scholarly journals Products of composition and n-th differentiation operators from α-bloch space to Qp space

Filomat ◽  
2013 ◽  
Vol 27 (5) ◽  
pp. 761-766
Author(s):  
Haiying Li ◽  
Cui Wang ◽  
Tianyu Xue ◽  
Xiangbo Zhang
Keyword(s):  
Unit Disk ◽  
Complex Plane ◽  
Composition Operators ◽  
Bloch Space ◽  
Open Unit ◽  
Open Unit Disk ◽  
Th Differentiation ◽  
Qp Space

Let ? be an analytic self-map of the open unit disk D on the complex plane and ? > 0, p ? 0, n ? N. In this paper, the boundedness and compactness of the products of composition operators and nth differentiation operators C?Dn from a-Bloch space B? and B?0 to Qp space are investigated.

scholarly journals SUMS OF WEIGHTED COMPOSITION OPERATORS ON COP

2012 ◽  
Vol 55 (1) ◽  
pp. 229-239 ◽  
Author(s):  
KEI JI IZUCHI ◽  
KOU HEI IZUCHI ◽  
YUKO IZUCHI
Keyword(s):  
Unit Disk ◽  
Composition Operators ◽  
Bloch Space ◽  
Open Unit ◽  
Weighted Composition Operators ◽  
Open Unit Disk ◽  
Disk Algebra ◽  
Weighted Composition ◽  
Little Bloch Space

AbstractLet COP =0∩H∞, where0is the little Bloch space on the open unit disk, andA() be the disk algebra on. For non-zero functionsu1,u2,. . .,uN∈A() and distinct analytic self-maps ϕ1,ϕ2,. . .,ϕNsatisfying ϕj∈A() and ∥ϕj∥∞=1 for everyj, it is given characterisations of which the sum of weighted composition operators ∑Nj=1ujCϕjmaps COP intoA().

On Algebras Generated by Composition Operators

1974 ◽  
Vol 26 (5) ◽  
pp. 1234-1241 ◽  
Author(s):  
J. A. Cima ◽  
W. R. Wogen
Keyword(s):  
Banach Spaces ◽  
Unit Disk ◽  
Complex Plane ◽  
Hardy Spaces ◽  
Composition Operators ◽  
Open Unit ◽  
Open Unit Disk ◽  
Group Of Automorphisms ◽  
Image Position

Let Δ be the open unit disk in the complex plane and let be the group of automorphisms of Δ onto Δ, define byThe Banach spaces Hp = Hp(Δ), 1 ≦ p < ∞, are the Hardy spaces of functions analytic in Δ with their integral p means bounded,

scholarly journals Characterisation of the isometric composition operators on the Bloch space

2005 ◽  
Vol 72 (2) ◽  
pp. 283-290 ◽  
Author(s):  
Flavia Colonna
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Composition Operator ◽  
Blaschke Product ◽  
Analytic Functions ◽  
Composition Operators ◽  
Bloch Space ◽  
Open Unit ◽  
Open Unit Disk ◽  
Identity Function

In this paper, we characterise the analytic functions ϕ mapping the open unit disk ▵ into itself whose induced composition operator Cϕ: f ↦ f ∘ ϕ is an isometry on the Bloch space. We show that such functions are either rotations of the identity function or have a factorisation ϕ = gB where g is a non-vanishing analytic function from Δ into the closure of ▵, and B is an infinite Blaschke product whose zeros form a sequence{zn} containing 0 and a subsequence satisfying the conditions , and

scholarly journals Strong Differential Subordination Results for Multivalent Analytic Functions Associated with Dziok-Srivastava Operator

2019 ◽  
pp. 159-168
Author(s):  
Abbas Kareem Wanas ◽  
Hala Abbas Mehdi
Keyword(s):  
Unit Disk ◽  
Complex Plane ◽  
Analytic Functions ◽  
Differential Subordination ◽  
Open Unit ◽  
Open Unit Disk ◽  
Closed Unit Disk ◽  
Strong Differential Subordination

In this paper, by making use of the principle of strong subordination, we establish some interesting properties of multivalent analytic functions defined in the open unit disk and closed unit disk of the complex plane associated with Dziok-Srivastava operator.

scholarly journals On a Uniqueness Theorem

1969 ◽  
Vol 35 ◽  
pp. 151-157 ◽  
Author(s):  
V. I. Gavrilov
Keyword(s):  
Unit Circle ◽  
Uniqueness Theorem ◽  
Unit Disk ◽  
Complex Plane ◽  
Open Unit ◽  
Open Unit Disk ◽  
Extended Complex ◽  
Extended Complex Plane

1. Let D be the open unit disk and r be the unit circle in the complex plane, and denote by Q the extended complex plane or the Rie-mann sphere.

A Class of Analytic Starlike Functions Associated with Petal Like Region on the Positive Half of Complex Plane

2020 ◽  
Vol 87 (3-4) ◽  
pp. 165
Author(s):  
Rajesh Kumar Maurya ◽  
Poonam Sharma
Keyword(s):  
Unit Disk ◽  
Complex Plane ◽  
Algebraic Structure ◽  
Starlike Functions ◽  
Open Mapping ◽  
Open Unit ◽  
Open Unit Disk ◽  
Hankel Determinant ◽  
Open Mapping Theorem ◽  
Positive Half

In the light of Riemann open mapping theorem, if we map open unit disk U conformally onto a region then depending on the geometry of boundary of we can always extract a subclass of H[a, n] by subordinating various functionals of the function f ∈ H[a, n]. Depending upon the geometry of the range set attempts have been made to find some algebraic structure in such classes, for that Hankel determinant of coefficients of functions pertaining to these classes have been studied, bounds of various coefficients have been determined and also based on the subordination principle we have determined radius |z| &lt; r ;z ∈ U for which f belongs to such a class. In this paper our focus would be on n−PS<sup>*</sup> defined as n − PS<sup>*</sup> = {f ∈ A : Re {zf<sup>'</sup>(z)/f(z)} &gt; 0,|(zf<sup>'</sup>(z)/f(z))<sup>n</sup> - 1|&lt;1}.

LIMITS OF FRACTIONAL DERIVATIVES AND COMPOSITIONS OF ANALYTIC FUNCTIONS

2016 ◽  
Vol 103 (1) ◽  
pp. 104-115 ◽  
Author(s):  
THOMAS H. MACGREGOR ◽  
MICHAEL P. STERNER
Keyword(s):  
Fractional Derivative ◽  
Unit Disk ◽  
Power Function ◽  
Complex Plane ◽  
Analytic Functions ◽  
Fractional Derivatives ◽  
Hadamard Product ◽  
Open Unit ◽  
Open Unit Disk

Suppose that the function $f$ is analytic in the open unit disk $\unicode[STIX]{x1D6E5}$ in the complex plane. For each $\unicode[STIX]{x1D6FC}>0$ a function $f^{[\unicode[STIX]{x1D6FC}]}$ is defined as the Hadamard product of $f$ with a certain power function. The function $f^{[\unicode[STIX]{x1D6FC}]}$ compares with the fractional derivative of $f$ of order $\unicode[STIX]{x1D6FC}$. Suppose that $f^{[\unicode[STIX]{x1D6FC}]}$ has a limit at some point $z_{0}$ on the boundary of $\unicode[STIX]{x1D6E5}$. Then $w_{0}=\lim _{z\rightarrow z_{0}}f(z)$ exists. Suppose that $\unicode[STIX]{x1D6F7}$ is analytic in $f(\unicode[STIX]{x1D6E5})$ and at $w_{0}$. We show that if $g=\unicode[STIX]{x1D6F7}(f)$ then $g^{[\unicode[STIX]{x1D6FC}]}$ has a limit at $z_{0}$.

scholarly journals Weighted Composition Operators from Hardy Spaces into Logarithmic Bloch Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Flavia Colonna ◽  
Songxiao Li
Keyword(s):  
Banach Space ◽  
Hardy Spaces ◽  
Composition Operators ◽  
Bloch Space ◽  
Weighted Composition Operator ◽  
Open Unit ◽  
Weighted Composition Operators ◽  
Open Unit Disk ◽  
Space Of Analytic Functions ◽  
Weighted Composition

The logarithmic Bloch spaceBlog⁡is the Banach space of analytic functions on the open unit disk&#x1D53B;whose elementsfsatisfy the condition∥f∥=sup⁡z∈&#x1D53B;(1-|z|2)log⁡  (2/(1-|z|2))|f'(z)|<∞. In this work we characterize the bounded and the compact weighted composition operators from the Hardy spaceHp(with1≤p≤∞) into the logarithmic Bloch space. We also provide boundedness and compactness criteria for the weighted composition operator mappingHpinto the little logarithmic Bloch space defined as the subspace ofBlog⁡consisting of the functionsfsuch thatlim⁡|z|→1(1-|z|2)log⁡  (2/(1-|z|2))|f'(z)|=0.

scholarly journals Composition Operators on Some Banach Spaces of Harmonic Mappings

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Munirah Aljuaid ◽  
Flavia Colonna
Keyword(s):  
Banach Spaces ◽  
Unit Disk ◽  
Besov Spaces ◽  
Analytic Functions ◽  
Composition Operators ◽  
Harmonic Mappings ◽  
Open Unit ◽  
Open Unit Disk ◽  
Zygmund Space ◽  
Spaces Of Analytic Functions

We study the composition operators on Banach spaces of harmonic mappings that extend several well-known Banach spaces of analytic functions on the open unit disk in the complex plane, including the α-Bloch spaces, the growth spaces, the Zygmund space, the analytic Besov spaces, and the space BMOA.

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 $|z|$.

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