Entire functions of several variables: behaviour of directional derivatives

Abstract

Let $h$ be a positive increasing on $[0;+\infty)$ function such that $h\Big(x+\frac{1}{h(x)}\Big)=O(h(x))$ as $x\to +\infty$. For a Lebesgue measurable set $E\subset[0;+\infty)$ of finite Lebesgue measure $\mathop{\rm meas}E=\int_{E}dx$, we define the asymptotic $h$-density of $E$ on $+\infty$ by \[{D}_{h}(E)= \varlimsup_{R \rightarrow +\infty} h(R)\cdot \mathop{\rm meas }(E \cap [R;+\infty)).\]

Consider the class $\mathcal{H}^{p}$ of an entire functions in $\mathbb{C}^{p}$, that are bounded in an arbitrary domain $\Pi_{R}=\{z=(z_1,\ldots ,z_p)\in\mathbb{C}^p\colon \text{Re} z_j<R_j\}$, $R=(R_{1},\ldots,R_{p})\in\mathbb{R}^{p}_{+}$ as well as in $G(r,A)=G+ rA$ for every fixed $A\in\mathbb{R}^p$ and for each $r>0$, where $G$ is a complete polylinear domain. For a function $F\in \mathcal{H}^p$ and $A\in\mathbb {R}^p$, let $F'_A(w)$ denotes the derivative of $F$ in the direction of $A$ at the point $w\in\mathbb{C}$. Let $F^{(k)}_A(w)=(F^{(k-1)}_A(w))'_A$ denotes the $k$th derivative in the direction of $A$ at the point $w\in\mathbb{C}$. We also denote \[S_F(r,A):=\sup\big\{|F(z)|\colon z\in G(r,A)\big\}=\sup\big\{|F(z)|\colon z\in \partial G(r,A)\big\},\]\[L_F(r,A)=(\ln S_F(r,a))'_+.\]

We prove the following statement. Let $F\in \mathcal{H}^p$ and $A\in\mathbb {R}^p$ be such that $L_F(r,A)\uparrow+\infty$ as $r\to+\infty$. Suppose that $\Phi$ is a positive increasing on $[0;+\infty)$ function satisfying $u(r)\ge \Phi(r)$ for all $r\ge r_0$, and $h(r)=o(\Phi(r))$ as $r\to +\infty$. Then there exists a set $E\subset\mathbb {R}_+$ of zero asymptotical $h$-density, i.e. $D_h(E)=0$, such that for every $k\in\mathbb{N}$ we have \[F^{(k)}_A(w)=(1+o(1))\, L^k_F(r,A)\, F(w) \;\;\text{as}\;\; r\to +\infty,\;\; r\in\mathbb {R}_+\setminus E,\] for all points $w\in\partial G(r,A)$ satisfying the inequality $|F(w)|\geq S_F(r,A)/(1+\varepsilon(r))$, where $\varepsilon(r)$ is a given arbitrary function such that $\varepsilon(r)\to + 0$ as $r\to +\infty$.