scholarly journals Analytic functions in the unit ball of bounded L-index in joint variables and of bounded 𝐿-index in direction: a connection between these classes

2019 ◽  
Vol 52 (1) ◽  
pp. 82-87 ◽  
Author(s):  
Andriy Bandura ◽  
Oleh Skaskiv
Keyword(s):  
Entire Function ◽  
Continuous Function ◽  
Unit Ball ◽  
Analytic Functions ◽  
Entire Functions ◽  
Negative Answer ◽  
Original Formulation ◽  
Positive Continuous Function

Abstract We give negative answer to the question of Bordulyak and Sheremeta for more general classes of entire functions than in the original formulation: Does index boundedness in joint variables for an entire function F imply index boundedness in the variable zj for the function F? This question is addressed for entire functions of bounded L-index in joint variables and entire functions of bounded L-index in direction. We also present a class of analytic functions in the unit ball which has bounded L-index in joint variables and has unbounded l-index in the variables z1 and z2 for any positive continuous function l : B2 → C.

COMPOSITION OF SLICE ENTIRE FUNCTIONS AND BOUNDED L-INDEX IN DIRECTION

2021 ◽  
Vol 9 (1) ◽  
pp. 29-38
Author(s):  
O. Skaskiv ◽  
A. Bandura
Keyword(s):  
Entire Function ◽  
Continuous Function ◽  
Entire Functions ◽  
Complex Line ◽  
Positive Continuous Function ◽  
Composite Function

We study the following question: "Let $f: \mathbb{C}\to \mathbb{C}$ be an entire function of bounded $l$-index, $\Phi: \mathbb{C}^n\to \mathbb{C}$ be a slice entire function, $n\geq2,$ $l:\mathbb{C}\to \mathbb{R}_+$ be a continuous function.We study the following question: "Let $f: \mathbb{C}\to \mathbb{C}$ be an entire function of bounded $l$-index, $\Phi: \mathbb{C}^n\to \mathbb{C}$ be a slice entire function, $n\geq2,$ $l:\mathbb{C}\to \mathbb{R}_+$ be a continuous function.What is a  positive continuous function $L:\mathbb{C}^n\to \mathbb{R}_+$ and a direction $\mathbf{b}\in\mathbb{C}^n\setminus\{\mathbf{0}\}$  such that the composite function $f(\Phi(z))$ has bounded $L$-index in the direction~$\mathbf{b}$?". In the present paper, early known results on boundedness of $L$-index in direction for the composition of entire functions$f(\Phi(z))$ are generalized to the case where  $\Phi: \mathbb{C}^n\to \mathbb{C}$ is a slice entire function, i.e.it is an entire function on a complex line $\{z^0+t\mathbf{b}: t\in\mathbb{C}\}$ for any $z^0\in\mathbb{C}^n$ andfor a given direction $\mathbf{b}\in\mathbb{C}^n\setminus\{\mathbf{0}\}$.These slice entire functions are not joint holomorphic in the general case. For~example, it allows consideration of functions which are holomorphic in variable $z_1$ and  continuous in variable $z_2.$

scholarly journals Note on composition of entire functions and bounded $L$-index in direction

2021 ◽  
Vol 55 (1) ◽  
pp. 51-56
Author(s):  
A. I. Bandura ◽  
O. B. Skaskiv ◽  
T. M. Salo
Keyword(s):  
Entire Function ◽  
Continuous Function ◽  
Entire Functions ◽  
Directional Derivative ◽  
Inner Function ◽  
Positive Continuous Function ◽  
Composite Function ◽  
Equal Zero

We study the following question: ``Let $f\colon \mathbb{C}\to \mathbb{C}$ be an entire function of bounded $l$-index, $\Phi\colon \mathbb{C}^n\to \mathbb{C}$ an be entire function, $n\geq2,$ $l\colon \mathbb{C}\to \mathbb{R}_+$ be a continuous function. What is a positive continuous function $L\colon \mathbb{C}^n\to \mathbb{R}_+$ and a direction $\mathbf{b}\in\mathbb{C}^n\setminus\{\mathbf{0}\}$ such that the composite function $f(\Phi(z))$ has bounded $L$-index in the direction~$\mathbf{b}$?'' In the present paper, early known result on boundedness of $L$-index in direction for the composition of entire functions $f(\Phi(z))$ is modified. We replace a condition that a directional derivative of the inner function $\Phi$ in a direction $\mathbf{b}$ does not equal zero. The condition is replaced by a construction of greater function $L(z)$ for which $f(\Phi(z))$ has bounded $L$-index in a direction. We relax the condition $|\partial_{\mathbf{b}}^k\Phi(z)|\le K|\partial_{\mathbf{b}}\Phi(z)|^k$ for all $z\in\mathbb{C}^n$,where $K\geq 1$ is a constant and ${\partial_{\mathbf{b}} F(z)}:=\sum\limits_{j=1}^{n}\!\frac{\partial F(z)}{\partial z_{j}}{b_{j}}, $ $\partial_{\mathbf{b}}^k F(z):=\partial_{\mathbf{b}}\big(\partial_{\mathbf{b}}^{k-1} F(z)\big).$ It is replaced by the condition $|\partial_{\mathbf{b}}^k\Phi(z)|\le K(l(\Phi(z)))^{1/(N(f,l)+1)}|\partial_{\mathbf{b}}\Phi(z)|^k,$ where $N(f,l)$ is the $l$-index of the function $f.$The described result is an improvement of previous one.

scholarly journals Approximation of a Function and its Derivatives by Entire Functions

2016 ◽  
Vol 59 (01) ◽  
pp. 87-94 ◽  
Author(s):  
Paul M. Gauthier ◽  
Julie Kienzle
Keyword(s):  
Entire Function ◽  
Continuous Function ◽  
Simple Proof ◽  
Entire Functions ◽  
Open Interval ◽  
Negative Integer ◽  
Positive Continuous Function ◽  
Approximation Of A Function

Abstract A simple proof is given for the fact that for m a non-negative integer, a function f ∊ C (m)(ℝ), and an arbitrary positive continuous function ∊, there is an entire function g such that |g(i)(x) − f (i)(x)| < ∊(x), for all x ∊ ℝ and for each i = 0, 1 …, m. We also consider the situation where ℝ is replaced by an open interval.

Boundedness of L-index in joint variables for composition of analytic functions in the unit ball

2020 ◽  
pp. 2150054
Author(s):  
Andriy Bandura
Keyword(s):  
Continuous Function ◽  
Unit Ball ◽  
Analytic Functions ◽  
Positive Continuous Function

In this paper, the following composite analytic functions [Formula: see text] and [Formula: see text] are considered, where [Formula: see text] [Formula: see text] [Formula: see text] We established conditions which provide equivalence of boundedness of the [Formula: see text]-index of the function [Formula: see text] and boundedness of the [Formula: see text]-index in joint variables of the function [Formula: see text] where [Formula: see text] is a continuous function, [Formula: see text] [Formula: see text] For the function [Formula: see text] with additional restrictions, the function [Formula: see text] is constructed such that [Formula: see text] has bounded [Formula: see text]-index in joint variables in the case when the function [Formula: see text] has bounded [Formula: see text]-index in the direction [Formula: see text], where [Formula: see text] is a positive continuous function. Our proofs are based on the application of analog of Hayman’s theorem for these classes of functions.

Weak-Star Continuous Analytic Functions

1995 ◽  
Vol 47 (4) ◽  
pp. 673-683 ◽  
Author(s):  
R. M. Aron ◽  
B. J. Cole ◽  
T. W. Gamelin
Keyword(s):  
Unit Ball ◽  
Finite Type ◽  
Analytic Functions ◽  
Approximation Property ◽  
Entire Functions ◽  
Open Unit ◽  
Closed Unit Ball ◽  
Open Unit Ball ◽  
Weakly Continuous ◽  
Weak Star

AbstractLet 𝒳 be a complex Banach space, with open unit ball B. We consider the algebra of analytic functions on B that are weakly continuous and that are uniformly continuous with respect to the norm. We show these are precisely the analytic functions on B that extend to be weak-star continuous on the closed unit ball of 𝒳**. If 𝒳* has the approximation property, then any such function is approximable uniformly on B by finite polynomials in elements of 𝒳*. On the other hand, there exist Banach spaces for which these finite-type polynomials fail to approximate. We consider also the approximation of entire functions by finite-type polynomials. Assuming 𝒳* has the approximation property, we show that entire functions are approximable uniformly on bounded sets if and only if the spectrum of the algebra of entire functions coincides (as a point set) with 𝒳**.

Composition operators on μ-Bloch spaces

2009 ◽  
Vol 61 (1) ◽  
pp. 50-75 ◽  
Author(s):  
Huaihui Chen ◽  
Paul Gauthier
Keyword(s):  
Continuous Function ◽  
Unit Ball ◽  
Composition Operator ◽  
Composition Operators ◽  
Sufficient Conditions ◽  
Positive Continuous Function ◽  
Bloch Spaces ◽  
Bloch Functions ◽  
Necessary And Sufficient

Abstract. Given a positive continuous function μ on the interval 0 < t ≤ 1, we consider the space of so-called μ-Bloch functions on the unit ball. If μ(t ) = t, these are the classical Bloch functions. For μ, we define a metric Fμz (u) in terms of which we give a characterization of μ-Bloch functions. Then, necessary and sufficient conditions are obtained in order that a composition operator be a bounded or compact operator between these generalized Bloch spaces. Our results extend those of Zhang and Xiao.

scholarly journals On approximation of homomorphisms of algebras of entire functions on Banach spaces

2019 ◽  
Vol 11 (1) ◽  
pp. 158-162
Author(s):  
H.M. Pryimak
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Algebra ◽  
Banach Spaces ◽  
Analytic Functions ◽  
Entire Functions ◽  
Commutative Banach Algebra ◽  
Uniformly Continuous ◽  
Positive Results

It is known due to R. Aron, B. Cole and T. Gamelin that every complex homomorphism of the algebra of entire functions of bounded type on a Banach space $X$ can be approximated in some sense by a net of point valued homomorphism. In this paper we consider the question about a generalization of this result for the case of homomorphisms to any commutative Banach algebra $A.$ We obtained some positive results if $A$ is the algebra of uniformly continuous analytic functions on the unit ball of $X.$

scholarly journals Vector-Valued Entire Functions of Several Variables: Some Local Properties

Axioms ◽  
2022 ◽  
Vol 11 (1) ◽  
pp. 31
Author(s):  
Andriy Ivanovych Bandura ◽  
Tetyana Mykhailivna Salo ◽  
Oleh Bohdanovych Skaskiv
Keyword(s):  
Continuous Function ◽  
Partial Derivative ◽  
Entire Functions ◽  
Positive Continuous Function ◽  
Several Variables ◽  
Local Properties ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Entire Vector

The present paper is devoted to the properties of entire vector-valued functions of bounded L-index in join variables, where L:Cn→R+n is a positive continuous function. For vector-valued functions from this class we prove some propositions describing their local properties. In particular, these functions possess the property that maximum of norm for some partial derivative at a skeleton of polydisc does not exceed norm of the derivative at the center of polydisc multiplied by some constant. The converse proposition is also true if the described inequality is satisfied for derivative in each variable.

scholarly journals The minimal growth of entire functions with given zeros along unbounded sets

2020 ◽  
Vol 54 (2) ◽  
pp. 146-153
Author(s):  
I. V. Andrusyak ◽  
P.V. Filevych
Keyword(s):  
Entire Function ◽  
Continuous Function ◽  
Entire Functions ◽  
Counting Function ◽  
Maximum Modulus ◽  
Positive Function ◽  
Minimal Growth ◽  
Complex Sequence ◽  
Growth Of Entire Functions ◽  
Necessary And Sufficient

Let $l$ be a continuous function on $\mathbb{R}$ increasing to $+\infty$, and $\varphi$ be a positive function on $\mathbb{R}$. We proved that the condition$$\varliminf_{x\to+\infty}\frac{\varphi(\ln[x])}{\ln x}>0$$is necessary and sufficient in order that for any complex sequence $(\zeta_n)$ with $n(r)\ge l(r)$, $r\ge r_0$, and every set $E\subset\mathbb{R}$ which is unbounded from above there exists an entire function $f$ having zeros only at the points $\zeta_n$ such that$$\varliminf_{r\in E,\ r\to+\infty}\frac{\ln\ln M_f(r)}{\varphi(\ln n_\zeta(r))\ln l^{-1}(n_\zeta(r))}=0.$$Here $n(r)$ is the counting function of $(\zeta_n)$, and $M_f(r)$ is the maximum modulus of $f$.

Linear directional differential equations in the unit ball: solutions of bounded L-index

2019 ◽  
Vol 69 (5) ◽  
pp. 1089-1098 ◽  
Author(s):  
Andriy Bandura ◽  
Oleh Skaskiv
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Differential Equations ◽  
Unit Ball ◽  
Analytic Functions ◽  
Entire Functions ◽  
Sufficient Conditions ◽  
Analytic Solutions ◽  
Directional Derivatives ◽  
Several Variables

Abstract We study sufficient conditions of boundedness of L-index in a direction b ∈ ℂn ∖ {0} for analytic solutions in the unit ball of a linear higher order non-homogeneous differential equation with directional derivatives. These conditions are restrictions by the analytic coefficients in the unit ball of the equation. Also we investigate asymptotic behavior of analytic functions of bounded L-index in the direction and estimate its growth. The results are generalizations of known propositions for entire functions of several variables.

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