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 On a Class of Composition Operators on Bergman Space

2007 ◽  
Vol 2007 ◽  
pp. 1-11
Author(s):  
Namita Das ◽  
R. P. Lal ◽  
C. K. Mohapatra
Keyword(s):  
Unit Disk ◽  
Composition Operator ◽  
Measurable Function ◽  
Analytic Functions ◽  
Composition Operators ◽  
Linear Operators ◽  
Open Unit ◽  
Open Unit Disk ◽  
Space Of Analytic Functions ◽  
Bounded Linear

Let&#x1D53B;={z∈ℂ:|z|<1}be the open unit disk in the complex planeℂ. LetA2(&#x1D53B;)be the space of analytic functions on&#x1D53B;square integrable with respect to the measuredA(z)=(1/π)dx dy. Givena∈&#x1D53B;andfany measurable function on&#x1D53B;, we define the functionCafbyCaf(z)=f(ϕa(z)), whereϕa∈Aut(&#x1D53B;). The mapCais a composition operator onL2(&#x1D53B;,dA)andA2(&#x1D53B;)for alla∈&#x1D53B;. Letℒ(A2(&#x1D53B;))be the space of all bounded linear operators fromA2(&#x1D53B;)into itself. In this article, we have shown thatCaSCa=Sfor alla∈&#x1D53B;if and only if∫&#x1D53B;S˜(ϕa(z))dA(a)=S˜(z), whereS∈ℒ(A2(&#x1D53B;))andS˜is the Berezin symbol of S.

scholarly journals On the second hankel determinant for the kth-root transform of analytic functions

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 227-245 ◽  
Author(s):  
Najla Alarifi ◽  
Rosihan Ali ◽  
V. Ravichandran
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Convex Functions ◽  
Analytic Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Hankel Determinant ◽  
Second Hankel Determinant ◽  
The Given ◽  
Logarithmically Convex Functions

Let f be a normalized analytic function in the open unit disk of the complex plane satisfying zf'(z)/f(z) is subordinate to a given analytic function ?. A sharp bound is obtained for the second Hankel determinant of the kth-root transform z[f(zk)/zk]1/k. Best bounds for the Hankel determinant are also derived for the kth-root transform of several other classes, which include the class of ?-convex functions and ?-logarithmically convex functions. These bounds are expressed in terms of the coefficients of the given function ?, and thus connect with earlier known results for particular choices of ?.

scholarly journals Radius of Star-Likeness for Certain Subclasses of Analytic Functions

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2448
Author(s):  
Caihuan Zhang ◽  
Mirajul Haq ◽  
Nazar Khan ◽  
Muhammad Arif ◽  
Khurshid Ahmad ◽  
...  
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Analytic Functions ◽  
Inverse Function ◽  
Open Unit ◽  
Open Unit Disk ◽  
Sine Function ◽  
Exponential Functions ◽  
Lemniscate Of Bernoulli ◽  
Rotation Function

In this paper, we investigate a normalized analytic (symmetric under rotation) function, f, in an open unit disk that satisfies the condition ℜfzgz>0, for some analytic function, g, with ℜz+1−2nzgz>0,∀n∈N. We calculate the radius constants for different classes of analytic functions, including, for example, for the class of star-like functions connected with the exponential functions, i.e., the lemniscate of Bernoulli, the sine function, cardioid functions, the sine hyperbolic inverse function, the Nephroid function, cosine function and parabolic star-like functions. The results obtained are sharp.

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 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.

scholarly journals Recursive interpolating sequences

2018 ◽  
Vol 16 (1) ◽  
pp. 461-468
Author(s):  
Francesc Tugores
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Blaschke Product ◽  
Bounded Sequence ◽  
Open Unit ◽  
Open Unit Disk ◽  
Separation Condition ◽  
Interpolating Sequences ◽  
Interpolation Problems ◽  
Suitable Sequence

AbstractThis paper is devoted to pose several interpolation problems on the open unit disk 𝔻 of the complex plane in a recursive and linear way. We look for interpolating sequences (zn) in 𝔻 so that given a bounded sequence (an) and a suitable sequence (wn), there is a bounded analytic function f on 𝔻 such that f(z1) = w1 and f(zn+1) = anf(zn) + wn+1. We add a recursion for the derivative of the type: f′(z1) = $\begin{array}{} w_1' \end{array} $ and f′(zn+1) = $\begin{array}{} a_n' \end{array} $ [(1 − |zn|2)/(1 − |zn+1|2)] f′(zn) + $\begin{array}{} w_{n+1}', \end{array} $ where ($\begin{array}{} a_n' \end{array} $) is bounded and ($\begin{array}{} w_n' \end{array} $) is an appropriate sequence, and we also look for zero-sequences verifying the recursion for f′. The conditions on these interpolating sequences involve the Blaschke product with zeros at their points, one of them being the uniform separation condition.

scholarly journals COMPACT DIFFERENCES OF COMPOSITION OPERATORS

2008 ◽  
Vol 77 (1) ◽  
pp. 161-165 ◽  
Author(s):  
ELKE WOLF
Keyword(s):  
Unit Disk ◽  
Composition Operator ◽  
Infinite Order ◽  
Composition Operators ◽  
Bergman Spaces ◽  
Weighted Bergman Spaces ◽  
Open Unit ◽  
Open Unit Disk ◽  
The Difference

AbstractLet ϕ and ψ be analytic self-maps of the open unit disk. Each of them induces a composition operator, Cϕ and Cψ respectively, acting between weighted Bergman spaces of infinite order. We show that the difference Cϕ−Cψ is compact if and only if both operators are compact or both operators are not compact and the pseudohyperbolic distance of the functions ϕ and ψ tends to zero if ∣ϕ(z)∣→1 or ∣ψ(z)∣→1.

scholarly journals On q-ANALOGUE of Differential Subordination Associated with Lemniscate of Bernoulli

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Mohsan Raza ◽  
Hira Naz ◽  
Sarfraz Nawaz Malik ◽  
Sahidul Islam
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Analytic Functions ◽  
Sufficient Conditions ◽  
Differential Subordination ◽  
Open Unit ◽  
Sufficient Condition ◽  
Open Unit Disk ◽  
Lemniscate Of Bernoulli

This article comprises the study of differential subordination with analogue of q -derivative. It includes the sufficient condition on γ for 1 + γ ∂ z q h z / h n z to be subordinated by 1 + A z / 1 + B z , − 1 ≤ B < A ≤ 1 , and implies that h z ≺ 1 + z , where h z is the analytic function in the open unit disk. Moreover, certain sufficient conditions for q -starlikeness of analytic functions related with lemniscate of Bernoulli are determined.

scholarly journals Sufficient Conditions for Janowski Starlikeness

2007 ◽  
Vol 2007 ◽  
pp. 1-7 ◽  
Author(s):  
Rosihan M. Ali ◽  
V. Ravichandran ◽  
N. Seenivasagan
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Analytic Functions ◽  
Sufficient Conditions ◽  
Open Unit ◽  
Open Unit Disk

LetA,B,D,E∈[−1,1]and letp(z)be an analytic function defined on the open unit disk,p(0)=1. Conditions onA,B,D, andEare determined so that1+βzp'(z)being subordinated to(1+Dz)/(1+Ez)implies thatp(z)is subordinated to(1+Az)/(1+Bz). Similar results are obtained by considering the expressions1+β(zp'(z)/p(z))and1+β(zp'(z)/p2(z)). These results are then applied to obtain sufficient conditions for analytic functions to be Janowski starlike.

scholarly journals A Generalized Class of Functions Defined by the q-Difference Operator

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2361
Author(s):  
Loriana Andrei ◽  
Vasile-Aurel Caus
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Analytic Functions ◽  
Difference Operator ◽  
Sufficient Conditions ◽  
Open Unit ◽  
Open Unit Disk ◽  
New Class ◽  
The Relationship ◽  
Class Of Functions

The goal of the present investigation is to introduce a new class of analytic functions (Kt,q), defined in the open unit disk, by means of the q-difference operator, which may have symmetric or assymetric properties, and to establish the relationship between the new defined class and appropriate subordination. We derived relationships of this class and obtained sufficient conditions for an analytic function to be Kt,q. Finally, in the concluding section, we have taken the decision to restate the clearly-proved fact that any attempt to create the rather simple (p,q)-variations of the results, which we have provided in this paper, will be a rather inconsequential and trivial work, simply because the added parameter p is obviously redundant.

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