Integration and Differentiation in Hardy Growth Spaces

Convex and alpha-prestarlike subordination chains

Let Ω be a domain in the complex plane \(\mathbb{C}\) containing the origin and let \({\cal P}_{n}^{0}(\Omega)\) denote the set of all polynomials P of degree ≤ n satisfying the conditions \(P(0)=0\) and \(P(\mathbb{D})\subset \Omega\), where \(\mathbb{D}=\left\{z\,:\,\,|z|<1\right\}\) is the unit disk in the complex plane. The maximal range, denoted by Ω n , is then defined as the union of all sets \(P(\mathbb{D}), P \in {\cal P}_{n}^{0}\) .We shall construct extremal polynomials and maximal polynomial range for a domain of intersection of two circular disks.

There are over one hundred articles in the literature dealing with second-order differential inequalities and differential subordinations in the complex plane, and only two articles dealing with very narrow classes of third-order results. This article addresses a much more general class of third-order differential inequalities and subordinations. Let Ω and Δ be given sets in , let p be an analytic in the unit disc with . The authors determine the conditions on an admissible class of operators , so that the third-order differential subordination implies . Applications of these results to third-order differential inequalities, third-order differential subordinations, univalent functions and Bessel functions are presented.

Recently, complex fuzzy sets have become a powerful tool for expanding the range of fuzzy sets that are located in a unit disk within complex plane. In this article, the complex fuzzy numbers are discussed and implemented to handle complex fractional fuzzy partial differential equation involving complex time-fractional fuzzy heat equation in sense of Hukuhara differentiability. After that, an explicit finite difference scheme known as the forward time center space are implemented to solve complex time-fractional fuzzy heat equations. The uncertainty present in our problem appears in the boundary and initial conditions as well as coefficients in both amplitude and phase terms simultaneously where the Convex normalized triangular fuzzy numbers are extended to unit disk in the complex plane. The properties and benefits of complex fuzzy set theory have been employed in the proposed numerical methods. A numerical experiment is used to showcase the efficiency and practicality of the presented method. The obtained outcomes are found to be consistent with the exact solution and related theoretical concepts.

In this article, we will discuss the type problem to judge whether a given infinitely sheeted covering surface of the complex sphere determined by the sequence of pasting arcs γ n in the complex plane is parabolic, i.e. , or not, in terms of Γ. Here the sequence of simple arcs γ n is supposed to satisfy the following two conditions: for every and the sequence of end points of γ n converges to ∞, the point at infinity of . Then is constructed by pasting to crosswise along γ n for every with γo = ∅ so that is either the unit disc or the complex plane. Then our type problem specifically asks whether the convergence to zero of the sequence of capacities cap of γ n relative to the region implies the parabolicity of or not. The purpose of this article is to answer affirmatively to the above question under the additional assumption that two end points of each pasting arc γ n is symmetric about the imaginary axis of for every .

State space methods for solving tangential interpolation problems for matrix functions in a given class have been developed extensively. These are interpolation problems in which the values in certain directions of the interpolant matrix functions, and several of their derivatives, are prescribed at given points in the complex plane. For the most part, the matrix functions under consideration are rational functions or polynomials (perhaps with symmetries); however, other important classes of matrix functions have been studied as well, notably, the Nevanlinna classes on the unit disk and the upper half-plane, and the Hardy class H/sub 2/ on the unit disk. This development is motivated by many important applications in systems and control. We present some recent results and open problems in this area.

INTRODUCTION. This note concerns Riemann mappings of rectangles onto the unit disk in the complex plane. Let's fix some notation for the discussion. For the unit circle and unit disk we'll use, respectively, S1 and D. For the rectangle parallel to the coordinate axes, centered at the origin 0 E C with height 1 and base length a > 0, so that the corners are at (?a/2, ? 1/2), we'll use R(a). In particular, it will be convenient to have R(b) c R(a) if b < a. Riemann's mapping theorem says that every simply connected plane domain Ql not equal to the whole plane can be mapped conformally one-to-one onto D. Thus this certainly holds for the interior of rectangles. These maps are thoroughly well understood from several points of view, primarily using the Schwarz-Christoffel formula (elliptic functions in a not too subtle disguise). The purpose of this note is to show that rectangles with different modulus (different values of a) can be seen as conformally different and completely characterized by sets of four distinct points on S with very little machinery. In fact, our only tools will be the Schwarz reflection principle, the Schwarz lemma (in other words, Schwarz seemed to have a pretty good sense about this sort of thing), some elementary conformal mappings, and the concept of normal families of analytic functions. Perhaps then the real purpose of this note is to serve as a reminder of the power of the profound, beautiful, and essentially elementary maximum modulus principle. The only reference necessary for this discussion is [A]. The theorem of Lowner may also be found in [C]. The proof given there is of a somewhat different flavor, and Lowner's proof [L] was roughly the same as the one given here.

In this paper, we give some sufficient and necessary conditions for subordination of the solution of the Briot–Bouquet differential equation: where δ and λ are real numbers with δ≠0, δ+λ>0, and U is the unit disc in the complex plane C. Using these subordinations, we obtain some sufficient and necessary conditions that the integral operators and preserve starlikeness, where α, β, δ, and γ are real numbers with α+δ=β+γ>0, αβ≠0.

We develop an approach to describe invariant subspaces of the integration operator on various scales of weighted spaces of holomorphic functions on the unit disk and the complex plane. It allows us to solve the problem for wide classes of Bergman, Bloch, Dirichlet, and Fock spaces, while all previous known results concern spaces defined by some weights of a special form. In addition, we also show that an analogous method works as well for the differentiation operator on weighted spaces of entire functions.

Let v be a radial weight function on the unit disc or on the complex plane. It is shown that for each analytic function $$f_0$$ in the Banach space $$H_v^\infty $$ of all analytic functions f such that v|f| is bounded, the distance of $$f_0$$ to the subspace $$H_v^0$$ of $$H_v^\infty $$ of all the functions g such that v|g| vanishes at infinity is attained at a function $$g_0 \in H_v^0$$ . Moreover a simple, direct proof of the formula of the distance of f to $$H_v^0$$ due to Perfekt is presented. As a consequence the corresponding results for weighted Bloch spaces are obtained.

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.

&lt;p&gt;Complex spherical fuzzy sets (CSFSs) are a theory that addresses confusing and unreliable information in real-life decision-making contexts by integrating elements of two theories: spherical fuzzy sets (SFSs) and complex fuzzy sets (CFSs). CSFSs are classified into three categories, represented by polar coordinates: membership, nonmember, and abstention. These grades are located on a complex plane within a unit disc. It is necessary for the total squares representing the real components of the grades for abstinence, membership, and non-membership to not surpass a certain interval. Several aspects of CSFS and the corresponding operational laws were examined in this work. The key components of this article were based on CSFs, including complex spherical fuzzy Schweizer-Sklar prioritized aggregation (CSFSSPA), complex spherical fuzzy Schweizer-Sklar weighted prioritized aggregation (CSFSSWPA), complex spherical fuzzy Schweizer-Sklar prioritized geometry (CSFSSPG), and complex spherical fuzzy Schweizer-Sklar prioritized weighted geometry (CSFSSWPG). Additionally, the suggested operators' specific instances were examined. The main outcome of this work includes new aggregation techniques for CSFS information, based on t-conorm and t-norm from Schweizer-Sklar (SS). The basic characteristics of the operators were established by this study. We looked at a numerical example centered on efficient mobile e-tourism selection to show the effectiveness and viability of the recommended approaches. Additionally, we carried out a thorough comparative analysis to assess the outcomes of the suggested aggregation approaches in comparison to the current methods. Last, we offer an overview of the planned study and talk about potential directions for the future.&lt;/p&gt;

Differential subordinations and inequalities in the complex plane

Let w(z) be regular in the unit disc U:|z|&lt;l, with w(0) = 0 and let h(r, s, t) be a complex function defined in a domain D of C3. The author determines conditions on h such that ifz∈U, then |w(z)|&lt; 1 for z ∈ U and n= 0, 1, 2, …. Here Dnw(z) = (z/(l-z)n+1*w(z), where * stands for the Hadamard product (convolution). Some applications of the results to certain differential equations are given.

We calculate the essential norm of some extensions of the generalized composition operators between kth weighted-type spaces on the unit disk in the complex plane, considerably extending some results in the literature.