Bounded analytic functions in the unit disk

1969 ◽  
pp. 1-9
Author(s):  
Kenneth Hoffman
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Bounded Analytic Functions

ESTIMATES OF THE SECOND DERIVATIVE OF BOUNDED ANALYTIC FUNCTIONS

2019 ◽  
Vol 100 (3) ◽  
pp. 458-469
Author(s):  
GANGQIANG CHEN
Keyword(s):  
Unit Disk ◽  
Complex Plane ◽  
Upper Bound ◽  
Analytic Functions ◽  
Open Unit ◽  
Second Derivative ◽  
Open Unit Disk ◽  
Bounded Analytic Functions ◽  
Schwarz’S Lemma

Assume a point $z$ lies in the open unit disk $\mathbb{D}$ of the complex plane $\mathbb{C}$ and $f$ is an analytic self-map of $\mathbb{D}$ fixing 0. Then Schwarz’s lemma gives $|f(z)|\leq |z|$, and Dieudonné’s lemma asserts that $|f^{\prime }(z)|\leq \min \{1,(1+|z|^{2})/(4|z|(1-|z|^{2}))\}$. We prove a sharp upper bound for $|f^{\prime \prime }(z)|$ depending only on $|z|$.

scholarly journals Global Holomorphic Functions in Several Non-Commuting Variables II

2018 ◽  
Vol 61 (3) ◽  
pp. 458-463 ◽  
Author(s):  
Jim Agler ◽  
John McCarthy
Keyword(s):  
Transfer Function ◽  
Unit Disk ◽  
Analytic Functions ◽  
Holomorphic Functions ◽  
Bounded Analytic Functions

AbstractWe give a new proof that bounded non-commutative functions on polynomial polyhedra can be represented by a realization formula, a generalization of the transfer function realization formula for bounded analytic functions on the unit disk.

scholarly journals Convolution properties of a class of bounded analytic functions

1992 ◽  
Vol 45 (1) ◽  
pp. 9-23 ◽  
Author(s):  
Zou Zhongzhu ◽  
Shigeyoshi Owa
Keyword(s):  
Integral Operator ◽  
Unit Disk ◽  
Analytic Functions ◽  
Bounded Analytic Functions ◽  
Class A ◽  
Convolution Properties ◽  
Class Of Functions

Let A be the class of functions f(z) which are analytic in the unit disk U with f(0) = f′(0) - 1 = 0. A subclass S(λ, M) (λ > 0, M > 0) of A is introduced. The object of the present paper is to prove some interesting convolution properties of functions f(z) belonging to the class S(λ, M). Also a certain integral operator J for f(z) in the class A is considered.

scholarly journals Factorization for bounded analytic functions in the unit disk and an application to prime ideals

2006 ◽  
Vol 99 (2) ◽  
pp. 168 ◽  
Author(s):  
Raymond Mortini
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Prime Ideals ◽  
Open Unit ◽  
Open Unit Disk ◽  
Finitely Generated ◽  
Bounded Analytic Functions

We prove several factorization theorems for bounded analytic functions in the open unit disk and present a very simple new proof of two conjectures of Frank Forelli and the author on the structure of finitely generated, respectively countably generated prime ideals in $H^{\infty}$.

scholarly journals Multipliers

Filomat ◽  
2013 ◽  
Vol 27 (7) ◽  
pp. 1277-1283 ◽  
Author(s):  
Miroljub Jevtic
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Mixed Norm ◽  
Bounded Analytic Functions ◽  
Mixed Norm Spaces ◽  
Spaces Of Analytic Functions

We describe the multiplier spaces (Hp,q,?,H?), and (Hp,q,?,H?,v,?), where Hp,q,? are mixed norm spaces of analytic functions in the unit disk D and H? is the space of bounded analytic functions in D. We extend some results from [7] and [3], particularly Theorem 4.3 in [3].

scholarly journals Approximation by invertible functions of $H^{\infty}$

2006 ◽  
Vol 99 (2) ◽  
pp. 287 ◽  
Author(s):  
Artur Nicolau ◽  
Daniel Suárez
Keyword(s):  
Banach Algebra ◽  
Unit Disk ◽  
Analytic Functions ◽  
Banach Algebras ◽  
Bounded Analytic Functions ◽  
Analytic Proof ◽  
Dense Image

We provide an analytic proof that if $H^\infty$ is the algebra of bounded analytic functions on the unit disk, $A$ is a Banach algebra and $f: H^\infty \rightarrow A$ is a Banach algebras morphism with dense image, then $f((H^\infty)^{-1})$ is dense in $A^{-1}$.

scholarly journals Hyponormality on a Weighted Bergman Space

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Houcine Sadraoui ◽  
Borhen Halouani ◽  
Mubariz T. Garayev ◽  
Adel AlShehri
Keyword(s):  
Hilbert Space ◽  
Unit Disk ◽  
Bergman Space ◽  
Analytic Functions ◽  
Toeplitz Operators ◽  
Sufficient Conditions ◽  
Weighted Bergman Space ◽  
Necessary Condition ◽  
Space Operator ◽  
Bounded Analytic Functions

A bounded Hilbert space operator T is hyponormal if T∗T−TT∗ is a positive operator. We consider the hyponormality of Toeplitz operators on a weighted Bergman space. We find a necessary condition for hyponormality in the case of a symbol of the form f+g¯ where f and g are bounded analytic functions on the unit disk. We then find sufficient conditions when f is a monomial.

scholarly journals REGION OF STARLIKENESS OF BOUNDED ANALYTIC FUNCTIONS

1995 ◽  
Vol 26 (1) ◽  
pp. 1-3
Author(s):  
S. R. KULKARNI ◽  
U. H. NAIK
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Radius Of Starlikeness ◽  
Bounded Analytic Functions

In this paper we obtain the radius of starlikeness for functions of the type $f(z) =a_1z +a_2z^2+ \cdots$ which are analytic and umvalent in the unit disk and satisfy $0<|f(z)|\le \alpha$ in $0<|z|<1$, where $\alpha$ is real.

scholarly journals WEIGHTED COMPOSITION OPERATORS ON H∞ ∩ o

2014 ◽  
Vol 57 (2) ◽  
pp. 475-480
Author(s):  
SHÛICHI OHNO
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Composition Operators ◽  
Open Unit ◽  
Weighted Composition Operators ◽  
Open Unit Disk ◽  
Bounded Analytic Functions ◽  
Disk Algebra ◽  
Closed Subalgebra ◽  
Weighted Composition

AbstractWe will characterize the boundedness and compactness of weighted composition operators on the closed subalgebra H∞ ∩ $\mathcal{B}$o between the disk algebra and the space of bounded analytic functions on the open unit disk.

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