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The drop of the smoothness of an analytic function compared to the smoothness of its modulus is discussed for the unit ball of ℂn. The paper is devoted to local aspects of the problem. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to consider a linear operator and define a certain class Ep(a, c, λ, γ: h) of analytic and multivalent functions in the open unit disk associated with differential subordination. Also, we discuss some geometric properties for this class. [ABSTRACT FROM AUTHOR]

This paper's findings are related to geometric function theory (GFT). We employ one of the most recent methods in this area, the fuzzy admissible functions methodology, which is based on fuzzy differential subordination, to produce them. To do this, the relevant fuzzy admissible function classes must first be defined. This work deals with fuzzy differential subordinations, ideas borrowed from fuzzy set theory and applied to complex analysis. This work examines the characteristics of analytic functions and presents a class of operators in the open unit disk J η , ς κ (a , e , x) for ς > − 1 , η > 0 , such that a , e ∈ R , (e − a) ≥ 0 , a > − x . The fuzzy differential subordination results are obtained using (GFT) concepts outside the field of complex analysis because of the operator's compositional structure, and some relevant classes of admissible functions are studied by utilizing fuzzy differential subordination. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to study certain geometric properties of a class of analytic functions involving the Barnes–Mittag-Leffler function. The main mathematical tools are the monotonicity patterns of some class of functions associated with the gamma and digamma functions. Furthermore, some consequences and examples are presented. [ABSTRACT FROM AUTHOR]

The main purpose of this paper, is to introduce a new subclass of analytic functions involving Polylogarithm functions and obtain coefficient inequalities, distortion properties, extreme points, radii of starlikeness and convexity, Hadamard product, and convolution and integral operators for the class. [ABSTRACT FROM AUTHOR]

The Hilbert transform plays an important role in the theory and practice of signal processing operations in continuous system theory because of its relevance to such problems as envelope detection and demodulation, as well as its use in relating the real and imaginary components, and the magnitude and phase components of spectra. The Hilbert transform is a multiplier operator and is widely used in the theory of Fourier transforms. It is also the main part of the theory of singular integral equations on the real line. Therefore, approximations of Hilbert transform are of great interest. Many papers have dealt with the numerical approximation of singular integrals in case of bounded intervals. On the other hand, the literature concerning the numerical integration on unbounded intervals is much sparser than the one on bounded intervals. There is very little literature concerning the case of Hilbert transform. This article is dedicated to the approximation of Hilbert transform in Hölder spaces by the operators introduced by V.R.Kress and E.Mortensen to approximate the Hilbert transform of analytic functions in a strip. [ABSTRACT FROM AUTHOR]

In this paper, we discuss a novel type of analytic bi-univalent functions by utilizing specialized q-Salagean differential operators. Then, we use the q-analogue of the logarithmic function to introduce definition and provide properties of a class of bi-univalent functions. Further, we use the subordination principle to estimate the initial Taylor and Maclaurin coefficients for these given univalent functions. Additionally, we introduce new operators to demonstrate practical applications of the existing theory and establish Fakte-Szego results for each function in the defined sets. Further, we discuss certain coefficient inequalities in detail. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the approximation of functions using a higher-order Hermite-Fej´er interpolation process on the unit circle. The system of nodes is composed of vertically projected zeros of Jacobi polynomials onto the unit circle with boundary points at ±1. Values of the polynomial and its first four derivatives are fixed by the interpolation conditions at the nodes. Convergence of the process is obtained for analytic functions on a suitable domain, and the rate of convergence is estimated. [ABSTRACT FROM AUTHOR]

In this paper, we shall find the order of starlikeness and convexity for integral operators Fαj,βj,λj,ζ (z) = {ζ ∫0z tζ−1 πnj=1 (Eαj,βj(t)/t )1/λj dt}1/ζ, where the functions Eαj,βj are the normalized Mittag-Leffler functions. [ABSTRACT FROM AUTHOR]

In this paper, we introduced two novel subclasses of bi-univalent functions, MΣ(α, B(x, ξ)) and HΣ(α, µ, B(x, ξ)), utilizing Lucas-Balancing polynomials. Within these function classes, we established bounds for the Taylor-Maclaurin coefficients |a2| and |a3|, addressing the Fekete-Szegö functional problems specific to functions within these new subclasses. Moreover, we illustrated how our primary findings could lead to various new outcomes through parameter specialization. [ABSTRACT FROM AUTHOR]

In the present paper, we are merging two interesting and well-known classes, namely those of Bazilevič and close-to-convex functions associated with a new derivative operator. We derive coefficient estimates for this broad category of analytic, univalent and bi-univalent functions and draw attention to the Fekete–Szegö inequalities relevant to functions defined within the open unit disk. Additionally, we identify several specific special cases of our results by specializing the parameters. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce two new subclasses R sin m (ℑ) and R sin (ℑ) of analytic functions by making use of subordination involving the sine function and the modified sigmoid activation function ℑ (v) = 2 1 + e − v , v ≥ 0 in the open unit disc E. Our purpose is to obtain some initial coefficients, Fekete–Szego problems, and upper bounds for the third- and fourth-order Hankel determinants for the functions belonging to these two classes. All the bounds that we will find here are sharp. We also highlight some known consequences of our main results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and investigate a subclass of analytic and bi-univalent functions which both f ⁢ (z) and f - 1 ⁢ (z) are m-fold symmetric on the open unit disk. Furthermore, we find upper bounds for the initial coefficients | a m + 1 | and | a 2 ⁢ m + 1 | for functions in this subclass. The results presented in this paper generalize and improve some recent works [ABSTRACT FROM AUTHOR]

In this existing paper, we examine a connection between certain class of harmonic univalent functions with starlike and convex harmonic functions define in the open unit disk by applying the convolution operator defined by Mittag-Leffler function. Several corollaries and consequences of the main results are also obtained. [ABSTRACT FROM AUTHOR]

In this paper, we consider universality in short intervals for the zeta-function attached to a normalized Hecke-eigen cusp form with respect to the modular group. For this, we apply a conjecture for the mean square in short interval on the critical strip for that zeta-function. The proof of the obtained universality theorem is based on a probabilistic limit theorem in the space of analytic functions. [ABSTRACT FROM AUTHOR]

In this paper, we present a new class of linear fractional differential operators that are based on classical Gaussian hypergeometric functions. Then, we utilize the new operators and the concept of differential subordination to construct a convex set of analytic functions. Moreover, through an examination of a certain operator, we establish several notable results related to differential subordination. In addition, we derive inclusion relation results by employing Briot–Bouquet differential subordinations. We also introduce a perspective study for developing subordination results using Gaussian hypergeometric functions and provide certain properties for further research in complex dynamical systems. [ABSTRACT FROM AUTHOR]

In the present paper, we consider the class A ¯ of functions f (z) of the form f (z) = z + ∑ k = 1 ∞ a 1 + k 3 z 1 + k 3 that are analytic in the open unit disc U. If a 1 + k 3 = 0 for k ≠ 3 n   (n = 1 , 2 , 3 , ⋯) , then f (z) is given by f (z) = z + ∑ k = 2 ∞ a k z k. 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. [ABSTRACT FROM AUTHOR]

In this study, using the ideas of subordination and the quantum-difference operator, we established a new subclass S*(δ,σ, q) of q-starlike functions and the subclass S*σ (δ,σ, q) of q-starlike bi-univalent functions associated with the vertical strip domain. We examined sharp bounds for the first two Taylor-Maclaurin coefficients, sharp Fekete-Szegö type problems, and coefficient inequalities for the function h that belong to S*(δ,σ, q), as well as sharp bounds for the inverse function h that belong to S*(δ,σ, q). We also investigated some results for the class of bi-univalent functions S*σ (δ,σ, q) and well-known corollaries were also highlighted to show connections between previous results and the findings of this paper. [ABSTRACT FROM AUTHOR]

Recently, Norvidas has introduced the general multidimensional Hermite sampling series, which involves samples from a function and its mixed and non-mixed partial derivatives. The convergence of this sampling series is slow unless the sample values |f(x)| rapidly decay as | x j | → ∞ for all 1 ≤ j ≤ n . In this paper, we investigate a modified version of this sampling series that utilizes a multivariate Gaussian multiplier to approximate functions from two classes of multivariate analytic functions using a complex approach. The first class comprises entire functions of exponential type in n variables that fulfill a decay condition, while the second class includes analytic functions in n variables defined on a multidimensional horizontal strip. It has a significantly higher convergence rate compared to the general multidimensional Hermite sampling series. [ABSTRACT FROM AUTHOR]

In this paper, we give upper bounds of the Hankel determinants H2,1(f) and H2,2(f) for the classes S*(λ,n), where f is analytic in the open unit disk Δ = {z ∈ C: |z| < 1g and normalized so that f(z) = z+a2z²+a3z³+... [ABSTRACT FROM AUTHOR]

Starlike and convex functions have gained increased prominence in both academic literature and practical applications over the past decade. Concurrently, logarithmic coefficients play a pivotal role in estimating diverse properties within the realm of analytic functions, whether they are univalent or nonunivalent. In this paper, we rigorously derive bounds for specific Toeplitz determinants involving logarithmic coefficients pertaining to classes of convex and starlike functions concerning symmetric points. Furthermore, we present illustrative examples showcasing the sharpness of these established bounds. Our findings represent a substantial contribution to the advancement of our understanding of logarithmic coefficients and their profound implications across diverse mathematical contexts. [ABSTRACT FROM AUTHOR]

Let A be a complex unital Banach algebra and D be a non-empty open connected subset of C. Let f be a holomorphic A-valued function on D. We set: ∑(f) = {z ∈ D | f(z) ∉ ∈ Inv(A)}, where Inv(A) is the set of all invertible elements in A. In this paper, we give a description of ∑(f) using the classical spectrum of f(z) (z ∈ D). Moreover, within the framework of this paper, we give a partial positive answer to the following problem which was posed by B. Aupetit: If the usual spectrum σ(f(z)) is polar for all z ∈ D, is it true that ∑(f) is polar? [ABSTRACT FROM AUTHOR]

Everyday problems are characterized by voluminous data and varying levels of ambiguity. Thereupon, it is critical to develop new mathematical approaches to dealing with them. In this context, the perfect functions are anticipated to be the best instrument for this purpose. Therefore, we investigate in this paper how to generate perfect functions using a variety of set operators. Symmetry is related to the interactions among specific types of perfect functions and their classical topologies. We can explore the properties and behaviors of classical topological concepts through the study of sets, thanks to symmetry. In this paper, we introduce a novel class of perfect functions in topological spaces that we term D-perfect functions and analyze them. Additionally, we establish the links between this new class of perfect functions and classes of generalized functions. Furthermore, while introducing the herein proposed D-perfect functions and analyzing them, we illustrate this new idea, explicate the associated relationships, determine the conditions necessary for their successful application, and give examples and counter-examples. Alternative proofs for the Hausdorff topological spaces and the D-compact topological spaces are also provided. For each of these functions, we examine the images and inverse images of specific topological features. Lastly, product theorems relating to these concepts have been discovered. [ABSTRACT FROM AUTHOR]

In a recent paper almost sure unisolvence of RBF interpolation at random points with no polynomial addition was proved, for Thin-Plate Splines and Radial Powers with noninteger exponent. The proving technique left unsolved the case of odd exponents. In this short note we prove almost sure polynomialfree unisolvence in such instances, by a deeper analysis of the interpolation matrix determinant and fundamental properties of analytic functions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a new subclass of starlike functions with respect to (2 ȷ , ℓ) -symmetric conjugate points using the principle of subordination. Several relationship with the well-known classes have been established. We have focussed on conic regions when it pertained to applications of our main results. Inclusion results, subordination property and coefficient inequality of the defined class are the main results of this paper. The applications of the results are presented here as corollaries, most of which are extensions of well-known results. [ABSTRACT FROM AUTHOR]

The logarithmic coeffcients of univalent functions play an important role in different estimates in the theory of univalent functions. In this paper, due to the significant importance of the study of these coeffcients, we find the upper bounds for some expressions associated with the logarithmic coeffcients of functions that belong to some classes defined by using the subordination. Moreover, we get the best upper bounds for the logarithmic coeffcients of some subclasses of analytic functions defined and studied in many earlier papers. [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of generalized Krätzel functions, which are the kernel functions of type-1 and type-2 pathway transforms. Various analytical properties such as Lipschitz continuity, fixed point property and integrability of these functions are investigated. Furthermore, the paper introduces two new inequalities associated with generalized Krätzel functions. The composition formulae for various fractional operators, such as Grünwald-Letnikov fractional differential operator, Riemann-Liouville fractional integral and differential operators with these functions are also obtained. Additionally, it has been demonstrated that these kernel functions are related to the Heaviside step function, the Dirac delta function, and the Riemann zeta function. A computable series representation of the kernel function is also obtained. Further scope of the work is pointed out by modifying the kernel function with the Mittag-Leffler function. [ABSTRACT FROM AUTHOR]

Let S denote the subclass of the analytic function and univalent functions in D, where D is defined as a unit disk, and having the Taylor representation form of S. In this paper, we will be obtaining the second Hankel determinants, in which the elements are the logarithmic coefficients mainly for the subclass of the close-to-convex functions in S. [ABSTRACT FROM AUTHOR]

In the Jack’s lemma it is considered q(z), an analytic function in | z | < 1 with q (0) = 0 for which |q(z)| attains its maximum value on the disc | z | ≤ r < 1 at the point z 0 , | z 0 | = r . Then z 0 q ′ (z 0) = k q (z 0) and k ≥ 1 . In this paper we try to say more about the number k in a generalizations of this lemma, where we consider max | arg { q (z) } | or min | Re { q (z) } | instead of |q(z)|. In these cases q(z) has another normalization at the origin. [ABSTRACT FROM AUTHOR]

The Ruscheweyh derivative operator is used in this paper to introduce and investigate interesting general subclasses of the function class Σ m of m-fold symmetric bi-univalent analytic functions. Estimates of the initial Taylor-Maclaurin coefficients | a m + 1 | and | a 2 m + 1 | are obtained for functions of the subclasses introduced in this study, and the consequences of the results are discussed. Additionally, the Fekete-Szegö inequalities for these classes are investigated. The results presented could generalize and improve some recent and earlier works. In some cases, our estimates are better than the existing coefficient bounds. Furthermore, within the engineering domain, the utilization of the Ruscheweyh derivative operator can encompass a broad spectrum of engineering applications, including the robotic manipulation control, optimizing optical systems, antenna array signal processing, image compression, and control system filter design. It emphasizes the potential for innovative solutions that can significantly enhance the reliability and effectiveness of engineering applications. [ABSTRACT FROM AUTHOR]

In the paper, we consider simultaneous approximation of a pair of analytic functions by discrete shifts ζuN (s + ikh1; a) and ζuN (s + ikh2, α; b) of the absolutely convergent Dirichlet series connected to the periodic zeta-function with multiplicative sequence a, and the periodic Hurwitz zeta-function, respectively. We suppose that uN → ∞ and uN ≪ N2 as N → ∞, and the set {(h1 log p: p ∈ P), (h2 log(m + α): m ∈ N0), 2π} is linearly independent over Q. [ABSTRACT FROM AUTHOR]

The purpose of this study is to establish new subclasses within the function class ∑m, which consists of analytic as well as m-fold symmetric bi-univalent functions expressed within the open unit disk Q. Additionally, for functions belonging to each of the newly established subclasses, this paper establishes estimates with regards to the Taylor-Maclaurin coefficients given by |am+1| as well as |a2m+1| Moreover, we take into consideration of specific as well as existing special cases for our respective findings. [ABSTRACT FROM AUTHOR]

In this paper, considering the various important applications of Miller–Ross functions in the fields of applied sciences, we introduced a new class of analytic functions f , utilizing the concept of Miller–Ross functions in the region of the Janowski domain. Furthermore, we obtained initial coefficients of Taylor series expansion of f , coefficient inequalities for f − 1 and the Fekete–Szegö problem. We also covered some key geometric properties for functions f in this newly formed class, such as the necessary and sufficient condition, convex combination, sequential subordination and partial sum findings. [ABSTRACT FROM AUTHOR]

In this paper we introduce a new class of ϑ -bi-pseudo-starlike functions with respect to symmetric points associated with telephone numbers and determine the bounds of the initial Taylor–Maclaurin coefficients and the Fekete–Szegö problem for functions belonging to this class. Some special cases of main results presented here are stated which are new and give better improvement to the initial Taylor-Maclaurin coefficients. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamics of an operator \mathcal T naturally associated to the so-called Collatz map , which maps an integer n \geq 0 to n / 2 if n is even and 3n + 1 if n is odd. This operator \mathcal T is defined on certain weighted Bergman spaces \mathcal B ^2 _\omega of analytic functions on the unit disk. Building on previous work of Neklyudov, we show that \mathcal T is hypercyclic on \mathcal B ^2 _\omega, independently of whether the Collatz Conjecture holds true or not. Under some assumptions on the weight \omega, we show that \mathcal T is actually ergodic with respect to a Gaussian measure with full support, and thus frequently hypercyclic and chaotic. [ABSTRACT FROM AUTHOR]

In this paper, we define new subclasses of analytic functions related to a modified sigmoid function and analytic univalent function. Then, we attempt to investigate the upper bounds of the third and fourth Hankel determinant in the special case. Further, bound on third Hankel determinant of its inverse function is also investigated. In addition, we attempt to obtain the Fekete-Szegö inequality for the classes. Then, we estimate the bounds of initial coefficients for the function belongs to some kind of new subclasses when its inverse function also belongs to these new subclasses. [ABSTRACT FROM AUTHOR]

In the present paper, we investigated some conditions to be in the class of Carathéodory functions by using the concept of the first-order differential subordinations. Moreover, various interesting special cases were considered in the geometric function theory as applications of main results presented here. [ABSTRACT FROM AUTHOR]

Let F (z 1 , ⋯ , z d) be the quotient of an analytic function with a product of linear functions. Working in the framework of analytic combinatorics in several variables, we compute asymptotic formulae for the Taylor coefficients of F using multivariate residues and saddle-point approximations. Because the singular set of F is the union of hyperplanes, we are able to make explicit the topological decompositions which arise in the multivariate singularity analysis. In addition to effective and explicit asymptotic results, we provide the first results on transitions between different asymptotic regimes, and provide the first software package to verify and compute asymptotics in non-smooth cases of analytic combinatorics in several variables. It is also our hope that this paper will serve as an entry to the more advanced corners of analytic combinatorics in several variables for combinatorialists. [ABSTRACT FROM AUTHOR]

Let P be the set of generalized prime numbers, and ζ P (s) , s = σ + i t , denote the Beurling zeta-function associated with P. In the paper, we consider the approximation of analytic functions by using shifts ζ P (s + i τ) , τ ∈ R . We assume the classical axioms for the number of generalized integers and the mean of the generalized von Mangoldt function, the linear independence of the set { log p : p ∈ P } , and the existence of a bounded mean square for ζ P (s) . Under the above hypotheses, we obtain the universality of the function ζ P (s) . This means that the set of shifts ζ P (s + i τ) approximating a given analytic function defined on a certain strip σ ^ < σ < 1 has a positive lower density. This result opens a new chapter in the theory of Beurling zeta functions. Moreover, it supports the Linnik–Ibragimov conjecture on the universality of Dirichlet series. For the proof, a probabilistic approach is applied. [ABSTRACT FROM AUTHOR]

This paper studies limit cycle bifurcation of an (n + 1) th order generalized Liénard differential system with an elliptic Hamiltonian H (x , y) = 1 2 x 2 + 1 2 y 2 - 1 4 x 4 , which bifurcates at least [ n 2 ] limit cycles near either a centre or a heteroclinic loop with two hyperbolic saddles for n ≤ 150 , where about two thirds of the limit cycles are large amplitude ones. To achieve this goal, we provide the precise expression of the third coefficient in the asymptotic expansion of the first order Melnikov function of an analytic near-Hamiltonian system near a polycycle with m (≥ 2) hyperbolic saddles, without any prescribed condition, whereas all the known results need. As well as we also characterize its higher order coefficients. Here [a] is the integer part function of a real number a. [ABSTRACT FROM AUTHOR]

Let A ¯ be the new general class of functions f (z) of the form f (z) = z + ∑ k = 1 ∞ a 1 + k 2 z 1 + k 2 that are analytic in the open unit disc U. In the present paper, for f (z) ∈ A ¯ , we consider classes S ∗ ¯ (α) ,   C ¯ (α) and R ¯ (α) and obtain some interesting properties of f (z) ∈ A ¯ concerning S ∗ ¯ (α) ,   C ¯ (α) and R ¯ (α) , applying subordinations of f (z). [ABSTRACT FROM AUTHOR]

In this paper, the approximation of analytic functions by shifts  ζ P (s + i τ)  of Beurling zeta-functions  ζ P (s)  of certain systems  P  of generalized prime numbers is discussed. It is required that the system of generalized integers  N P  generated by  P  satisfies  ∑ m ⩽ x , m ∈ N 1 = a x + O (x δ) ,  a > 0 ,  0 ⩽ δ < 1 , and the function  ζ P (s)  in some strip lying in  σ ^ < σ < 1 ,  σ ^ > δ , which has a bounded mean square. Proofs are based on the convergence of probability measures in some spaces. [ABSTRACT FROM AUTHOR]

Adequate mathematical description of magnetization curves is indispensable in engineering. The accuracy of the description has a significant impact on the design of electric machines and devices. The aim of this paper was to analyze the capability of Bézier curves systematically, to describe the nonlinear static magnetic properties of non-oriented electrical steels, and to compare this approach versus the established mathematical descriptions. First, analytic functions versus measurements were analyzed. The Bézier curves were then compared systematically with the most adequate analytic functions. Next, the most suitable orders of Bézier curves were determined for the approximation of nonlinear magnetic properties, where the influence of the range of the input measurement dataset on the approximation process was analyzed. Last, the extrapolation capabilities of the Bézier curves and analytic functions were evaluated. The general conclusion is that Bézier curves have adequate flexibility and significant potential for the approximation and extrapolation of nonlinear properties of non-oriented electrical steels. [ABSTRACT FROM AUTHOR]

In this paper, we consider the representation and extension of the analytic functions of three variables by special families of functions, namely branched continued fractions. In particular, we establish new symmetric domains of the analytical continuation of Lauricella–Saran's hypergeometric function  F K  with certain conditions on real and complex parameters using their branched continued fraction representations. We use a technique that extends the convergence, which is already known for a small domain, to a larger domain to obtain domains of convergence of branched continued fractions and the PC method to prove that they are also domains of analytical continuation. In addition, we discuss some applicable special cases and vital remarks. [ABSTRACT FROM AUTHOR]

In this study, we introduce a new class of normalized analytic and bi-univalent functions denoted by D Σ (δ , η , λ , t , r) . These functions are connected to the Bazilevič functions and the λ -pseudo-starlike functions. We employ Sakaguchi Type Functions and Horadam polynomials in our survey. We establish the Fekete-Szegö inequality for the functions in D Σ (δ , η , λ , t , r) and derive upper bounds for the initial Taylor–Maclaurin coefficients | a 2 | and | a 3 | . Additionally, we establish connections between our results and previous research papers on this topic. [ABSTRACT FROM AUTHOR]