Sets of Uniqueness for Univalent Functions

2000 ◽  
Vol 43 (1) ◽  
pp. 105-107
Author(s):  
Marius Overholt
Keyword(s):  
Univalent Functions ◽  
Dirichlet Space ◽  
Holomorphic Functions ◽  
Set Of Uniqueness ◽  
Sets Of Uniqueness

AbstractWe observe that any set of uniqueness for the Dirichlet space 𝐷 is a set of uniqueness for the class S of normalized univalent holomorphic functions.

scholarly journals Schiffer Comparison Operators and Approximations on Riemann Surfaces Bordered by Quasicircles

2020 ◽  
Author(s):  
Eric Schippers ◽  
Mohammad Shirazi ◽  
Wolfgang Staubach
Keyword(s):  
Riemann Surface ◽  
Bergman Space ◽  
Riemann Surfaces ◽  
Compact Riemann Surface ◽  
Dirichlet Space ◽  
Holomorphic Functions ◽  
Finite Collection ◽  
Approximation Theorems ◽  
Arbitrary Genus ◽  
Simply Connected

Abstract We consider a compact Riemann surface R of arbitrary genus, with a finite number of non-overlapping quasicircles, which separate R into two subsets: a connected Riemann surface $$\Sigma $$ Σ , and the union $$\mathcal {O}$$ O of a finite collection of simply connected regions. We prove that the Schiffer integral operator mapping the Bergman space of anti-holomorphic one-forms on $$\mathcal {O}$$ O to the Bergman space of holomorphic forms on $$\Sigma $$ Σ is an isomorphism onto the exact one-forms, when restricted to the orthogonal complement of the set of forms on all of R. We then apply this to prove versions of the Plemelj–Sokhotski isomorphism and jump decomposition for such a configuration. Finally we obtain some approximation theorems for the Bergman space of one-forms and Dirichlet space of holomorphic functions on $$\Sigma $$ Σ by elements of Bergman space and Dirichlet space on fixed regions in R containing $$\Sigma $$ Σ .

Perfect Sets of Uniqueness on the Group 2ω

1982 ◽  
Vol 34 (3) ◽  
pp. 759-764 ◽  
Author(s):  
Kaoru Yoneda
Keyword(s):  
Haar Measure ◽  
Measure Zero ◽  
Countable Subset ◽  
Walsh Series ◽  
Dyadic Group ◽  
Set Of Uniqueness ◽  
Image Position ◽  
Sets Of Uniqueness

Let ω0, ω1, … denote the Walsh-Paley functions and let G denote the dyadic group introduced by Fine [3]. Recall that a subset E of G is said to be a set of uniqueness if the zero series is the only Walsh series ∑ akωk which satisfiesA subset E of G which is not a set of uniqueness is called a set of multiplicity.It is known that any subset of G of positive Haar measure is a set of multiplicity [5] and that any countable subset of G is a set of uniqueness [2]. As far as uncountable subsets of Haar measure zero are concerned, both possibilities present themselves. Indeed, among perfect subsets of G of Haar measure zero there are sets of multiplicity [1] and there are sets of uniqueness [5].

Application of strip domain and parabolic region on univalent holomorphic functions

2021 ◽  
Vol 26 (5) ◽  
pp. 66-71
Author(s):  
Shahram Najafzadeh
Keyword(s):  
Convex Functions ◽  
Univalent Functions ◽  
Holomorphic Functions ◽  
Uniformly Convex ◽  
Uniformly Convex Functions ◽  
Strip Domain

In this paper, by using univalent functions connected with the strip domain,parabolic starlike and parabolic uniformly convex functions are introduced.Some relations between these classes are proved.

scholarly journals A property of univalent functions in A_{p}

2000 ◽  
Vol 42 (1) ◽  
pp. 121-124 ◽  
Author(s):  
David Walsh
Keyword(s):  
Besov Space ◽  
Unit Disc ◽  
Univalent Functions ◽  
Dirichlet Space ◽  
Subject Classification ◽  
Image Domain

The univalent functions in the diagonal Besov space A_{p}, where 1<p<\infty , are characterized in terms of the distance from the boundary of a point in the image domain. Here A_{2} is the Dirichlet space. A consequence is that there exist functions in A_{p},\ p>2, for which the area of the complement of the image of the unit disc is zero.1991 Mathematics Subject Classification 30C99, 46E35.

scholarly journals Hyperbolic linear invariance and hyperbolic k-convexity

1995 ◽  
Vol 58 (1) ◽  
pp. 73-93 ◽  
Author(s):  
Wancang Ma ◽  
David Minda
Keyword(s):  
Unit Disk ◽  
Function Theory ◽  
Univalent Functions ◽  
Holomorphic Functions ◽  
Geometric Function Theory ◽  
Related Concept ◽  
Invariant Functions ◽  
Geometric Function ◽  
Definition Of ◽  
New Phenomena

AbstractPommerenke initiated the study of linearly invariant families of locally schlicht holomorphic functions defined on the unit disk The concept of linear invariance has proved fruitful in geometric function theory. One aspect of Pommerenke's work is the extension of certain results from classical univalent function theory to linearly invariant functions. We propose a definition of a related concept that we call hyperbolic linear invariance for locally schlicht holomorphic functions that map the unit disk into itself. We obtain results for hyperbolic linearly invariant functions which generalize parts of the theory of bounded univalent functions. There are many similarities between linearly invariant functions and hyperbolic linearly invariant functions, but some new phenomena also arise in the study of hyperbolic linearly invariant functions.

scholarly journals Some results of Fekete–Szegö type for Bavrin’s families of holomorphic functions in $${\mathbb {C}}^{n}$$

2021 ◽  
Author(s):  
Renata Długosz ◽  
Piotr Liczberski
Keyword(s):  
Complex Variable ◽  
Univalent Functions ◽  
Holomorphic Functions ◽  
Symmetry Condition ◽  
Type Problem ◽  
Geometric Properties ◽  
Several Variables ◽  
Starlike Mappings ◽  
Complex Valued

AbstractIn the paper there is considered a generalization of the well-known Fekete–Szegö type problem onto some Bavrin’s families of complex valued holomorphic functions of several variables. The definitions of Bavrin’s families correspond to geometric properties of univalent functions of a complex variable, like as starlikeness and convexity. First of all, there are investigated such Bavrin’s families which elements satisfy also a (j, k)-symmetry condition. As application of these results there is given the solution of a Fekete–Szegö type problem for a family of normalized biholomorphic starlike mappings in $${\mathbb {C}}^{n}.$$ C n .

scholarly journals The regularity of the space of germs of Fréchet valued holomorphic functions and the mixed Hartog's theorem

2006 ◽  
Vol 99 (1) ◽  
pp. 119 ◽  
Author(s):  
Thai Thuan Quang
Keyword(s):  
Holomorphic Function ◽  
Sufficient Conditions ◽  
Holomorphic Functions ◽  
Schwartz Space ◽  
Necessary And Sufficient Conditions ◽  
Fréchet Space ◽  
Compact Set ◽  
Frechet Space ◽  
Set Of Uniqueness ◽  
Necessary And Sufficient

It is shown that $H(K, F)$ is regular for every reflexive Fréchet space $F$ with the property ($\mathrm{LB}_\infty)$ where $K$ is a compact set of uniqueness in a Fréchet-Schwartz space $E$ such that $E \in (\Omega)$. Using this result we give necessary and sufficient conditions for a Fréchet space $F$, under which every separately holomorphic function on $K \times F^*$ is holomorphic, where $K$ is as above.

scholarly journals The Study of the New Classes of m-Fold Symmetric bi-Univalent Functions

Mathematics ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 75
Author(s):  
Daniel Breaz ◽  
Luminiţa-Ioana Cotîrlă
Keyword(s):  
Unit Disk ◽  
Univalent Functions ◽  
Holomorphic Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Function Classes

In this paper, we introduce three new subclasses of m-fold symmetric holomorphic functions in the open unit disk U, where the functions f and f−1 are m-fold symmetric holomorphic functions in the open unit disk. We denote these classes of functions by FSΣ,mp,q,s(d), FSΣ,mp,q,s(e) and FSΣ,mp,q,s,h,r. As the Fekete-Szegö problem for different classes of functions is a topic of great interest, we study the Fekete-Szegö functional and we obtain estimates on coefficients for the new function classes.

Sets of uniqueness for trigonometric series and integrals

1950 ◽  
Vol 46 (4) ◽  
pp. 538-548 ◽  
Author(s):  
R. Henstock
Keyword(s):  
Trigonometric Series ◽  
Set Of Uniqueness ◽  
Point Set ◽  
Image Position ◽  
Sets Of Uniqueness

It is well known that, if a trigonometric seriesconverges to zero in 0 ≤ x < 2π then all the coefficients are zero. To generalize this property of the series, sets of uniqueness have been defined. A point-set E in 0 ≤ x < 2π is a set of uniqueness if every series (0·1), converging to zero in [0, 2π) − E, has zero coefficients. Otherwise E is a set of multiplicity. For example, every enumerable set is a set of uniqueness. An account of the theory may be found in Zygmund (2), chapter 11, pp. 267 et seq.

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