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The existence of smooth but nowhere analytic functions is well-known (du Bois-Reymond [Math. Ann. 21 (1883), no. 1, pp. 109–117]). However, smooth solutions to the heat equation are usually analytic in the space variable. It is also well-known (Kowalevsky [Crelle 80 (1875), pp. 1–32]) that a solution to the heat equation may not be time-analytic at t=0 even if the initial function is real analytic. Recently, it was shown by Dong and Pan [J Math. Fluid Mech. 22 (2020), no. 4, Paper No. 53]; Dong and Zhang [J. Funct. Anal. 279 (2020), no. 4, Paper No. 108563]; Zhang [Proc. Amer. Math. Soc. 148 (2020), no. 4, pp. 1665–1670] that solutions to the heat equation in the whole space, or in the half space with zero boundary value, are analytic in time under an essentially optimal growth condition. In this paper, we show that time analyticity is not always true in domains with general boundary conditions or without suitable growth conditions. More precisely, we construct two bounded solutions to the heat equation in the half plane which are nowhere analytic in time. In addition, for any \delta >0, we find a solution to the heat equation on the whole plane, with exponential growth of order 2+\delta, which is nowhere analytic in time. [ABSTRACT FROM AUTHOR]

In this paper, we consider and study in detail the generalized Fox–Wright function Ψ ˜ q p introduced in our recent work as an extension of the Fox–Wright function Ψ q p . This special function can be seen as an important case of the so-called I-functions of Rathie and H ¯ -functions of Inayat-Hussain, that in turn extend the Fox H-functions and appear to include some Feynman integrals in statistical physics, in polylogarithms, in Riemann Zeta-type functions and in other important mathematical functions. Depending on the parameters, Ψ ˜ q p is an entire function or is analytic in an open disc with a final radius. We derive its basic properties, such as its order and type, and its images under the Laplace transform and under classical fractional-order integrals. Particular cases of Ψ ˜ q p are specified, including the Mittag-Leffler and Le Roy-type functions and their multi-index analogues and many other special functions of Fractional Calculus. The corresponding results are illustrated. Finally, we emphasize the role of these new generalized hypergeometric functions as eigenfunctions of operators of new Fractional Calculus with specific I-functions as singular kernels. This paper can be considered as a natural supplement to our previous surveys "Going Next after 'A Guide to Special Functions in Fractional Calculus': A Discussion Survey", and "A Guide to Special Functions of Fractional Calculus", published recently in this journal. [ABSTRACT FROM AUTHOR]

This document is a summary of a special issue of the journal "Mathematics" that focuses on new trends in complex analysis research. The issue includes 14 papers that cover various aspects of complex-valued functions of one or several complex variables. The papers explore topics such as coefficient estimates, starlikeness and convexity of analytic functions, holomorphic and bi-univalent functions, and integral operators. The research presented in the papers aims to contribute to the development of complex analysis and inspire further studies in the field. The document also acknowledges the authors and reviewers who contributed to the special issue's success. [Extracted from the article]

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]

This document is a summary of a special issue of the journal Axioms titled "New Developments in Geometric Function Theory II." The special issue contains 14 research papers that explore various topics related to complex-valued functions in the field of Geometric Function Theory. The papers cover subjects such as coefficient estimates, subordination theories, hypergeometric functions, and differential operators. Each paper presents new findings and results that contribute to the development of Geometric Function Theory. The special issue is recommended for researchers and scholars interested in this field of study. The document also acknowledges the authors, reviewers, and editors involved in the creation of the special issue. [Extracted from the article]

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]

In this paper, we present a new method for isolating real roots of a bivariate nonlinear system. It is a subdivision method based on analyzing the local geometrical properties of the given system. We propose the concept of opposite monotone system in a box and use it to determine the existence and the uniqueness of a simple real zero of the system in the box. We have implemented our method and the experiments show the effectivity and efficiency of our approach, especially for high degree sparse polynomial systems. This paper is an extended version of the ISSAC'19 conference paper (Cheng and Wen (2019)). [ABSTRACT FROM AUTHOR]

Let n ∈ N 0 , ψ be an analytic self-map on D and u be an analytic function on D. The single operator D u , ψ n acting on various spaces of analytic functions has been a subject of investigation for many years. It is defined as (D u , ψ n f) (z) = u (z) f (n) (ψ (z)) , f ∈ H (D) . However, the study of the operator P u → , ψ k , which represents a finite sum of these operators with varying orders, remains incomplete. The boundedness, compactness and essential norm of the operator P u → , ψ k on the Bloch space are investigated in this paper, and several characterizations for these properties are provided. [ABSTRACT FROM AUTHOR]

For a fixed analytic function g on the unit disc, we consider the analytic paraproducts induced by g, which are formally defined by T g f (z) = ∫ 0 z f (ζ) g ′ (ζ) d ζ , S g f (z) = ∫ 0 z f ′ (ζ) g (ζ) d ζ , and M g f (z) = g (z) f (z). We are concerned with the study of the boundedness of operators in the algebra A g generated by the above operators acting on Hardy, or standard weighted Bergman spaces on the disc. The general question is certainly very challenging, since operators in A g are finite linear combinations of finite products (words) of T g , S g , M g which may involve a large amount of cancellations to be understood. The results in [1] show that boundedness of operators in a fairly large subclass of A g can be characterized by one of the conditions g ∈ H ∞ , or g n belongs to B M O A or the Bloch space, for some integer n > 0. However, it is also proved that there are many operators, even single words in A g whose boundedness cannot be described in terms of these conditions. The present paper provides a considerable progress in this direction. Our main result provides a complete quantitative characterization of the boundedness of an arbitrary word in A g in terms of a "fractional power" of the symbol g , that only depends on the number of appearances of each of the letters T g , S g , M g in the given word. [ABSTRACT FROM AUTHOR]

This paper introduces a generalization of the Hasse derivative in the sense of the complex conformable derivative. The definition coincides with the classical version of the Hasse derivative of order i$$ i $$. Accordingly, a new base, named complex conformable Hasse derivative bases (CCHDBs), is defined. We investigate the existence of expansions of analytic functions in a series of CCHDBs in Fréchet space on closed and open disks, open regions surrounding closed disks, for all entire functions and at the origin. Moreover, an upper bound for the order and type of the CCHDBs is obtained and proved to be attainable. The 핋ρ‐property of CCHDBs is also discussed. Our results improve and extend the analog results in the complex analysis related to the classical complex derivative of a base of polynomials (BPs). The obtained results clarify several implications for the CCHDBs of special functions such as Euler, Bernoulli, Chebyshev, Bessel, and Gontcharaff polynomials. [ABSTRACT FROM AUTHOR]

AbstractThis paper is devoted to develop the concepts of semi-c-periodicity, c-uniform recurrence and almost automorphy for functions defined on vertical strips in the complex plane, where c is a non-zero complex number. As an extension of the study performed for functions defined on the real axis, this work aims to investigate the main properties of these classes of functions and establish their connections with the more known class of c-almost periodic functions defined on vertical strips. In fact, we resolve an open problem which was raised in 2020 for the real case. Additionally, we also use this approach to introduce an asymptotic version of these new classes of functions. [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]

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]

Let E be an Anderson A -module over C ∞. The period lattice of E is related to its module of special functions by means of a non-canonical isomorphism introduced by the authors in [GM21]. In this paper, we explain how a modification of the inverse map is canonical by interpreting it as a residue morphism along the characteristic graph. This phenomenon has already been observed in various situations. The main innovation of this text is that of costability (costable admissible opens, costable site, etc.) which provides a convenient framework to develop the notion of sheaves of E (C ∞) -valued meromorphic functions on the rigid analytic plane. [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 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 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 many high-impact applications, it is important to ensure the quality of the output of a machine learning algorithm as well as its reliability in comparison to the complexity of the algorithm used. In this paper, we have initiated a mathematically rigorous theory to decide which models (algorithms applied on data sets) are close to each other in terms of certain metrics, such as performance and the complexity level of the algorithm. This involves creating a grid on the hypothetical spaces of data sets and algorithms so as to identify a finite set of probability distributions from which the data sets are sampled and a finite set of algorithms. A given threshold metric acting on this grid will express the nearness (or statistical distance) of each algorithm and data set of interest to any given application. A technically difficult part of this project is to estimate the so-called metric entropy of a compact subset of functions of infinitely many variables that arise in the definition of these spaces. [ABSTRACT FROM AUTHOR]

In this article, we first define and then propose to systematically study some new subclasses of the class of analytic and bi-concave functions in the open unit disk. For this purpose, we make use of a combination of the binomial series and the confluent hypergeometric function. Among some other properties and results, we derive the estimates on the initial Taylor-Maclaurin coefficients | a 2 | and | a 3 | for functions in these analytic and bi-concave function classes, which are introduced in this paper. We also derive a number of corollaries and consequences of our main results in this paper. [ABSTRACT FROM AUTHOR]

The paper concerns a nonlinear second-order parabolic evolution equation, one of the well-known objects of mathematical physics, which describes the processes of high-temperature thermal conductivity, nonlinear diffusion, filtration of liquid in a porous medium and some other processes in continuum mechanics. A particular case of it is the well-known porous medium equation. Unlike previous studies, we consider the case of several spatial variables. We construct and study solutions that describe disturbances propagating over a zero background with a finite speed, usually called 'diffusion-wave-type solutions'. Such effects are atypical for parabolic equations and appear since the equation degenerates on manifolds where the desired function vanishes. The paper pays special attention to exact solutions of the required type, which can be expressed as either explicit or implicit formulas, as well as a reduction of the partial differential equation to an ordinary differential equation that cannot be integrated in quadratures. In this connection, Cauchy problems for second-order ordinary differential equations arise, inheriting the singularities of the original formulation. We prove the existence of continuously differentiable solutions for them. A new example, an analog of the classic example by S.V. Kovalevskaya for the considered case, is constructed. We also proved a new existence and uniqueness theorem of heat-wave-type solutions in the class of piece-wise analytic functions, generalizing previous ones. During the proof, we transit to the hodograph plane, which allows us to overcome the analytical difficulties. [ABSTRACT FROM AUTHOR]

In this research paper, we utilize the q-derivative concept to formulate specific differential and integral operators denoted as R q n , m , λ , F q n , m , λ and G q n , m , λ . These operators are introduced with the aim of generalizing the class of Ruscheweyh operators within the set of univalent functions. We extract certain properties and characteristics of the set of differential subordinations employing specific techniques. By utilizing the newly defined operators, this paper goes on to establish subclasses of analytic functions defined on an open unit disc. Additionally, we delve into the convexity properties of the two recently introduced q-integral operators, F q n , m , λ and G q n , m , λ . Special cases of the primary findings are also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we present a novel class of analytic functions in the form h (z) = z p + ∑ k = p + 1 ∞ a k z k in the unit disk. These functions establish a connection between the extended Mittag–Leffler function and the quantum operator presented in this paper, which is denoted by ℵ q , p n (L , a , b) and is also an extension of the Raina function that combines with the Jackson derivative. Through the application of differential subordination methods, essential properties like bounds of coefficients and the Fekete–Szegő problem for this class are derived. Additionally, some results of special cases to this study that were previously studied were also highlighted. [ABSTRACT FROM AUTHOR]

A branch of complex analysis with a rich history is geometric function theory, which first appeared in the early 20th century. The function theory deals with a variety of analytical tools to study the geometric features of complex-valued functions. The main purpose of this paper is to estimate more accurate bounds for the coefficient | a n | of the functions that belong to a class of bi-univalent functions with missing coefficients that are defined by using the subordination. The significance of our present results consists of improvements to some previous results concerning different recent subclasses of bi-univalent functions, and the aim of this paper is to improve the results of previous outcomes. In addition, important examples of some classes of such functions are provided, which can help to understand the issues related to these functions. [ABSTRACT FROM AUTHOR]

The object of the present paper is to derive some argument properties associated with meromorphic close-to-starlike and Bazilević functions in the punctured open unit disk. Furthermore, we investigate integral-preserving properties in a sector with some special cases of the main results presented here. [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]

The paper explores the impact of Palmer's spontaneous gesture of a red stripe that completed his abstract painting—emotionally and aesthetically—in response to a difficult analytic relationship. The red stripe arises from his search for greater openness, emotional freedom, and creative responses to the flow of communication with his patient. The analyst's ethical intentions are inward to be open and available to the influx of the patient's emotional world, and outward, found in actions and gestures with the patient striving for honesty, transparency, a willingness to take risks and accept accidents in a search for emotion truth and authentically being present. The analyst inhabits, gathers, and concentrates feeling that provides a medium for feelings to live, take shape, and be recognized by an analytic function called "the aesthetic" that then fosters "aesthetic experiences"—intense passages of shared living that are transformative. [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]

We prove that there exists a unique solution to the initial value problem describing the motion of a particle subject to a unilateral constraint and Coulomb friction, if the external force acting on the particle is an analytic function of time and of the particle's position and velocity. Previous work claimed that this problem has a local series solution that corresponds to an analytic function, after any impacts have been resolved. However, a counterexample to that claim was recently discovered, involving a particle starting to slide, in which the series solution is divergent, and thus does not correspond to an analytic function. This paper corrects previous arguments by considering a general formal series solution for a particle that is starting to slide. [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]

The usual process for solving a nonhomogeneous system of linear differential equations is to find the general complementary homogeneous solution first and then construct a particular solution from it. When can we flip the script on this process and compute a particular solution first? This paper argues this is possible when the nonhomogeneous forcing function is analytic. It presents two different series expansions for this particular particular solution using power series methods, and reveals a surprising connection to the more familiar variation of parameters formula. [ABSTRACT FROM AUTHOR]

This paper consists of two parts. The first part is devoted to the study of the Beltrami equation with a polar singularity in a circle centred at the origin, with a cut along the positive semiaxis. The coefficients of the equation have a first-order pole at the origin and do not belong to the class $ L_2(G). $ L 2 (G). Therefore, despite having a unique form, this equation is not covered in the analysis of I.N.Vekua [Generalized analytic functions. Nauka: Moscow; 1988. Russian] and needs to be independently studied. In the second part of the article, the coefficients of the equation are chosen so that the resulting solutions are continuous in a circle without a cut. These results can be used in the theory of infinitesimal bendings of surfaces of positive curvature with a flat point and in constructing a conjugate isometric coordinate system on a surface of positive curvature with a planar point. [ABSTRACT FROM AUTHOR]

In the paper, we prove a general universality theorem for the Riemann zeta-function ζ (s) on approximation of a class of analytic functions by generalized shifts ζ (s + ia(τ)). Here a(τ) is a real-valued continuous increasing to +∞ function, uniformly distributed modulo 1 and such that |ζ(σ + ia(τ) + it)|2, for σ > 1/2, has a traditional mean estimate for every t ∈ ℝ. For example, the function tv(τ), v > 0, where t(τ) is the Gram function, satisfies the hypotheses of the proved theorem. For the proof, the method of weak convergence of probabilistic measures in the space of analytic functions is developed. [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]

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]

This document titled "Fractional Calculus and Hypergeometric Functions in Complex Analysis" explores the impact of fractional calculus on various scientific and engineering disciplines. It emphasizes the significance of fractional operators in the study of fractional calculus and their applications in complex analysis research, specifically in the theory of univalent functions. The document also introduces hypergeometric functions and their connection to the theory of univalent functions. It compiles 12 research papers that cover topics such as geometric properties of fractional differential operators, logarithmic-related problems of univalent functions, and the study of generalized bi-subordinate functions. This document serves as a valuable resource for researchers interested in these subjects and their applications in complex analysis. Additionally, it provides a summary of three articles published in the Special Issue on "Fractional Calculus and Hypergeometric Functions in Complex Analysis." The first article explores the use of the Sălăgean q-differential operator for meromorphic multivalent functions, introducing new subclasses of functions. The second article presents three general double-series identities using Whipple transformations for terminating generalized hypergeometric functions, which can be used to derive additional identities. The third article defines a new generalized domain based on the quotient of two analytic functions and investigates the upper bounds of certain coefficients and determinants. The authors anticipate that these findings will inspire further research in the field. [Extracted from the article]

Let φ be an analytic selfmap of the open unit disk D and g be an analytic function on D. The Volterra-type composition operators induced by the maps g and φ are defined as (Ig φ f) (z) = ∫0 z f' (φ(ζ))g(ζ)dζ and (Tg φf) (z) = ∫0 z f(φ(ζ))g′ (ζ)dζ. For 1 ≤ p < ∞, S p (D) is the space of all analytic functions on D whose first derivative f ′ lies in the Hardy space Hp (D), endowed with the norm ∥f∥Sp = |f(0)| + ∥f′ ∥Hp . Let µ : (0, 1] → (0, ∞) be a positive continuous function on D such that for z ∈ D we define µ(z) = µ(|z|). The weighted Zygmund space Zµ(D) is the space of all analytic functions f on D such that supz∈D µ(z)|f ′′(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 space [ABSTRACT FROM AUTHOR]

A class $ \mathcal {F} $ F consisting of analytic functions $ f(z)=\sum _{n=0}^{\infty }a_nz^n $ f (z) = ∑ n = 0 ∞ a n z n in the unit disc $ \mathbb {D}=\{z\in \mathbb {C}:|z| D = { z ∈ C : | z | < 1 } satisfies a Bohr phenomenon if there exists an $ r_f>0 $ r f > 0 such that \[ \sum_{n=1}^{\infty}|a_n|r^n\leq{d}\left(f(0),\partial f(\mathbb{D})\right) \] ∑ n = 1 ∞ | a n | r n ≤ d (f (0) , ∂f (D)) for every function $ f\in \mathcal {F} $ f ∈ F , and $ |z|=r\leq r_f $ | z | = r ≤ r f . The largest radius $ r_f $ r f is the Bohr radius and the inequality $ \sum _{n=1}^{\infty }|a_n|r^n\leq {d}\left (f(0),\partial f(\mathbb {D})\right) $ ∑ n = 1 ∞ | a n | r n ≤ d (f (0) , ∂f (D)) is Bohr inequality for the class $ \mathcal {F} $ F , where 'd' is the Euclidean distance. In this paper, we prove sharp refinement of the Bohr–Rogosinski inequality for certain classes of harmonic mappings. [ABSTRACT FROM AUTHOR]

In this paper, we study the corona theorem and Wolff theorem for the multiplier algebra of the Besov–Morrey space $ \mathcal {B}_p^{\lambda }(s) $ B p λ (s) when $ 0 0 < λ , s < 1 < p < ∞. Here the Besov–Morrey space $ \mathcal {B}_p^{\lambda }(s) $ B p λ (s) is the space consisting of those analytic functions on the unit disc $ {\mathbb D} $ D such that $ |f'(z)|^p(1-|z|^2)^{p+s-2}{\rm d}A(z) $ | f ′ (z) | p (1 − | z | 2) p + s − 2 d A (z) is a $ \lambda s $ λs -Carleson measure. [ABSTRACT FROM AUTHOR]

This paper deals with the error analysis of the trapezoidal rule for the computation of Fourier type integrals, based on two double exponential transformations. The theory allows to construct algorithms in which the steplength and the number of nodes can be a priori selected. The analysis is also used to design an automatic integrator that can be employed without any knowledge of the function involved in the problem. Several numerical examples, which confirm the reliability of this strategy, are reported. • Error analysis through the theory of analytic functions. • Error estimates for double exponential trapezoidal rules for Fourier type integrals. • Automatic integrator for Fourier type integrals. [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]

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]