Approximation by Inner Functions

In this paper, we study the computation of the moments associated to rational weight functions given as a power spectrum with known or unknown poles of any order in the interior of the unit disc. A recursive algebraic procedure is derived that computes the moments in a finite number of steps. We also study the associated interpolatory quadrature formulas with equidistant nodes on the unit circle. Explicit expressions are given for the positive quadrature weights in the case of a polynomial weight function. For rational weight functions with simple poles, mostly real or uniformly distributed on a circle in the open unit disc, we also obtain expressions for the quadrature weights and sufficient conditions that guarantee that they are positive. The Poisson kernel is a simple example of a rational weight function, and in the last section, we derive an asymptotic expansion of the quadrature error.

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.

For a proper holomorphic map F from a ball B n into another ball B N , we give a criterion when a rational map F is equivalent to a polynomial one.As application, we show that proper rational holomorphic maps from B 2 into B N of degree two are equivalent to polynomial maps, and we also give an example of a rational holomorphic maps of degree 3 that are 'almost' linear but are not equivalent to holomorphic polynomial maps.

A T T -symmetric univalent function is a complex valued function that is conformally mapping the unit disk onto itself and satisfies the symmetry condition ϕ [ T ] ( ζ ) = [ ϕ ( ζ T ) ] 1 ∕ T {\phi }^{\left[T]}\left(\zeta )={\left[\phi \left({\zeta }^{T})]}^{1/T} for all ζ \zeta in the unit disk. In other words, it is a complex function that preserves the unit disk’s shape and orientation and is symmetric about the unit circle. They are used in the study of geometric function theory and the theory of univalent functions. In recent effort, we extend the class of fractional anomalous diffusion equations in a symmetric complex domain. we aim to present the analytic univalent solution for such a class using special functions technique. Our analysis and comparative findings are further supported by the geometric simulations for the univalent solution such as the convexity and starlikeness of the diffusion. As a consequence of illustration of a list of conditions yielding the univalent solutions (normalize analytic function in the open unit disk), the normalization of diffusion shape is achieved.

A function with complex values and at every point of the specific domain contains a derivative is commonly known as analytic functions which is also referred as holomorphic functions. We begin by interpreting \(A\) as the class for all holomorphic functions \(L(\xi)\) with a Taylor series expansion written in the form: \[L(\xi) = \xi + \sum_{i=2}^{\infty} x_i \xi^i\] where \(x_i \in \mathbb{C}\) and \(\xi \in D\). \(D\) is the open unit disk where \(D = \{\xi : \xi \in \mathbb{C}, |\xi| < 1 \}\). Furthermore, we suggest the subclass of \(A\) that is univalent in \(D\) represented as \(S\). It is commonly known that the main subclasses of class \(S\) are the class of starlike functions and the class of convex functions. To develop and analyze the Fekete-Szegö problems, the theory of geometric function contributes significantly to this. Moreover, the frequent use of \(q\)-calculus as a general direction of research among mathematicians has caught our attention. In this research, we attained the initial coefficients, \(x_2\) and \(x_3\), and the upper bound for the functional \(|x_3 - \nu x_2^2|\) of functions \(L\) in the two new subclasses that are introduced by involving the Sălăgeăn \(q\)-differential operator, \(M_q^\eta L(\xi)\) and the definition of subordination.

We study the structure of inductive limits of weighted spaces of harmonic and holomorphic functions defined on the open unit disk of ℂ, and of the associated weighted locally convex spaces. Using a result of Lusky we prove, for certain radial weights on the open unit disk D of ℂ, that the spaces of harmonic and holomorphic functions are isomorphic to complemented subspaces of the corresponding Köthe sequence spaces. We also study the spaces of harmonic functions for certain non-radial weights on D. We show, under a natural sufficient condition for the weights, that the spaces of harmonic functions on D are isomorphic to corresponding spaces of continuous or bounded functions on ∂D.

Let 1 ≤ p < ∞, let be the open unit ball of lp and let Δ be the open unit disc in ℂ. We prove that there is a holomorphic map such that F(Δ) is dense in and such that whenever .

We show that for every connected analytic subvariety V there is a pseudoconvex set Ω such that every bounded matrix-valued holomorphic function on V extends isometrically to Ω. We prove that if V is two analytic discs intersecting at one point, if every bounded scalar valued holomorphic function extends isometrically to Ω, then so does every matrix-valued function. In the special case that Ω is the symmetrized bidisc, we show that this cannot be done by finding a linear isometric extension from the functions that vanish at one point.

Assume that there are given a simply connected domain Gin the complex plane containing the open unit disk, but not the closed unit disk and an infinite triangular matrix A. Under suitable conditions on A, which are even weaker than its P-regularity, in this paper we construct a noncontinuable holomorphic function Φ on Gsuch that the A-transforms of its Taylor expansion around any point of Gtend to a scalar multiple of Φ compactly in Gand exhibit universal properties in the complement of Gwith respect to uniform approximation on compact sets and to almost everywhere approximation on measurable sets.

A Herglotz function is a holomorphic map from the open complex unit disk into the closed complex right halfplane. A classical Herglotz function has an integral representation against a positive measure on the unit circle. We prove a free analytic analogue of the Herglotz representation and describe how our representations specialize to the free probabilistic case. We also show that the set of representable Herglotz functions arising from noncommutative conditional expectations must be closed in a natural topology.

Let be a subharmonic function on the open unit disc , centered at the origin of the complex plane, and let be a holomorphic function such that . A classical result, known as Littlewood subordination principle, states , where and are integral means over the circle of radius centered at the origin, of the functions and , respectively. In this note, we obtain an unexpected improvement of Littlewood subordination principle in the case when the function is univalent, by proving that where . We also list some applications of this result including an improved variant of Rogosinski theorem with univalent symbol.

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.

Recently J. Miao proved that if is a holomorphic function with Hadamard gaps on the open unit disc D then f ∈ Xp if and only if f ∈ Bp if and only if if and if only if where Xp, Bp and denote respectively the class of holomorphic functions on D which satisfy |f′(z)|p(1 − |z|2)p − 1dxdy is a finite measure, a Carleson measure and a little Carleson measure on D. In this paper we give a higher-dimensional version of Miao's result.

Products of multiplication composition and differentiation operators on weighted Bergman spaces

Abstract. AsequenceofcompositionoperatorsonHardyspaceiscon-sidered. Weprovethat,bynumericalrangeproperties,itisSOT-conver-gencebutnotconverge. 1. introductionLet ϕ be a holomorphic self-map of the unit disc U := {z ∈ C : |z| < 1}.The function ϕ induces the composition operator C ϕ , defined on the space ofholomorphic functions on Uby C ϕ f = f ◦ ϕ. The restriction of C ϕ to variousBanach spaces of holomorphic functions on U has been an active subject ofresearch for more than three decades and it will continue to be for decadesto come (see [11], [12] and [4]). Let H 2 denote the Hardy space of analyticfunctions on the open unit disc with square summable Taylor coefficients. Inrecent yearsthe study ofcomposition operatorson the Hardy space has receivedconsiderable attention.In this paper we consider the numerical range of elliptic composition opera-tors on H 2 . The numerical range of a bounded linear operator A on a Hilbertspace H is the set of complex numbersW(A) := {hAx,xi : x ∈ H,kxk = 1},where h·,· denotes the inner product in H.In [9] V. Matache determined the shape W(C