The Reflection Principle for Banach Space-Valued Analytic Functions

1969 ◽  
Vol 21 ◽  
pp. 1189-1191
Author(s):  
Mark Finkelstein
Keyword(s):  
Banach Space ◽  
Analytic Function ◽  
Power Series ◽  
Analytic Functions ◽  
Sufficient Conditions ◽  
Reflection Principle ◽  
Complex Numbers ◽  
Convergent Power Series ◽  
Regular Subset ◽  
Schwarz Reflection Principle

We give sufficient conditions for the continuation of an analytic function with values in a Branch space. For analytic functions taking complex numbers as values, the principle is known as the Schwarz Reflection Principle.A function defined on a domain of the complex plane with values in a Banach space X is analytic if it possesses at each point Z0 of the domain a convergent power series in z, with coefficients in X.THEOREM. Let D be a domain in the upper half-plane, and E a regular subset of the boundary of D. Suppose that E is an interval of the real axis (a,b). Let f be an analytic function defined on D, continuous up to E, taking values in a Banach space X. Let the image of D under f be Ω, and let Γ be the part of the boundary of Ω which is the image of E under f.

scholarly journals Classes of Entire Analytic Functions of Unbounded Type on Banach Spaces

Axioms ◽  
2020 ◽  
Vol 9 (4) ◽  
pp. 133
Author(s):  
Andriy Zagorodnyuk ◽  
Anna Hihliuk
Keyword(s):  
Banach Space ◽  
Analytic Function ◽  
Banach Spaces ◽  
Analytic Functions ◽  
Sufficient Conditions ◽  
Main Question ◽  
Dimensional Banach Space ◽  
Infinite Dimensional ◽  
Infinite Dimensional Banach Space ◽  
Taylor Polynomials

In this paper we investigate analytic functions of unbounded type on a complex infinite dimensional Banach space X. The main question is: under which conditions is there an analytic function of unbounded type on X such that its Taylor polynomials are in prescribed subspaces of polynomials? We obtain some sufficient conditions for a function f to be of unbounded type and show that there are various subalgebras of polynomials that support analytic functions of unbounded type. In particular, some examples of symmetric analytic functions of unbounded type are constructed.

scholarly journals Some Sufficient Conditions for Analytic Functions to Belong to Space

2008 ◽  
Vol 2008 ◽  
pp. 1-9 ◽  
Author(s):  
Xiaoge Meng
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Analytic Functions ◽  
Sufficient Conditions

This paper gives some sufficient conditions for an analytic function to belong to the space consisting of all analytic functions on the unit disk such

scholarly journals On equivalence of analytic functions to rational regular functions

1987 ◽  
Vol 43 (2) ◽  
pp. 279-286
Author(s):  
Wojciech Kucharz
Keyword(s):  
Analytic Function ◽  
Analytic Functions ◽  
Regular Function ◽  
Sufficient Conditions ◽  
Regular Functions

AbstractWe give sufficient conditions for an analytic function from Rnto R to be analytically equivalent to a rational regular function.

scholarly journals On Some Properties of Cowen-Douglas Class of Operators

2018 ◽  
Vol 2018 ◽  
pp. 1-6 ◽  
Author(s):  
Parastoo Heiatian Naeini ◽  
Bahmann Yousefi
Keyword(s):  
Hilbert Space ◽  
Analytic Function ◽  
Essential Spectrum ◽  
Analytic Functions ◽  
Sufficient Conditions ◽  
Multiplication Operator ◽  
Multiplication Operators ◽  
Space Of Analytic Functions ◽  
Class Operator ◽  
Necessary And Sufficient

We will consider multiplication operators on a Hilbert space of analytic functions on a domainΩ⊂C. For a bounded analytic functionφonΩ, we will give necessary and sufficient conditions under which the complement of the essential spectrum ofMφinφΩbecomes nonempty and this gives conditions for the adjoint of the multiplication operatorMφbelongs to the Cowen-Douglas class of operators. Also, we characterize the structure of the essential spectrum of a multiplication operator and we determine the commutants of certain multiplication operators. Finally, we investigate the reflexivity of a Cowen-Douglas class operator.

scholarly journals Applications of Borel Distribution Series on Analytic Functions

2020 ◽  
pp. 71-82
Author(s):  
Abbas Kareem Wanas ◽  
Jubran Abdulameer Khuttar
Keyword(s):  
Power Series ◽  
Analytic Functions ◽  
Sufficient Conditions ◽  
Extreme Points ◽  
Open Unit ◽  
Open Unit Disk ◽  
Coefficient Estimates ◽  
Sigma Delta ◽  
The Family ◽  
Necessary And Sufficient

The purpose of the present paper is to determine the necessary and sufficient conditions for the power series B_{\mu} whose coefficients are probabilities of the Borel distribution to be in the family H(\lambda, \sigma, \delta, \mu) of analytic functions which defined in the open unit disk. We derive a number of important geometric properties, such as, coefficient estimates, integral representation, radii of starlikeness and convexity. Also we discuss the extreme points and neighborhood property for functions belongs to this family.

scholarly journals Properties of power series of analytic in a bidisc functions of bounded $\mathbf{L}$-index in joint variables

2017 ◽  
Vol 9 (1) ◽  
pp. 6-12 ◽  
Author(s):  
A.I. Bandura ◽  
N.V. Petrechko
Keyword(s):  
Continuous Function ◽  
Power Series ◽  
Series Expansion ◽  
Analytic Functions ◽  
Homogeneous Polynomial ◽  
Necessary Conditions ◽  
Power Series Expansion ◽  
Sufficient Conditions ◽  
Universal Quantifier ◽  
Existential Quantifier

We generalized some criteria of boundedness of $\mathbf{L}$-index in joint variables for analytic in a bidisc functions, where $\mathbf{L}(z)=(l_1(z_1,z_2),$ $l_{2}(z_1,z_2)),$ $l_j:\mathbb{D}^2\to \mathbb{R}_+$ is a continuous function, $j\in\{1,2\},$ $\mathbb{D}^2$ is a bidisc $\{(z_1,z_2)\in\mathbb{C}^2: |z_1|<1,|z_2|<1\}.$ The propositions describe a behaviour of power series expansion on a skeleton of a bidisc. We estimated power series expansion by a dominating homogeneous polynomial with the degree that does not exceed some number depending only from radii of bidisc. Replacing universal quantifier by existential quantifier for radii of bidisc, we also proved sufficient conditions of boundedness of $\mathbf{L}$-index in joint variables for analytic functions which are weaker than necessary conditions.

scholarly journals On q-ANALOGUE of Differential Subordination Associated with Lemniscate of Bernoulli

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Mohsan Raza ◽  
Hira Naz ◽  
Sarfraz Nawaz Malik ◽  
Sahidul Islam
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Analytic Functions ◽  
Sufficient Conditions ◽  
Differential Subordination ◽  
Open Unit ◽  
Sufficient Condition ◽  
Open Unit Disk ◽  
Lemniscate Of Bernoulli

This article comprises the study of differential subordination with analogue of q -derivative. It includes the sufficient condition on γ for 1 + γ ∂ z q h z / h n z to be subordinated by 1 + A z / 1 + B z , − 1 ≤ B < A ≤ 1 , and implies that h z ≺ 1 + z , where h z is the analytic function in the open unit disk. Moreover, certain sufficient conditions for q -starlikeness of analytic functions related with lemniscate of Bernoulli are determined.

scholarly journals Sufficient Conditions for Janowski Starlikeness

2007 ◽  
Vol 2007 ◽  
pp. 1-7 ◽  
Author(s):  
Rosihan M. Ali ◽  
V. Ravichandran ◽  
N. Seenivasagan
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Analytic Functions ◽  
Sufficient Conditions ◽  
Open Unit ◽  
Open Unit Disk

LetA,B,D,E∈[−1,1]and letp(z)be an analytic function defined on the open unit disk,p(0)=1. Conditions onA,B,D, andEare determined so that1+βzp'(z)being subordinated to(1+Dz)/(1+Ez)implies thatp(z)is subordinated to(1+Az)/(1+Bz). Similar results are obtained by considering the expressions1+β(zp'(z)/p(z))and1+β(zp'(z)/p2(z)). These results are then applied to obtain sufficient conditions for analytic functions to be Janowski starlike.

Criteria for Algebraicity of Analytic Functions

2019 ◽  
Author(s):  
Jacek Bochnak ◽  
Janusz Gwoździewicz ◽  
Wojciech Kucharz
Keyword(s):  
Analytic Function ◽  
Power Series ◽  
Formal Power Series ◽  
Open Subset ◽  
Analytic Functions ◽  
Algebraic Curves ◽  
Polynomial Function ◽  
Algebraic Set ◽  
Formal Power

Abstract We consider functions defined on an open subset of a nonsingular, either real or complex, algebraic set. We give criteria for an analytic function to be a Nash (resp. regular, resp. polynomial) function. Our criteria depend only on the behavior of such a function along irreducible nonsingular algebraic curves passing through a given point. In the proofs we use results on algebraicity of formal power series, which are also established in this paper.

scholarly journals A Generalized Class of Functions Defined by the q-Difference Operator

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2361
Author(s):  
Loriana Andrei ◽  
Vasile-Aurel Caus
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Analytic Functions ◽  
Difference Operator ◽  
Sufficient Conditions ◽  
Open Unit ◽  
Open Unit Disk ◽  
New Class ◽  
The Relationship ◽  
Class Of Functions

The goal of the present investigation is to introduce a new class of analytic functions (Kt,q), defined in the open unit disk, by means of the q-difference operator, which may have symmetric or assymetric properties, and to establish the relationship between the new defined class and appropriate subordination. We derived relationships of this class and obtained sufficient conditions for an analytic function to be Kt,q. Finally, in the concluding section, we have taken the decision to restate the clearly-proved fact that any attempt to create the rather simple (p,q)-variations of the results, which we have provided in this paper, will be a rather inconsequential and trivial work, simply because the added parameter p is obviously redundant.

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