On some problem for entire functions of unbounded index in any direction

Hural, I. M.

doi:10.15330/ms.51.1.107-110

2019

DOI: 10.15330/ms.51.1.107-110

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On some problem for entire functions of unbounded index in any direction

I. M. Hural¹

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Cited by 4 publications

(2 citation statements)

References 11 publications

(17 reference statements)

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“…In addition, if n = 1 and L = l, then it becomes the definition of univariate entire function of bounded l-index [16], and if, finally, l ≡ 1, then we obtain the definition of the entire function having a bounded (finite) index [11]. Entire functions of bounded L-index in direction are considered in [4,8].…”

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confidence: 99%

Uniform estimates for local properties of analytic functions in a complete Reinahrdt domain

Bandura,

Salo

2024

Mat. Stud.

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Using recent estimates of maximum modulus for partial derivatives of the analytic functions with bounded $\mathbf{L}$-index in joint variables we describe maximum modulus of these functions at the polydisc skeleton with given radii by the maximum modulus with lesser radii. Such a description is sufficient and necessary condition of boundedness of $\mathbf{L}$-index in joint variables for functions which are analytic in a complete Reinhardt domain. The vector-valued function $\mathbf{L}$ is a positive and continuous function in the domain and its values at a point is greater than reciprocal of distance from the point to the boundary of the Reinhardt domain multiplied by some constant.

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confidence: 99%

Uniform estimates for local properties of analytic functions in a complete Reinahrdt domain

Bandura,

Salo

2024

Mat. Stud.

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“…The first approach is based on a directional derivative in a definition of function having bounded index and leads to the notion of function of bounded L-index in direction where L is some positive continuous function defined in a unit ball or in C n . They implemented this approach for entire functions of several complex variables [7,17] and for functions analytic in the unit ball (see [2,6]). 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.…”

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Entire Bivariate Functions of Exponential Type II

Bandura

Nuray

2023

Mat. Stud.

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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.

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Some Results on Composition of Analytic Functions in a Unit Polydisc

Bandura,

Kurliak,

Skaskiv

2024

Universal Journal of Mathematics and Applications

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The manuscript is an attempt to consider all methods which are applicable to investigation a directional index for composition of an analytic function in some domain and an entire function. The approaches are applied to find sufficient conditions of the $L$-index boundedness in a direction $\mathbf{b}\in\mathbb{C}^n\setminus\{\mathbf{0}\}$, where the continuous function $L$ satisfies some growth condition and the condition of positivity in the unit polydisc. The investigation is based on a counterpart of the Hayman Theorem for the class of analytic functions in the polydisc and a counterpart of logarithmic criterion describing local conduct of logarithmic derivative modulus outside some neighborhoods of zeros. The established results are new advances for the functions analytic in the polydisc and in multidimensional value distribution theory.

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On some problem for entire functions of unbounded index in any direction

Cited by 4 publications

References 11 publications

Uniform estimates for local properties of analytic functions in a complete Reinahrdt domain

Uniform estimates for local properties of analytic functions in a complete Reinahrdt domain

Entire Bivariate Functions of Exponential Type II

Some Results on Composition of Analytic Functions in a Unit Polydisc

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