \mathbb{C}^n\to\mathbb{C}$ is of bounded frame index in the frame $\mathbf{b}(z)$ if and only if
for each $\eta>0$ there exist
$n_{0}=n_{0}(\eta)\in \mathbb{Z}_{+}$ and $P_{1}=P_{1}(\eta)\geq 1$
such that for every $z\in \mathbb{C}^{n}$ there exists $k_{0}=k_{0}(z)\in \mathbb{Z}_{+},$\
$0\leq k_{0}\leq n_{0},$ for which inequality
$\max\{|\partial_{\mathbf{b}(z)}^{k_{0}} F(z+t\mathbf{b}(z))|: |t|\leq\eta \}\leq P_{1}|\partial_{\mathbf{b(z)}}^{k_{0}}F(z)|$
holds.