scholarly journals A NEW CHARACTERIZATION OF HOLOMORPHIC FUNCTIONS IN THE UNIT DISK

2018 ◽  
Vol 25 (1) ◽  
pp. 134-147
Author(s):  
Valerii V. Volchkov ◽  
Vitalii V. Volchkov
Keyword(s):  
Unit Disk ◽  
Holomorphic Functions

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 Coefficients problems for families of holomorphic functions related to hyperbola

2020 ◽  
Vol 70 (3) ◽  
pp. 605-616
Author(s):  
Stanisława Kanas ◽  
Vali Soltani Masih ◽  
Ali Ebadian
Keyword(s):  
Unit Disk ◽  
Holomorphic Functions ◽  
Hankel Determinant ◽  
Coefficient Problem

AbstractWe consider a family of analytic and normalized functions that are related to the domains ℍ(s), with a right branch of a hyperbolas H(s) as a boundary. The hyperbola H(s) is given by the relation $\begin{array}{} \frac{1}{\rho}=\left( 2\cos\frac{\varphi}{s}\right)^s\quad (0 \lt s\le 1,\, |\varphi| \lt (\pi s)/2). \end{array}$ We mainly study a coefficient problem of the families of functions for which zf′/f or 1 + zf″/f′ map the unit disk onto a subset of ℍ(s) . We find coefficients bounds, solve Fekete-Szegö problem and estimate the Hankel determinant.

A Remark on Extensions of CR Functions from Hyperplanes

2008 ◽  
Vol 51 (1) ◽  
pp. 21-25
Author(s):  
Luca Baracco
Keyword(s):  
Radon Transform ◽  
Integral Geometry ◽  
Holomorphic Functions ◽  
Holomorphic Extension ◽  
General Statement ◽  
Cr Geometry ◽  
Cr Functions ◽  
Natural Context ◽  
Extension Of Functions

AbstractIn the characterization of the range of the Radon transform, one encounters the problem of the holomorphic extension of functions defined on ℝ2 \ Δℝ (where Δℝ is the diagonal in ℝ2) and which extend as “separately holomorphic” functions of their two arguments. In particular, these functions extend in fact to ℂ2 \ Δℂ where Δℂ is the complexification of Δℝ. We take this theorem from the integral geometry and put it in the more natural context of the CR geometry where it accepts an easier proof and amore general statement. In this new setting it becomes a variant of the celebrated “edge of the wedge” theorem of Ajrapetyan and Henkin.

scholarly journals Boundary behaviour of holomorphic functions on the cardioid domain with some applications

2019 ◽  
Vol 38 (7) ◽  
pp. 203-218
Author(s):  
Shatha Sami Alhily ◽  
_ Deepmala
Keyword(s):  
General Solution ◽  
Holomorphic Function ◽  
Unit Disk ◽  
Laplace Equation ◽  
Holomorphic Functions ◽  
Research Paper ◽  
Boundary Behaviour ◽  
Polar Function

The objective of this research paper is to show how the Bennan'sconjecture  become a useful tool  to construct a holomorphic function on the cardioid domain, and on the boundary of unit disk. Moreover , we have addressed some applications on the existence of cusp on the boundary of arising from integrability of conformalmaps through one of the polar function in the general solution of Laplace equation.

The characterization of orders for regular functions

1992 ◽  
Vol 111 (2) ◽  
pp. 299-307 ◽  
Author(s):  
C. N. Linden
Keyword(s):  
Unit Disk ◽  
Regular Functions ◽  
Logarithmic Means ◽  
Image Position

For a given function f, regular in the unit disk D(0, 1), logarithmic means of order p may be defined in terms of integrals by the formulaefor 0 < p < , and bywhen 0 < r < 1. Associated p-orders are subsequently defined byfor 0 < p .

scholarly journals Composition operators on some Möbius invariant Banach spaces

2000 ◽  
Vol 62 (1) ◽  
pp. 1-19 ◽  
Author(s):  
Shamil Makhmutov ◽  
Maria Tjani
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Unit Disk ◽  
Composition Operator ◽  
Composition Operators ◽  
Bloch Space ◽  
Holomorphic Functions ◽  
Mean Oscillation

We characterise the compact composition operators from any Mobius invariant Banach space to VMOA, the space of holomorphic functions on the unit disk U that have vanishing mean oscillation. We use this to obtain a characterisation of the compact composition operators from the Bloch space to VMOA. Finally, we study some properties of hyperbolic VMOA functions. We show that a function is hyperbolic VMOA if and only if it is the symbol of a compact composition operator from the Bloch space to VMOA.

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