Open Unit Disk Research Articles

Littlewood-type theorems for Hardy spaces in infinitely many variables

Littlewood's theorem is one of the pioneering results in random analytic functions over the open unit disk. In this paper, we prove some analogues of this theorem for Hardy spaces in infinitely many variables. Our results not only cover finite-variable setting, but also apply in cases of Dirichlet series.

On a Certain Subclass of Analytic Functions Defined by Bessel Functions

In this work, we introduce and investigate a new subclass of analytic functions in the open unit disc with negative coefficients. The object of the present paper is to determine the coefficient estimates, extreme points, integral means inequalities and subordination results for this class.

Sharpening Littlewood subordination principle with univalent symbol

AbstractLet 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.

Phase retrieval on circles and lines

Abstract Let f and g be analytic functions on the open unit disk ${\mathbb D}$ such that $|f|=|g|$ on a set A. We give an alternative proof of the result of Perez that there exists c in the unit circle ${\mathbb T}$ such that $f=cg$ when A is the union of two lines in ${\mathbb D}$ intersecting at an angle that is an irrational multiple of $\pi $ , and from this, deduce a sequential generalization of the result. Similarly, the same conclusion is valid when f and g are in the Nevanlinna class and A is the union of the unit circle and an interior circle, tangential or not. We also provide sequential versions of this result and analyze the case $A=r{\mathbb T}$ . Finally, we examine the most general situation when there is equality on two distinct circles in the disk, proving a result or counterexample for each possible configuration.

New Properties of Analytic Functions

In the present paper, we consider the class A¯ of functions f(z) of the form f(z)=z+∑k=1∞a1+k3z1+k3 that are analytic in the open unit disc U. If a1+k3=0 for k≠3n(n=1,2,3,⋯), then f(z) is given by f(z)=z+∑k=2∞akzk. For such functions f(z)∈A¯, some interesting properties for subordinations and strongly starlike functions are given. Also, some interesting examples for the results are shown.

Volterra-Composition Operators Acting on $S^{p}$ Spaces and Weighted Zygmund Spaces

Let $\varphi$ be an analytic selfmap of the open unit disk $\mathbb{D}$ and $g$ be an analytic function on $\mathbb{D}$. The Volterra-type composition operators induced by the maps $g$ and $\varphi$ are defined as $$\left(I_{g}^{\varphi}f\right)(z)= \int_{0}^{z} f^{\prime}(\varphi(\zeta)) g(\zeta) d\zeta \hspace{0.07in} \text{and} \hspace{0.07in} \left(T_{g}^{\varphi}f\right)(z)= \int_{0}^{z} f(\varphi(\zeta)) g^{\prime}(\zeta) d\zeta .$$For $1\leq p<\infty$, $S^p(\mathbb{D})$ is the space of all analytic functions on $\mathbb{D}$ whose first derivative $f^{\prime}$ lies in the Hardy space $H^p(\mathbb{D})$, endowed with the norm $\displaystyle \|f\|_{S^p}=|f(0)|+\|f^{\prime}\|_{H^p}$. Let $\mu: (0,1] \rightarrow (0, \infty)$ be a positive continuous function on $\mathbb{D}$ such that for $z\in\mathbb{D}$ we define $\mu(z) = \mu(|z|)$. The weighted Zygmund space $\mathcal{Z}_{\mu}(\mathbb{D})$ is the space of all analytic functions $f$ on $\mathbb{D}$ such that $\sup_{z\in \mathbb{D}} \mu(z) |f^{\prime\prime}(z)|$ is finite. In this paper, we characterize the boundedness and compactness of the Volterra-type composition operators that act between $S^p$ spaces and weighted Zygmund spaces.

Initial Coefficient Bounds Analysis for Novel Subclasses of Bi-Univalent Functions Linked with Lucas-Balancing Polynomials

We investigate some subclasses of regular and bi-univalent functions in the open unit disk that are associated with Lucas-Balancing polynomials in this work. For functions that belong to these subclasses, we obtain upper bounds on their initial coefficients. The Fekete–Szegö problem is also discussed. Along with presenting some new results, we also explore pertinent connections to earlier findings.

On α-Pseudo Spiralike Functions Associated with Exponential Pareto Distribution (EPD) and Libera Integral Operator

The present study aims at investigating some characterizations of a new subclass Gα(μ,τ) and obtaining the bounds on the first two Taylor–Maclaurin coefficients for functions belonging to the newly introduced subclass. In order to achieve this, a compound function Lx,nσ(z) is derived from the convolution of the analytic function f(z) and a modified exponential Pareto distribution G(x) in conjunction with the famous Libera integral operator L(ζ). With the aid of the derived function, the aforementioned subclass Gα(μ,τ) is introduced, while some properties of functions belonging to this subclass are considered in the open unit disk.

LOG HARMONIC MAPPINGS ASSOCIATED WITH THE SINE FUNCTION

In this paper, we define new subclasses $\mathcal{ST}_{lh}(s)$ and $\mathcal{CST}_{lh}(s)$ of sin starlike log-harmonic mappings and sin close to starlike log-harmonic mappings, respectively, defined in the open unit disc $\de$. We investigate representation theorem and integral representation theorem for functions $f \in \mathcal{ST}_{lh}(s)$. Further, we determine radii of starlikeness for such functions.

On a Class of Multivalent Functions with Negative Coefficients Involving (r, q)− Calculus

In this research, we focused on presenting a novel subclass of multivalent analytic functions situated in the open unit disk, characterized by the use of Jackson’s derivative operator. Our investigation systematically establishes the requisite inclusion conditions in this class, offering detailed coefficient characterizations. The exploration encompasses an array of significant properties intrinsic to this subclass, encompassing coefficient estimates, growth and distortion theorems, identification of extreme points, and the determination of the radius of starlikeness and convexity for functions falling within this specialized category. Expanding the preliminary findings, this research extended the inquiry to delve deeper into the intriguing features and implications associated with this new subclass of multivalent analytic functions. The research concentrated the light on the nuanced intricacies of coefficient estimates, providing a comprehensive understanding of how these functions evolve within the open unit disk through exploring the growth and distortion theorems, unraveling the underlying mathematical principles governing the behavior of functions in this subclass as they extend beyond the unit disk. The findings of this research contribute to the broader understanding of multivalent analytic functions, paving the way for further exploration and applications in diverse mathematical contexts.

Ozaki-Type Bi-Close-to-Convex and Bi-Concave Functions Involving a Modified Caputo’s Fractional Operator Linked with a Three-Leaf Function

The focus of the present work is on the establishment and investigation of the coefficient estimates of two new subclasses of bi-close-to-convex functions and bi-concave functions; these are of an Ozaki type and involve a modified Caputo’s fractional operator that is associated with three-leaf functions in the open unit disc. The classes are defined using the notion of subordination based on the previously established fractional integral operators and classes of starlike functions associated with a three-leaf function. For functions in these classes, the Fekete-Szegö inequalities and the initial coefficients, |a2| and |a3|, are discussed. Several new implications of the findings are also highlighted as corollaries.

Measure induced Hankel and Toeplitz type operators on weighted Dirichlet spaces

For a complex Borel measure μ \mu on the open unit disk, and for a weighted Dirichlet space H s \mathcal {H}_s with 0 > s > 1 0>s>1 , we characterize the boundedness of the measure induced Hankel type operator H μ , s : H s → H s ¯ H_{\mu ,s}: \mathcal {H}_s \to \overline {\mathcal {H}_s} , extending the results of Xiao [Bull. Austral. Math. Soc. 62 (2000), pp. 135–140] for the classical Hardy space H 2 = H 1 H^2=\mathcal {H}_1 , and of Arcozzi, Rochberg, Sawyer, and Wick [J. Lond. Math. Soc. (2) 83 (2011), pp. 1–18] for the classical Dirichlet space D = H 0 \mathcal {D}= \mathcal {H}_0 . Our approach relies on some recent results about weak products of complete Nevanlinna-Pick reproducing kernel Hilbert spaces. We also include some related results on Hankel measures, Carleson measures, and Toeplitz type operators on weighted Dirichlet spaces H s \mathcal {H}_s , 0 > s > 1 0>s>1 .

Coefficient estimates for a subclass of bi-univalent functions associated with the Salagean differential operator

In this paper, we present and examine a novel subset of the function class ∑, which consists of analytic and bi-univalent functions defined in the open unit disk U and connected to the Salagean differential operator. Additionally, we determine estimates for the Taylor-Maclaurin coefficients |a2| and |a3| functions within this new subclass and enhance some recent findings. 2010 Mathematics Subject Classification. 30C45; 30C50.

On the convolution of convex 2-gons

We study the convolution of functions of the formfα(z):=(1+z1−z)α−12α, which map the open unit disk of the complex plane onto polygons of 2 edges when α∈(0,1). Inspired by a work of Cima, we study the limits of convolutions of finitely many fα and the convolution of arbitrary unbounded convex mappings. The analysis for the latter is based on the notion of angle at infinity, which provides an estimate for the growth at infinity and determines whether the convolution is bounded or not. A generalization to an arbitrary number of factors shows that the convolution of n randomly chosen unbounded convex mappings has a probability of 1/n! of remaining unbounded. We provide the precise asymptotic behavior of the coefficients of the functions fα.

An equation for complex fractional diffusion created by the Struve function with a T-symmetric univalent solution

Abstract 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.

Estimating the Second Order Hankel Determinant for the Subclass of Bi-Close-to-Convex Function of Complex Order

The article aims to estimate the coefficient bounds for the second Hankel determinant by using the class of bi-close-to-convex functions of a complex order in the open unit disk. Making the direct application of Carathéodory function along with the closely related properties of starlike functions, we obtain the upper bound for the second Hankel determinant via certain subclass of bi-close-to-convex functions of complex order. The study discusses the maximization of the second Hankel determinant in both conventional graph and analytic methods. Moreover, we explore and modify some results on the study of bi-close-to-convex functions and its second degree Hankel determinant. At the end of the article, we remark on improvement in the earlier work of some researchers and discover a better value than the one they obtained.

محدد هنكل الثاني لفئة معينة من الدوال المعرفة بواسطة مؤثر تفاضلي

The objective of this paper is to obtain an upper bound of second Hankel determinant for a class of functions M_(α,β,λ,δ)^k (φ) defined in the open unit disc U by using the differential operator〖 D〗_(α,β,λ,δ)^k f(z). In addition, we give particular values to the parameters A,B and k to study special cases of the results of this article. The class M_(α,β,λ,δ)^k (φ) and the differential operator〖 D〗_(α,β,λ,δ)^k f(z). Key words: differential operator, Second Hankel determinant, Starlike functions, Subordination property

Norms of Composition Operators from Weighted Harmonic Bloch Spaces into Weighted Harmonic Zygmund Spaces

This article examines the norms of composition operators from the weighted harmonic Bloch space BHλ,0<λ<∞ to the weighted harmonic Zygmund space ZHβ,0<β<∞. The critical norm is on the open unit disk. We first give necessary and sufficient conditions where the composition operator between BHλ and ZHβ is bounded. Secondly, we will study the compactness case of the composition operator between BHλ and ZHβ. Finally, we will estimate the essential norms of the composition operator between BHλ and ZHβ.

Initial Coefficient Bounds Analysis for Novel Subclasses of Bi-Univalent Functions linked with Horadam Polynomials

In this work, we investigate some subclasses of bi-univalent and regular functions associated with Horadam polynomials in the open unit disk $\mathfrak{U}=\{\varsigma\in\mathbb{C}:|\varsigma| <1\}$. For functions that belong to these subclasses, we find bounds on their initial coefficients. The functional problem of Fekete-Szegö is also examined. Along with presenting some new results, we also talk about pertinent connections to earlier findings.

Initial Coefficient Bounds Analysis for Novel Subclasses of Bi-Univalent Functions linked with Horadam Polynomials

In this work, we investigate some subclasses of bi-univalent and regular functions associated with Horadam polynomials in the open unit disk $\mathfrak{U}=\{\varsigma\in\mathbb{C}:|\varsigma| <1\}$. For functions that belong to these subclasses, we find bounds on their initial coefficients. The functional problem of Fekete-Szegö is also examined. Along with presenting some new results, we also talk about pertinent connections to earlier findings.