α-Bloch Spaces Research Articles

Landau's theorem for harmonic Bloch functions on simply connected domains

For α ∈ ( 0 , ∞ ) , let B H , Ω ( α ) denote the class of harmonic α-Bloch mappings on a proper simply connected domain Ω ⊆ C . In this article, we study coefficient estimates for harmonic α-Bloch mappings on a proper simply connected domain containing the unit disk. Furthermore, we establish the Landau's theorem and coefficient estimates for the harmonic α-Bloch space B H , Ω γ ( α ) on the shifted disk Ω γ , where Ω γ := { z ∈ C : | z + γ 1 − γ | < 1 1 − γ } and 0 ≤ γ < 1 .

Boundedness and Compactness of Weighted Composition Operators from α-Bloch Spaces to Bers-Type Spaces on Generalized Hua Domains of the First Kind

We address weighted composition operators ψCϕ from α-Bloch spaces to Bers-type spaces of bounded holomorphic functions on Y, where Y is a generalized Hua domain of the first kind, and obtain some necessary and sufficient conditions for the boundedness and compactness of those operators.

Difference of weighted composition operators from α-Bloch spaces to β-Bloch spaces

We study the boundedness and compactness of the differences of two weighted composition operators acting from α-Bloch space to β-Bloch space on the open unit disk. Such study has a relationship to the topological structure of weighted composition from α-Bloch space to β-Bloch space.

New characterizations for differences of integral-type operators from α-Bloch space to β-Bloch-Orlicz space

New characterizations for differences of integral-type operators from α-Bloch space to β-Bloch-Orlicz space

α-Bloch空间到β-Bloch空间的广义积分算子的差分

α-Bloch空间到β-Bloch空间的广义积分算子的差分

Properties of $\beta$-Cesàro operators on $\alpha$-Bloch space

For each α>0, the α-Bloch space consists of all analytic functions f on the unit disk satisfying sup|z|<1(1−|z|2)α|f′(z)|<+∞. We consider the following complex integral operators, namely the β-Cesàro operator C β ( f ) ( z ) = ∫ 0 z f ( w ) w ( 1 − w ) β d w and its generalization, acting from the α-Bloch space to itself, where f(0)=0 and β∈ℝ. We investigate the boundedness and compactness of the β-Cesàro operators and their generalizations. Also we calculate the essential norm and spectrum of these operators.

Extreme points and support points of harmonic $\alpha$-Bloch mappings

We consider the problem of existence of the extreme points and support points of harmonic α-Bloch mappings and little harmonic α-Bloch mappings. First, we prove a necessary condition for a little harmonic α-Bloch mapping to be an extreme point of the unit ball of the normalized little harmonic α-Bloch spaces in the unit disk 𝔻, and we also show that a harmonic α-Bloch unit-valued set consists of several simple closed pairwise disjoint analytic curves and several isolated points. Then we show that a harmonic α-Bloch mapping f is a support point of the unit ball of the normalized harmonic α-Bloch spaces in 𝔻 if and only if the α-Bloch unit-valued set of f is not empty. We also give a characterization for the support points of the unit ball of the harmonic α-Bloch spaces in 𝔻.

Composition Operators on Some Banach Spaces of Harmonic Mappings

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.

Characterizations of Bloch-Type Spaces of Harmonic Mappings

We study the Banach space BHα (α&gt;0) of the harmonic mappings h on the open unit disk D satisfying the condition supz∈D⁡(1-z2)α(hzz+hz¯z)&lt;∞, where hz and hz¯ denote the first complex partial derivatives of h. We show that several properties that are valid for the space of analytic functions known as the α-Bloch space extend to BHα. In particular, we prove that for α&gt;0 the mappings in BHα can be characterized in terms of a Lipschitz condition relative to the metric defined by dH,α(z,w)=sup⁡{hz-hw:h∈BHα,hBHα≤1}. When α&gt;1, the harmonic α-Bloch space can be viewed as the harmonic growth space of order α-1, while for 0&lt;α&lt;1, BHα is the space of harmonic mappings that are Lipschitz of order 1-α.

Second-Order Linear Differential Equations with Solutions in Analytic Function Spaces

This research is concerned with second-order linear differential equation f′′+A(z)f=0, where A(z) is an analytic function in the unit disc. On the one hand, some sufficient conditions for the solutions to be in α-Bloch (little α-Bloch) space are found by using exponential type weighted Bergman reproducing kernel formula. On the other hand, we find also some sufficient conditions for the solutions to be in analytic Morrey (little analytic Morrey) space by using the representation formula.

Differences of weighted differentiation composition operators from α-Bloch space to H∞ space

This paper characterizes the boundedness and compactness of the differences of weighted differentiation composition operators acting from the ?-Bloch space B? to the space H? of bounded holomorphic functions on the unit disk D.

Q_K-Type Spaces of Quaternion-Valued Functions

In this paper we develop the necessary tools to generalize the Q_K-type function classes to the case of the monogenic functions defined in the unit ball of R^3, some important basic properties of these functions are also considered. Further, we show some relations between Q_K(p,q) and α-Bloch spaces of quaternion-valued functions.

Maximal functions and properties of the weighted composition operators acting on the Korenblum, &amp;alpha;-Bloch and &amp;alpha;-Zygmund spaces

Using certain maximal analytic functions, we obtain new characterizations of the continuity and compactness of the weighted composition operators when acts between Korenblum spaces, α-Bloch spaces and when acts from certain weighted Banach spaces of analytic functions with a logarithmic weight into α-Bloch spaces. As consequence of our results, we obtain a new characterization of the continuity and compactness of composition operators acting between α-Zygmund spaces.

Operator norms and essential norms of weighted composition operators between Banach spaces of analytic functions

In this work, we characterize the bounded and compact weighted composition operators from a large class of Banach space of analytic functions on the open unit disk into weighted Banach spaces, determine the operator norm exactly and obtain estimates on the essential norm. We obtain an exact formula of the essential norm when such operators map the Hardy spaces, the weighted Bergman spaces, the Bloch space and the little Bloch space into the Banach space Hμ∞ (where the weight μ is a continuous function). In this general setting we also study such operators when the target space is an α-Bloch space for any α>0 and more generally, any Bloch-type space. As special cases, we derive norm and essential norm approximations for the operators acting on the Hardy spaces and the weighted Bergman spaces. Immediate corollaries are results on certain integral operators.

Generalized weighted composition operators from α-Bloch spaces into weighted-type spaces

Some criteria for the boundedness, as well as for the compactness, of the generalized weighted composition operator $D^{n}_{\varphi, u}$ from α-Bloch spaces into weighted-type spaces are given. Estimates for the norm and the essential norm of the operator are also given. Our results extend and complement some results in the literature.

A NEW ESSENTIAL NORM ESTIMATE OF COMPOSITION OPERATORS FROM α-BLOCH SPACES INTO μ-BLOCH SPACES

Let μ be any weight function defined on the unit disk 𝔻 and let ϕ be an analytic self-map of 𝔻. In this paper, we show that the essential norm of composition operator Cϕmapping from the α-Bloch space, with α &gt; 0, to μ-Bloch space [Formula: see text] is comparable to [Formula: see text] where, for a ∈ 𝔻, σais a certain special function in α-Bloch space. As a consequence of our estimate, we extend recent results, about the compactness of composition operators, due to Tjani in [Compact composition operators on Besov spaces, Trans. Amer. Math. Soc.355(11) (2003) 4683–4698] and Malavé Ramírez and Ramos-Fernández in [On a criterion for continuity and compactness of composition operators acting on α-Bloch spaces, C. R. Math. Acad. Sci. Paris351 (2013) 23–26, http://dx.doi.org/10.1016/j.crma.2012.11.013 ].

Weighted composition operators from the Besov spaces into the weighted-type space [formula omitted

Let μ denote a positive continuous function on the open unit disk D. In this work, we characterize the bounded weighted composition operators from the analytic Besov spaces Bp (1≤p<∞) into the weighted-type space Hμ∞ consisting of the analytic functions on D such that supz∈Dμ(z)|f(z)|<∞ and determine their operator norms. We also determine the essential norm of the bounded weighted composition operators acting on the Dirichlet space and obtain explicit estimates when p=1. In the general case when the operator maps Bp into Hμ∞, we derive an approximation of the essential norm that yields a characterization of the compact weighted composition operators. Finally, we derive characterizations of the bounded and the compact weighted composition operators from the spaces of antiderivatives of functions in Bp and BMOA into the α-Bloch spaces.

WEIGHTED LIPSCHITZ CONTINUITY, SCHWARZ–PICK'S LEMMA AND LANDAU–BLOCH'S THEOREM FOR HYPERBOLIC-HARMONIC MAPPINGS IN ℂN

In this paper, we discuss some properties on hyperbolic-harmonic functions in the unit ball of ℂ n . First, we investigate the relationship between the weighted Lipschitz functions and the hyperbolic-harmonic Bloch spaces. Then we establish the Schwarz–Pick type theorem for hyperbolic-harmonic functions and apply it to prove the existence of Landau-Bloch constant for functions in α-Bloch spaces.

On a criterion for continuity and compactness of composition operators acting on α-Bloch spaces

In this Note we give a new criterion for the continuity and compactness of composition operators acting on α-Bloch spaces.

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

Let ? be an analytic self-map of the open unit disk D on the complex plane and ? &gt; 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.