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An iterated Neumann boundary value problem for the polyanalytic operator is investigated in the unit disc of the complex plane. Although this problem can be treated for domains in the complex plane having a harmonic Green function, the unit disk is particularly interesting as it is the origin of complex boundary value problems and the prototype of simply connected domains. Moreover, the resulting formulas for solutions and for solvability conditions are expressed by integral formulas with explicit kernels. By the way, the result verifies the findings for domains with harmonic Green function. [ABSTRACT FROM AUTHOR]

Motivated by the relationship between the area of the image of the unit disk under a holomorphic mapping h and that of zh, we study various L2 norms for Tϕ(h), where Tϕ is the Toeplitz operator with symbol ϕ. In Theorem 2.1, given polynomials p and q we find a symbol ϕ such that Tϕ(p) = q. We extend some of our results to the polydisc. [ABSTRACT FROM AUTHOR]

There is considered an interpolation problem f(λn) = bn in the class of holomorphic in the unit disk U(0; 1) = {z ε C: |z| < 1} functions of finite η-type, i.e such that... where η: [1;+∞) → [0;+∞) is an increasing convex function with respect to ln t and ln t = o (η(t)) (t → +∞). There were received sufficient conditions of the interpolation problem solvability in terms of the counting functions.... Earlier, in 2004, necessary conditions were obtained (Ukr. Math. J., 56 (2004), №3) in these terms. For the moderate growth of f (when the majorant η = ψ satisfies the condition ψ (2x) = O (ψ (x)), x → +∞) that problem was solved in J. Math. Anal. Appl., 414 (2014), №1. In this paper, we remove any restrictions on the growth of η and construct an interpolation function f such that... [ABSTRACT FROM AUTHOR]

In this paper we extend some previously known results in the unit ball and in the unit disk on Bergman projections in bounded strongly pseudoconvex domains and in tubular domains over symmetric cones. It should be also added that on the other hand all our new results are also related with analytic spaces in product domains. Below first we give some definitions and basic lemmas and facts on bounded strongly pseudoconvex domains and then in tubular domains over symmetric cones. [ABSTRACT FROM AUTHOR]

We provide new sharp embeddig theorem in Bergman type spaces in tubular domains over symmetric cones related with so called diagonal map. Related embeddings with a certain condition on Bergman kernel will be also provided. [ABSTRACT FROM AUTHOR]

The boundedness and compactness of weighted composition operators have been extensively investigated on various analytic function spaces. In this paper, we study the boundedness and compactness of two several variables differences weighted composition operators on some analytic function spaces.

We provide a complete description of zero sets of some new analytic area Nevanlinna type spaces in the unit disk. Our theorems extend recent sharp results of E. Rodikova to larger analytic function spaces of area Nevanlinna type. [ABSTRACT FROM AUTHOR]

The boundedness and compactness of weighted composition operators have been extensively investigated on various analytic function spaces. In this paper, we study the boundedness and compactness of two several variables differences weighted composition operators on some analytic function spaces. [ABSTRACT FROM AUTHOR]

The role played by explicit formulas for solving boundary value problems for elliptic equations and systems is well known. In this paper, explicit formulas for a general solution of the Dirichlet problem for second-order elliptic systems in the unit disk are given. In addition, an iterative method for solving this problem for systems with respect to two unknown functions is described, and an integral representation of the Poisson type is obtained by applying this method. [ABSTRACT FROM AUTHOR]

Convolution operators have profited in various areas of science. They are utilized in the investigations of computing techniques. A new convolution operator linked to a specific class of multi-valent meromorphic functions in the punctured unit disk (symmetric domain) is formulated. This analysis uncovers certain properties on the connections as well as the power series. We study a novel class of holomorphic functions concerning the recommended new operator. The second part of the outcome concerns the boundedness of the suggested difference structure given by the proposed operator. We focus on the Bloch space of meromorphic functions in the open unit disk. In this case, we use the spherical derivative. To obtain the maximum value of the polar derivative of the polynomials created by their partial sums, we use their partial sums as applications of the suggested operator. [ABSTRACT FROM AUTHOR]

We consider the diffusion asymptotics of a coupled model arising in radiative transfer in a unit disk in R 2 with one-speed velocity. The model consists of a kinetic equation satisfied by the specific intensity of radiation coupled a nonhomogeneous heat equation satisfied by the material temperature. For the well-prepared data, we prove the existence and the nonequilibrium diffusion limit of solutions to the initial boundary problem for the coupled model. [ABSTRACT FROM AUTHOR]

We prove a sharp Schwarz lemma type inequality for the Weierstrass–Enneper parameterization of minimal disks. It states the following. If F : D → Σ is a conformal harmonic parameterization of a minimal disk Σ ⊂ R 3 , where D is the unit disk and | Σ | = π R 2 , then | F x (z) | (1 − | z | 2) ≤ R. If for some z the previous inequality is equality, then the surface is an affine image of a disk, and F is linear up to a Möbius transformation of the unit disk. [ABSTRACT FROM AUTHOR]

We provide new sharp embeddig theorem in Bergman type spaces in tubular domains over symmetric cones related with so called diagonal map. Related embeddings with a certain condition on Bergman kernel will be also provided. [ABSTRACT FROM AUTHOR]

The paper continues the study of boundary properties of normal subharmonic functions defined in the unit disk . Theorems are obtained on the existence of angular boundary limits for normal subharmonic functions almost everywhere on the arc of the unit disk . [ABSTRACT FROM AUTHOR]

We obtain some new uniform estimates and maximal theorems in classical Hardy spaces in the unit disk related with Bergman pro jection extending some previously well-known inequalities for Hardy spaces. [ABSTRACT FROM AUTHOR]

The theory of geometric functions was first introduced by Bernard Riemann in 1851. In 1916, with the concept of normalized function revealed by Bieberbach, univalent function concept has found application area. Assume f(z)=z+Σn>2(anzn converges for all complex numbers z with 1,|z|, and f(z)is one-to-one on the set of such z. Convex and starlike functions f(z) and g(z) are discussed with the help of subordination. The f(z) and g(z) are analytic in unit disc and f(0)0,f'(0)=1, and g(0)=0,g'(0)-1=0. A single valued function f(z) is said to be univalent (or schlict or one-to-one) in domain DCC never gets the same value twice; that is, if f(x1)-fz2 for all z1 and z2 with z1= z2. Let A be the class of analytic functions in the unit disk U={z:|z|1} that are normalized with f(0)=0,f'(0)=1. In this paper we give the some necessary conditions for f(z)E S*[a,a²] and 0< a²

We provide some new sharp assertions on the action of the Toeplitz operator Tφ in new Fp,qα type spaces of analytic functions of several complex variables extending previously known assertions proved by various authors. [ABSTRACT FROM AUTHOR]

Abstract. In Geometric Function Theory of Complex Analysis, there have been many interesting and fruitful usages of a wide variety of differential subordinations for holomorphic functions in the unit disk ...: ... Here, in this article, we derive some properties of the first-order differential subordinations for holomorphic functions which are defined in the unit ball ℝ: ... by using a certain class of admissible functions. We also make use of the theory of biholomorphic functions in our investigation here. [ABSTRACT FROM AUTHOR]

The main object of this paper is to extend the univalent condition for two general integral operators. A number of known univalent condition would follow upon specializing the parameters involved in our main results.

Let E be an arbitrary closed set on the unit circle ∂ⅅ and u be a harmonic function on the unit disk ⅅ satisfying |u(z)| ≾ (1 − |z|)γρ−q(z)where ρ(z) = dist (z, E), γ, q are some real constants, γ ≤ q. We establish an estimate of the conjugate u ˜ of the same type which is sharp in some sense, and in the case E = ∂ⅅ coincides with known estimates. As an application we describe growth classes defined by the non-radial condition |u(z)| ≾ ρ−q(z) in terms of smoothness of the Stieltjes measure associated to the harmonic function u. [ABSTRACT FROM AUTHOR]

In current realization, our object is to use the convolution product in terms of the notion quantum calculus to deliver a propagated q-derivative factor taking a more generalized Sàlàgean formula. By joining both the new factor together with the Janowski formula, we designate a special category of analytic factors in domain of unit disk. Finally, we deliberate a set of significant inequalities involving these classes. As applications, we seek the q-differential translator to generalize a denomination of differential equations species Briot-Bouquet and formulate its upper analytic solution using the subordination idea. This application can be employed in information theory and thermo dynamical systems.

Abstract In our current investigation, we apply the idea of quantum calculus and the convolution product to amend a generalized Salagean q-differential operator. By considering the new operator and the typical version of the Janowski function, we designate definite new classes of analytic functions in the open unit disk. Significant properties of these modules are considered, and recurrent sharp consequences and geometric illustrations are realized. Applications are considered to find the existence of solutions of a new class of q-Briot–Bouquet differential equations.

Operator theory is a magnifcent tool for studying the geometric behaviors of holomorphic functions in the open unit disk. Recently, a combination between two well known differential operators, Ruscheweyh derivative and Sălăgean operator are suggested by Lupas in [10]. In this effort, we shall follow the same principle, to formulate a generalized differential-difference operator. We deliver a new class of analytic functions containing the generalized operator. Applications are illustrated in the sequel concerning some differential subordinations of the operator. [ABSTRACT FROM AUTHOR]

We introduce new general spaces of subharmonic functions in the unit disk and prove some new sharp parametric representation results for them expanding some previously known assertions. Some related issues in upper half plane and product domains will be discussed. [ABSTRACT FROM AUTHOR]

In this paper we introduce a new general integral operator for analytic functions in the open unit disk U and we obtain sufficient conditions for univalence of this integral operator. [ABSTRACT FROM AUTHOR]

In this paper, we define a new differential-difference operator in the open unit disk. This operator describes the analytic geometric representation of the solution of the well known Beltrami differential equation in a complex domain, which this type of equations usually employs in image processing. Moreover, topological properties such as boundedness and compactness are introduced in different spaces such as Hardy space and Blog space. [ABSTRACT FROM AUTHOR]

We study {0, 1} and {−1, 1} polynomials ƒ(z), called Newman and Littlewood polynomials, that have a prescribed number N(ƒ) of zeros in the open unit disk D = {z ∈ C: |z| < 1}. For every pair (k, n) ∈ N2, where n ≥ 7 and k ∈ [3, n−3], we prove that it is possible to find a {0, 1}-polynomial ƒ(z) of degree °{ƒ} = n with non-zero constant term ƒ(0) ≠ 0, such that N(ƒ) = k and ƒ(z) ≠ 0 on the unit circle ∂ D. On the way to this goal, we answer a question of D. W. Boyd from 1986 on the smallest degree Newman polynomial that satisfies |ƒ(z)| > 2 on the unit circle ∂ D. This polynomial is of degree 38 and we use this special polynomial in our constructions. We also identify (without a proof) all exceptional (k, n) with k ∈ {1, 2, 3, n−3, n−2, n−1}, for which no such {0, 1}-polynomial of degree n exists: such pairs are related to regular (real and complex) Pisot numbers. Similar, but less complete results for {−1, 1} polynomials are established. We also look at the products of spaced Newman polynomials and consider the rotated large Littlewood polynomials. Lastly, based on our data, we formulate a natural conjecture about the statistical distribution of N(ƒ) in the set of Newman and Littlewood polynomials. [ABSTRACT FROM AUTHOR]

We introduce new general spaces of subharmonic functions in the unit disk and prove some new parametric representation results for them expanding some previously known assertions. [ABSTRACT FROM AUTHOR]

The purpose of this article is to introduce a new general family of normalized analytic fractal function in the open unit disk. We employ this class to define a fractional differential operator of two fractals. This operator, under some conditions involves the well known Salagean differential operator. Our method is based on the Hadamard product and its generalization of functions with negative coefficients. [ABSTRACT FROM AUTHOR]

Inequality study is a magnificent field for investigating the geometric behaviors of analytic functions in the open unit disk calling the subordination and superordination. In this work, we aim to formulate a generalized differential-difference operator. We introduce a new class of analytic functions having the generalized operator. Some subordination results are included in the sequel. [ABSTRACT FROM AUTHOR]

Characteristic between species is an important feature of animal investigation and preservation. For turtles, different analysis is possibly respected implements for identification. Shell shape is an essential component of analysis and classification in turtles and can be simply styled and measured by conformal geometric (CG). Here, we focus on the shell shape of a Southeast Asian leaf turtle by utilizing CG approaches. CG approaches recognized important differences in shell shape of turtles by using the convexity part of the shell structure. The symmetry of the sail shape in this type of turtles is a very important recognition. Therefore, we introduce a symmetric differential operator using its roots to study the shell shape. 2D printing tool paths are recognized for these shells. [ABSTRACT FROM AUTHOR]

A recent paper in this journal contained speculation about the potential for improving some of its novel high-order cubature schemes. Those schemes are for approximating uniformly weighted integrals over the unit disk. The two highest order rules were (i) a degree-65 solution with 733 cubature points and (ii) a degree-77 solution with 1021 cubature points. Both schemes possessed six-fold rotational symmetry and it is reported here that, as speculated, one point can indeed be dropped from each segment so that the end results have just 727 and 1015 cubature points, respectively. The details for these new schemes are now presented along with further discussion of this type of solution and additional characterization of related options of even higher orders. [ABSTRACT FROM AUTHOR]

Abstract 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.

In current realization, our object is to use the convolution product in terms of the notion quantum calculus to deliver a propagated q-derivative factor taking a more generalized Sàlàgean formula. By joining both the new factor together with the Janowski formula, we designate a special category of analytic factors in domain of unit disk. Finally, we deliberate a set of significant inequalities involving these classes. As applications, we seek the q-differential translator to generalize a denomination of differential equations species Briot-Bouquet and formulate its upper analytic solution using the subordination idea. This application can be employed in information theory and thermo dynamical systems. [ABSTRACT FROM AUTHOR]

In our current investigation, we apply the idea of quantum calculus and the convolution product to amend a generalized Salagean q-differential operator. By considering the new operator and the typical version of the Janowski function, we designate definite new classes of analytic functions in the open unit disk. Significant properties of these modules are considered, and recurrent sharp consequences and geometric illustrations are realized. Applications are considered to find the existence of solutions of a new class of q-Briot–Bouquet differential equations. [ABSTRACT FROM AUTHOR]

We derive bounds on the extremal singular values and the condition number of N × K , with N ⩾ K , Vandermonde matrices with nodes in the unit disk. The mathematical techniques we develop to prove our main results are inspired by a link—first established by Selberg [1] and later extended by Moitra [2] —between the extremal singular values of Vandermonde matrices with nodes on the unit circle and large sieve inequalities. Our main conceptual contribution lies in establishing a connection between the extremal singular values of Vandermonde matrices with nodes in the unit disk and a novel large sieve inequality involving polynomials in z ∈ C with | z | ⩽ 1. Compared to Bazán's upper bound on the condition number [3] , which, to the best of our knowledge, constitutes the only analytical result—available in the literature—on the condition number of Vandermonde matrices with nodes in the unit disk, our bound not only takes a much simpler form, but is also sharper for certain node configurations. Moreover, the bound we obtain can be evaluated consistently in a numerically stable fashion, whereas the evaluation of Bazán's bound requires the solution of a linear system of equations which has the same condition number as the Vandermonde matrix under consideration and can therefore lead to numerical instability in practice. As a byproduct, our result—when particularized to the case of nodes on the unit circle—slightly improves upon the Selberg–Moitra bound. [ABSTRACT FROM AUTHOR]

In this paper we introduce a new general integral operator for analytic functions in the open unit disk U and we obtain sufficient conditions for univalence of this integral operator. [ABSTRACT FROM AUTHOR]

Functions selected from convex functions complex number of analytical and univalent conditions and unit D disk |z| < 1 was used. Whether the new function obtained by addition process is convex function is examined. Although f (z) ∈ K and g(z) ∈ K are convex function which is shown to be f (z)+g(z)/∈ K, it is proved that the sums of two functions are not convex functions. [ABSTRACT FROM AUTHOR]