H(ϕ) Spaces

1986 ◽  
Vol 29 (3) ◽  
pp. 295-301 ◽  
Author(s):  
W. Deeb ◽  
M. Marzuq
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Almost Everywhere ◽  
Subadditive Function

AbstractLet ψ be a non-decreasing continuous subadditive function defined on [0, ∞) and satisfy ψ(x) = 0 if and only if x = 0. The space H(ψ) is defined as the set of analytic functions in the unit disk which satisfyand the space H+ (ψ) is the space of a f ∊ H(ψ) for whichwhere almost everywhere.In this paper we study the H(ψ) spaces and characterize the continuous linear functionals on H+ (ψ).

Radial variation of analytic functions with non-tangential boundary limits almost everywhere

1990 ◽  
Vol 108 (2) ◽  
pp. 371-379 ◽  
Author(s):  
D. J. Hallenbeck ◽  
K. Samotij
Keyword(s):  
Analytic Function ◽  
Asymptotic Behaviour ◽  
Unit Disk ◽  
Analytic Functions ◽  
Radial Variation ◽  
The Third ◽  
Almost Everywhere ◽  
Image Position ◽  
Boundary Limits ◽  
Tangential Limits

The purpose of this paper is to investigate the asymptotic behaviour as r → 1− of the integralsand f is an analytic function on the unit disk Δ which has non-tangential limits at almost every point on ∂Δ. The paper is divided into three parts. In the first part we consider the case where λ ≠ 1/k, in the second the somewhat more delicate case when λ = 1/k and in the third part we concentrate on some problems related to the case λ = k = 1.

scholarly journals OnF-AlgebrasMp  (1

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Romeo Meštrović
Keyword(s):  
Unit Disk ◽  
Complex Plane ◽  
Topological Structure ◽  
Holomorphic Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Continuous Linear ◽  
Linear Functionals

We consider the classesMp (1<p<∞)of holomorphic functions on the open unit disk𝔻in the complex plane. These classes are in fact generalizations of the classMintroduced by Kim (1986). The spaceMpequipped with the topology given by the metricρpdefined byρp(f,g)=f-gp=∫02π‍logp1+Mf-gθdθ/2π1/p, withf,g∈MpandMfθ=sup0⩽r<1⁡f(reiθ), becomes anF-space. By a result of Stoll (1977), the Privalov spaceNp (1<p<∞)with the topology given by the Stoll metricdpis anF-algebra. By using these two facts, we prove that the spacesMpandNpcoincide and have the same topological structure. Consequently, we describe a general form of continuous linear functionals onMp(with respect to the metricρp). Furthermore, we give a characterization of bounded subsets of the spacesMp. Moreover, we give the examples of bounded subsets ofMpthat are not relatively compact.

scholarly journals Difference formula defined by a new differential symmetric operator for a class of meromorphically multivalent functions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Rabha W. Ibrahim ◽  
Ibtisam Aldawish
Keyword(s):  
Differential Operator ◽  
Boundary Value Problems ◽  
Unit Disk ◽  
Spectral Theory ◽  
Analytic Functions ◽  
Special Class ◽  
Symmetric Operator ◽  
New Class ◽  
Difference Formula ◽  
Symmetric Operators

AbstractSymmetric operators have benefited in different fields not only in mathematics but also in other sciences. They appeared in the studies of boundary value problems and spectral theory. In this note, we present a new symmetric differential operator associated with a special class of meromorphically multivalent functions in the punctured unit disk. This study explores some of its geometric properties. We consider a new class of analytic functions employing the suggested symmetric differential operator.

scholarly journals Extending Edgar's ordering to locally convex spaces

1992 ◽  
Vol 34 (2) ◽  
pp. 175-188
Author(s):  
Neill Robertson
Keyword(s):  
Convex Space ◽  
Dual Pair ◽  
Locally Convex Space ◽  
Vector Spaces ◽  
Locally Convex Spaces ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Hausdorff Topological Vector Space ◽  
Locally Convex ◽  
Mackey Topologies

By the term “locally convex space”, we mean a locally convex Hausdorff topological vector space (see [17]). We shall denote the algebraic dual of a locally convex space E by E*, and its topological dual by E′. It is convenient to think of the elements of E as being linear functionals on E′, so that E can be identified with a subspace of E′*. The adjoint of a continuous linear map T:E→F will be denoted by T′:F′→E′. If 〈E, F〈 is a dual pair of vector spaces, then we shall denote the corresponding weak, strong and Mackey topologies on E by α(E, F), β(E, F) and μ(E, F) respectively.

scholarly journals QKtype spaces of analytic functions

2006 ◽  
Vol 4 (1) ◽  
pp. 73-84 ◽  
Author(s):  
Hasi Wulan ◽  
Jizhen Zhou
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Sufficient Conditions ◽  
Nondecreasing Function ◽  
Necessary And Sufficient Conditions ◽  
Type Space ◽  
Necessary And Sufficient ◽  
Spaces Of Analytic Functions

For a nondecreasing functionK:[0,8)?[0,8)and0<p<8,-2<q<8, we introduceQK(p,q), aQKtype space of functions analytic in the unit disk and study the characterizations ofQK(p,q). Necessary and sufficient conditions onKsuch thatQK(p,q)become some known spaces are given.

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