The functions of Bounded L-Index in the Collection of Variables Analytic in 𝔻×ℂ

The concept of bounded $L$-index in a direction $\mathbf{b}=(b_1,\ldots,b_n)\in\mathbb{C}^n\setminus\{\mathbf{0}\}$ is generalized for a class of analytic functions in the unit polydisc, where $L$ is some continuous function such that for every $z=(z_1,\ldots,z_n)\in\mathbb{D}^n$ one has $L(z)>\beta\max_{1\le j\le n}\frac{|b_j|}{1-|z_j|},$ $\beta=\mathrm{const}>1,$ $\mathbb{D}^n$ is the unit polydisc, i.e. $\mathbb{D}^n=\{z\in\mathbb{C}^n: |z_j|\le 1, j\in\{1,\ldots,n\}\}.$ For functions from this class we obtain sufficient and necessary conditions providing boundedness of $L$-index in the direction. They describe local behavior of maximum modulus of derivatives for the analytic function $F$ on every slice circle $\{z+t\mathbf{b}: |t|=r/L(z)\}$ by their values at the center of the circle, where $t\in\mathbb{C}.$ Other criterion describes similar local behavior of the minimum modulus via the maximum modulus for these functions. We proved an analog of the logarithmic criterion desribing estimate of logarithmic derivative outside some exceptional set by the function $L$. The set is generated by the union of all slice discs $\{z^0+t\mathbf{b}: |t|\le r/L(z^0)\}$, where $z^0$ is a zero point of the function $F$. The analog also indicates the zero distribution of the function $F$ is uniform over all slice discs. In one-dimensional case, the assertion has many applications to analytic theory of differential equations and infinite products, i.e. the Blaschke product, Naftalevich-Tsuji product. Analog of Hayman's Theorem is also deduced for the analytic functions in the unit polydisc. It indicates that in the definition of bounded $L$-index in direction it is possible to remove the factorials in the denominators. This allows to investigate properties of analytic solutions of directional differential equations.

Section: Introductionmentioning

confidence: 99%

Analytic in the unit polydisc functions of bounded L-index in direction

Bandura,

Salo

2023

“…Another approach used all possible partial derivatives in a definition of a function having bounded index and led to the notion of functions of bounded L-index in joint variables, where L is some vector-valued positive continuous function. This approach was proposed for entire functions of several complex variables [3,8,9], analytic functions in the unit ball [1], for analytic functions in the polydisc [4], for functions analytic in the Cartesian product of the complex plane and the unit ball [5]. These functions have applications in analytic theory of ordinary differential equations [11,[28][29][30], partial differential equations [6] and their system [9,20].…”

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confidence: 99%

Entire Bivariate Functions of Exponential Type II

Bandura

Nuray

2023

Let $f(z_{1},z_{2})$ be a bivariate entire function and $C$ be a positive constant. If $f(z_{1},z_{2})$ satisfies the following inequality for non-negative integer $M$, for all non-negative integers $k,$ $l$ such that $k+l\in\{0, 1, 2, \ldots, M\}$, for some integer $p\ge 1$ and for all $(z_{1},z_{2})=(r_{1}e^{\mathbf{i}\theta_{1}},r_{2}e^{\mathbf{i}\theta_{2}})$ with $r_1$ and $r_2$ sufficiently large:\begin{gather*}\sum_{i+j=0}^{M}\frac{\left(\int_{0}^{2\pi}\int_{0}^{2\pi}|f^{(i+k,j+l)}(r_{1}e^{\mathbf{i}\theta_{1}},r_{2}e^{\mathbf{i}\theta_{2}})|^{p}d\theta_{1}\theta_{2}\right)^{\frac{1}{p}}}{i!j!}\ge \\\ge \sum_{i+j=M+1}^{\infty}\frac{\left(\int_{0}^{2\pi}\int_{0}^{2\pi}|f^{(i+k,j+l)}(r_{1}e^{\mathbf{i}\theta_{1}},r_{2}e^{\mathbf{i}\theta_{2}})|^{p}d\theta_{1}\theta_{2}\right)^{\frac{1}{p}}}{i!j!},\end{gather*}then $f(z_{1},z_{2})$ is of exponential type not exceeding\[2+2\log\Big(1+\frac{1}{C}\Big)+\log[(2M)!/M!].\]If this condition is replaced by related conditions, then also $f$ is of exponential type.

“…In recent years, analytic functions of several variables with bounded index have been intensively investigated. The main objects of investigations are such function classes: entire functions of several variables [2,3,4,9,19,20], functions analytic in a polydisc [10], in a ball [5,6,12] or in the Cartesian product of the complex plane and the unit disc [11]. For entire functions and analytic functions in a ball there were proposed two approaches to introduce a concept of index boundedness in a multidimensional complex space.…”

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confidence: 99%

“…Using the definition of index boundedness in frame, the definition of q(η), inequalities (11) and (13), for k z p ≤ N we have…”

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confidence: 99%

Entire functions of bounded index in frame

Bandura

2020

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 $|t|=\eta$: {\it 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 iffor 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$$\max\left\{\left|{\partial_{\mathbf{b}(z)}^{k_{0}} F(z+t\mathbf{b}(z))}\right|\colon\ |t|\leq\eta \right\}\leqP_{1}\left|\partial_{\mathbf{b}(z)}^{k_{0}}{F(z)}\right|$$holds.