Punctured Unit Disk Research Articles

New Criteria for Meromorphic Bazilević Functions Associated with Linear Operator

In this work, was proposed and characterized a new linear operator and having this linear operator establishing, we benefited from it to define class of meromorphic Bazilević functions in the punctured unit disk . What's more, we obtain some adequate conditions for functions having a place with this class.

A New Investigation into a Linear Operator Connected to Gaussian Hypergeometric Functions

In this work we introduce new subclasses of meromorphic functions in punctured unit disk and analyzes numerous connections and various features of these subclasses by making use of the linear operator that is connected to Gaussian hypergeometric functions.

Logarithmic pinpricks in wavefunctions

Waves in the plane, punctured by excision of a small disk with radius much smaller than the wavelength, can be modified by being forced to vanish on the boundary of the disk. Such waves exhibit a logarithmically thin ‘pinprick’, and logarithmically weak oscillations persisting far away. As the radius vanishes, these modifications become asymptotically invisible. Examples are punctured plane waves, and a punctured unit disk; in the latter case, the pinprick causes a logarithmic shift in the eigenvalues. It is conjectured that the plane can be densely covered with asymptotically invisible pinpricks, and that there are analogous phenomena in higher dimensions. The curious phenomenon of pinpricks is not hard to understand, and would be worth presenting in graduate courses on waves.

Commuting Toeplitz operators on weighted harmonic Bergman spaces and hyponormality on the Bergman space of the punctured unit disk

<abstract><p>We first describe commuting Toeplitz operators with harmonic symbols on weighted harmonic Bergman spaces. Then, a sufficient condition for hyponormality on weighted Bergman spaces of the punctured unit disk, when the analytic part of the symbol is a monomial, is shown.</p></abstract>

Fractional Zernike functions

We consider and provide an accurate study for the fractional Zernike functions on the punctured unit disc, generalizing the classical Zernike polynomials and their associated β-restricted Zernike functions. Mainly, we give the spectral realization of the latter ones and show that they are orthogonal L2-eigenfunctions for certain perturbed magnetic (hyperbolic) Laplacian. The algebraic and analytic properties for the fractional Zernike functions to be established include the connection to special functions, their zeros, their orthogonality property, as well as the differential equations, recurrence and operational formulas they satisfy. Integral representations are also obtained. Their regularity as poly-meromorphic functions is discussed and their generating functions including a bilinear one of “Hardy–Hille type” are derived. Moreover, we prove that a truncated subclass defines a complete orthogonal system in the underlying Hilbert space giving rise to a specific Hilbertian orthogonal decomposition in terms of a class of generalized Bergman spaces.

Some properties for meromorphic functions associated with integral operators

In the present paper we aim at proving some subordinations properties for meromorphic functions analytic in the punctured unit disc Delta ^*={z:0<|z|<1} with a simple pole at the origin. The functions under investigation are associated with two integral operators mathcal {P}_sigma ^gamma and mathcal {Q}_sigma ^gamma (see Lashin in Comput Math Appl 59:524–531, 2010, https://doi.org/10.1016/j.camwa.2009.06.015). Several other results and numerical examples are also obtained.

On a Subclass of Meromorphic multivalent Functions associated with q-differential operator

In this present paper, we have introduced and defined a new subclass of meromorphic multivalent function using the q-differential operator in punctured unit disk We have obtained some geometric properties like coefficient inequality, a sufficient and necessary condition for the meromorphic multivalent functions to be in the class and we have get a corollary from that theorem. Also, we have acquired the growth and distortion theorems bounds for the class Subsequently we have fixed the radii of starlikeness and convexity theorems. After that we have proved the class is closed under convex linear combinations. Also, we have introduced the extreme points theorem and proved that. Finally, we have obtained arithmetic mean theorem in the same class.

Boundedness in the Bloch Space of Symmetric Domain for a Class of Multi-Valent Meromorphic Functions Given by a Fractional Integral

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.

Applications of the Symmetric Quantum-Difference Operator for New Subclasses of Meromorphic Functions

Our goal in this article is to use ideas from symmetric q-calculus operator theory in the study of meromorphic functions on the punctured unit disc and to propose a novel symmetric q-difference operator for these functions. A few additional classes of meromorphic functions are then defined in light of this new symmetric q-difference operator. We prove many useful conclusions regarding these newly constructed classes of meromorphic functions, such as convolution, subordination features, integral representations, and necessary conditions. The technique presented in this article may be used to produce a wide variety of new types of generalized symmetric q-difference operators, which can subsequently be used to investigate a wide variety of new classes of analytic and meromorphic functions related to symmetric quantum calculus.

On Meromorphic Parabolic Starlike Functions with Fixed Point Involving the q-Hypergeometric Function and Fixed Second Coefficients

This article defines a new class of meromorphic parabolic starlike functions in the punctured unit disc D*={z∈C:0&lt;|z|&lt;1} that includes fixed second coefficients of class As,cdψ,τ,ν,η and the q- hypergeometric functions. For the function belonging to the class As,cdψ,τ,ν,η, some properties are obtained, including the coefficient inequalities, closure theorems, and the radius of convexity.

Applications of Some Subclasses of Meromorphic Functions Associated with the q-Derivatives of the q-Binomials

In this article, we make use of the q-binomial theorem to introduce and study two new subclasses ℵ(αq,q) and ℵ(α,q) of meromorphic functions in the open unit disk U, that is, analytic functions in the punctured unit disk U∗=U\{0}={z:z∈Cand0&lt;z&lt;1}. We derive inclusion relations and investigate an integral operator that preserves functions which belong to these function classes. In addition, we establish a strict inequality involving a certain linear convolution operator which we introduce in this article. Several special cases and corollaries of our main results are also considered.

Pascu-Rønning Type Meromorphic Functions Based on Sălăgean-Erdély–Kober Operator

In the present investigation, we introduce a new class of meromorphic functions defined in the punctured unit disk Δ*:={ϑ∈C:0&lt;|ϑ|&lt;1} by making use of the Erdély–Kober operator Iς,ϱτ,κ which unifies well-known classes of the meromorphic uniformly convex function with positive coefficients. Coefficient inequalities, growth and distortion inequalities, in addition to closure properties are acquired. We also set up a few outcomes concerning convolution and the partial sums of meromorphic functions in this new class. We additionally state some new subclasses and its characteristic houses through specializing the parameters that are new and no longer studied in association with the Erdély–Kober operator thus far.

A-harmonic equation and cavitation

Suppose that \(f\) is a homeomorphism from the punctured unit disk \(D \setminus \{0\}\) onto the annulus \(A(r') = \{r' &lt; |z| &lt;1 \}\), \(r' \geq 0\), and \(f\) is quasiconformal in every \(A(r)\), \(r&gt; 0\), but not in \(D\). If \(r' &gt; 0\) then \(f\) has cavitation at \(0\) and no cavitation if \(r' = 0\). The singular factorization problem is to find harmonic functions \(h\) in \(A(r')\) such that \(h \circ f\) satisfies the elliptic PDE associated with \(f\) with a singularity at \(0\). Sufficient conditions in terms of the dilatation \(K_{f^{-1}}(z)\) together with the properties of \(h\) are given to the factorization problem, to the continuation of \(h \circ f\) to \(0\) and to the regularity of \(h \circ f\). We also give sufficient conditions for cavitation and non-cavitation in terms of the complex dilatation of \(f\) and demonstrate both cases with several examples.

Differential subordination for analytic and meromorphic multivalent functions

Certain differential subordination implications for multivalent functions defined on the unit disc [Formula: see text] and meromorphic multivalent functions on the punctured unit disc [Formula: see text] are established. The results obtained generalize several earlier known results on differential inequalities.

CERTAIN SUBCLASSES OF MEROMORPHICALLY P-VALENT FUNCTIONS WITH POSITVE OR NEGATIVE COEFICIENTS USING DIFFERENTIAL OPERATOR

In this paper, we have introduced two subclasses and of meromorphically p-valent functions with positive and negative coefficients, defined by differential operator in the punctured unit disk and obtain some sharp results including coefficient inequality, distortion theorem, radii of starlikeness and convexity, closure theorems of these subclasses of meromorphically p-valent functions. We also derive some interesting results for the Hadamard products of functions belonging to the classes and .

A new subclass of meromorphic univalent functions associated with linear operator Lk involving complex order I

In this paper, we introduce a new subclass of analytic and meromorphic univalent function associated with liner operator involving complex order in the punctured unit disk. Then we characterized these functions and obtained some of properties of this subclass.

Admissible Classes of Multivalent Meromorphic Functions Defined by a Linear Operator

The results from this paper are related to the geometric function theory. In order to obtain them, we use the technique based on differential subordination, one of the newest techniques used in the field, also known as the technique of admissible functions. For that, the appropriate classes of admissible functions are first defined. Based on these classes, we obtain some differential subordination and superordination results for multivalent meromorphic functions, analytic in the punctured unit disc, related to a linear operator ℑρ,τp(ν), for τ&gt;0,ν,ρ∈C, such that Re(ρ−ν)≧0, Re(ν)&gt;τp,(p∈N). Moreover, taking into account both subordination and superordination results, we derive a sandwich-type theorem. The connection with some other known results and an example are also provided.

On harmonic homeomorphisms from the punctured unit disc onto its exterior

The purpose of this paper is to study the class of harmonic sense-preserving homeomorphisms from the punctured unit disc onto its planar exterior which extend homeomorphically between the unit circle and itself and admit a simple pole at the origin.

Geometric Studies on Mittag-Leffler Type Function Involving a New Integrodifferential Operator

The generalized exponential function in a complex domain is called the Mittag-Leffler function (MLF). The implementations of MLF are significant in diverse areas of science. Over the past few decades, MLF and its analysis with generalizations have become an increasingly rich research area in mathematics and its allied fields. In the geometric theory of meromorphic functions, the main contribution to this discipline of study is to enrich areas of operator theory on complex punctured domains and differential complex inequalities, namely, subordination theory. This effort presents integrodifferential operator of meromorphic functions in the punctured unit disk. It is formulated by combining the differential operator and the integral operator correlating with the extended generalized Mittag-Leffler function. Furthermore, some interesting geometric features in terms of the subordination principle are investigated.

Convergence of volume forms on a family of log Calabi–Yau varieties to a non-Archimedean measure

We study the convergence of volume forms on a degenerating holomorphic family of log Calabi–Yau varieties to a non-Archimedean measure, extending a result of Boucksom and Jonsson. More precisely, let (X, B) be a holomorphic family of sub log canonical, log Calabi–Yau complex varieties parameterized by the punctured unit disk. Let \(\eta \) be a meromorphic form on X with poles along B such that the restriction of \(\eta \) is a top-dimensional form on each of the fibers. We show that the (possibly infinite) measures induced by the restriction of \(|\eta \wedge \overline{\eta }|\) to a fiber converge to a measure on the Berkovich analytification of \(X \setminus B\) as we approach the puncture. The convergence takes place on a hybrid space, which is obtained by filling in the space \(X \setminus B\) with the aforementioned Berkovich space over the puncture.