Integral-type operators from weighted Bloch spaces into Bergman-type spaces

In this paper we study bounded and compact weighted composition operator, induced by a fixed analytic function and an analytic self-map of the open unit disk, from Bergman space into weighted Bloch space. As a corollary, obtain the characterization of composition operator from Bergman space into weighted Bloch space.

Based on recent results on boundedness of Bergman projection with positive Bergman kernel in analytic spaces in various types of domains inCn, we extend our previous sharp results on distances obtained for analytic Bergman type spaces in unit disk to some new Bergman type spaces in Lie ball, bounded symmetric domains of tube type, Siegel domains, and minimal bounded homogeneous domains.

Let \({\mathbb{D}}\) be the open unit disk in the complex plane \({\mathbb{C}}\) and \({H(\mathbb{D})}\) the class of all analytic functions on \({\mathbb{D}}\). Let \({\phi}\) be an analytic self-map of \({\mathbb{D}}\) and \({u \in H(\mathbb{D})}\). As a class of new space, we first study the properties of the logarithmic Bergman-type space. We also use Young’s function to define another new space as a generalization of Zygmund space, called the Zygmund–Orlicz space. We study some of its properties and show that the Zygmund–Orlicz space is isometrically equal to certain Zygmund-type space for a very special weight. This allows us to define the little Zygmund–Orlicz space which is a closed subspace of the Zygmund–Orlicz space. As some applications of properties obtained, we characterize the boundedness and compactness of the product-type operators \({D^{n} M_{u}C_{\phi}}\), \({D^{n}C_{\phi}M_{u}}\), \({C_\phi D^n M_u}\), \({M_uD^nC_{\phi}}\), \({M_{u}C_{\phi}D^{n}}\), and \({C_{\phi}M_{u}D^{n}}\) from the logarithmic Bergman-type space to the Zygmund–Orlicz space and the little Zygmund–Orlicz space.

We study the following two integral operators equation image where g is an analytic function on the open unit disk in the complex plane. The boundedness and compactness of these two operators between the α ‐Bloch space Bα and the Besov space are discussed in this paper (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

In this paper, we characterize the bonudedness and compactness of weighted composition operators from weighted Bergman spaces to weighted Bloch spaces. Also, we investigate weighted composition operators on weighted Bergman spaces and extend the obtained results in the unit ball of $mathbb{C}^n$.

The boundedness and the compactness of the two integral operators;, where is an analytic function on the open unit disk in the complex plane, on the Zygmund space are studied.

A classical result due to Blaschke states that for every analytic self-map f of the open unit disc of the complex plane there exists a Blaschke product B such that the zero sets of f and B agree. Indeed, a sequence is the zero set of an analytic self-map of the open unit disc if and only if it satisfies the simple geometric condition known as the Blaschke condition. In contrast, the critical sets of analytic self-maps of the open unit disc have not been completely described yet. In this paper, we show that for every analytic self-map f of the open unit disc there is even an indestructible Blaschke product B such that the critical sets of f and B coincide. We further relate the problem of describing the critical sets of bounded analytic functions to the problem of characterizing the zero sets of some weighted Bergman space as well as to the Berger–Nirenberg problem from differential geometry. By solving the Berger–Nirenberg problem in a special case, we identify the critical sets of bounded analytic functions with the zero sets of the weighted Bergman space 𝒜12.

Let $$\mathbb {D}$$ be the open unit disk in the complex plane $$\mathbb {C}$$, and let $$\varphi $$ be a holomorphic function from disk $$\mathbb {D}$$ into $$\mathbb {D}$$. We study the composition operator $$C_{\varphi }$$ on the variable exponent Bergman space in unit disk, and prove that this operator is bounded on $$A^{p(\cdot )}(\mathbb {D})$$. We give a sufficient condition for the compactness of this operator on $$A^{p(\cdot )}(\mathbb {D})$$ as well.

On a class of starlike functions related to a shell-like curve connected with Fibonacci numbers

Recently, the generalized Mittag-Leffler (EGML) function is extended by utilizing the extended Beta function. Based on this type of function, we provide a new integral operator in the open unit disk. The present article discusses several applications of differential subordination for certain normalized analytic functions in the open unit disk, which are acted upon by Salagean type integral operator.

Let 𝔻 be the open unit disk in the complex plane ℂ and \(dA(z) = \frac {1}{\pi }dx dy\), the normalized area measure on 𝔻. Let \({L_{a}^{2}}(\mathbb {D}, dA)\) be the Bergman space consisting of analytic functions on 𝔻 that are also in \(L^{2}(\mathbb {D}, dA)\). Let \(\mathcal {L}({L_{a}^{2}}(\mathbb {D}))\) be the set of all bounded linear operators from the Hilbert space \({L_{a}^{2}}(\mathbb {D})\) into itself. For \(T\in \mathcal {L}({L_{a}^{2}}(\mathbb {D}))\), let \(\widetilde {T}\) denote the Berezin transform of T. In this paper, we find conditions on positive operators \(S, T\in \mathcal {L}({L_{a}^{2}}(\mathbb {D}))\) such that \(\widetilde {S}(z)\geq \widetilde {T}(z)\) for all z ∈ 𝔻.

Consider an open unit disk D = z ∈ ℂ : z < 1 in the complex plane ℂ , ξ a holomorphic function on D , and ψ a holomorphic self-map of D . For an analytic function f , the weighted composition operator is denoted and defined as follows: W ξ , ψ f z = ξ z f ψ z . We estimate the essential norm of this operator from Dirichlet-type spaces to Bers-type spaces and Bloch-type spaces.

Let \( A^2 (\mathbb{D}) \) be the Bergman space over the open unit disk \( \mathbb{D} \) in the complex plane. Korenblum conjectured that there is an absolute constant \( c \in (0,1) \), such that whenever \( |f(z)| \leq |g(z)| (f, g \in A^2 (\mathbb{D})) \) in the annulus \( c < |z| < 1 \), then \( ||f|| \leq ||g|| \). In 1999 Hayman proved Korenblum’s conjecture. But the sharp value of c (we use γ to denote this sharp value and call it Korenblum’s constant) is still unknown. In this paper we give an upper bound on γ, that is, γ < 0.685086, which improves an earlier result of the author.

Let D be the open unit disk in the complex plane, and let C be its boundary, the unit circle.If x £ C, then by an arc at x we mean a simple arc y with one end point at x such that y-{x}£ö.If/is a function mapping D into some metric space M, then the set of curvilinear convergence of/is defined to be {x e C: there exists an arc y at x and there exists a point p e M such that f(z) ->p as z -»■ x along y}.If ^ is a function whose domain is a subset F of the set of curvilinear convergence of/ then <f> is called a boundary function for/if, and only if, for each x e E there exists an arc y at x such that/(z) -*■ <f>(x) as z -> x along y.Let 5 be another metric space.We shall say that a function <f> is of Baire class ^ 1(5, M) if (i) domain <£ = S, (ii) range ^ £ M, and(iii) there exists a sequence {<£"} of continuous functions, each mapping S into M, such that <f>n^>-<p pointwise on 5. We shall say that (f> is of honorary Baire class ^ 2(S, M) if (i) domain <f> = S, (ii) range ^£M, and (iii) there exists a countable set TVs 5 and there exists a function y!r of Baire class ^ 1(5, M) such that <f>(x) = tp(x) for every xe S-N.It is known that if/is a continuous function mapping F into the Riemann sphere, then the set of curvilinear convergence off is of type Fa6, and any boundary function for/is of honorary Baire class ^2(C, Riemann sphere).(See [3], [4], [5], [6],[9].) J. E. McMillan [6] posed the following problem.If A is a given set in C of type Fao, and if </> is a function of honorary Baire class Ú2(A, Riemann sphere), does there always exist a continuous function / mapping D into the Riemann sphere such that A is the set of curvilinear convergence of/ and <f> is a boundary function for /?The purpose of this paper is to give an affirmative answer to McMillan's question.However, the corresponding question for real-valued functions remains open.(See Problems 1 and 2 at the end of this paper.)In proving our result, we first give a proof under the assumption that <j> is a bounded complexvalued function, and we then use a certain device to transfer the theorem to the Riemann sphere.As we shall indicate in an appendix, the same device can be

The main purpose of this paper is to obtain an upper bound for the second Hankel determinant for functions belonging to a subclass of bi-univalent functions in the open unit disk in the complex plane. Furthermore, the presented results in this work improve or generalize the recent works of other authors.