Theory Of Analytic Functions Research Articles

Upper Bounds of the Third Hankel Determinant for Bi-Univalent Functions in Crescent-Shaped Domains

This paper investigates the third Hankel determinant, denoted H3(1), for functions within the subclass RS∑*(λ) of bi-univalent functions associated with crescent-shaped regions φ⦅z=z+1+z2. The primary aim of this study is to establish upper bounds for H3(1). By analyzing functions within this specific geometric context, we derive precise constraints on the determinant, thereby enhancing our understanding of its behavior. Our results and examples provide valuable insights into the properties of bi-univalent functions in crescent-shaped domains and contribute to the broader theory of analytic functions.

Application of Hayman’s Theorem to Directional Differential Equations With Analytic Solutions in the Unit Ball

In this paper, we investigate analytic solutions of higher order linear non-homogeneous directional differential equations whose coefficients are analytic functions in the unit ball. We use methods of theory of analytic functions in the unit ball having bounded L-index. Our proofs are based on application of inequalities from analog of Hayman’s theorem for analytic functions in the unit ball. There are presented growth estimates of their solutions which contain parameters depending on the coefficients of the equations. Also, we obtained sufficient conditions that every analytic solution of the equation has bounded L-index in the direction. The deduced results are also new in one-dimensional case, i.e. for functions analytic in the unit disc. Keywords: Analytic function, analytic solution, slice function, unit ball, directional differential equation, growth estimate, bounded L-index in direction.

Fuzzy Subordination Results for Meromorphic Functions Connected with a Linear Operator

The concept of subordination is expanded in this study from the fuzzy sets theory to the geometry theory of analytic functions with a single complex variable. This work aims to clarify fuzzy subordination as a notion and demonstrate its primary attributes. With this work’s assistance, new fuzzy differential subordinations will be presented. The first theorems lead to intriguing corollaries for specific aspects chosen to exhibit fuzzy best dominance. The work introduces a new integral operator for meromorphic functions and uses the newly developed integral operator, which is starlike and convex, respectively, to obtain conclusions on fuzzy differential subordination.

The Hilbert problem in a half-plane for generalized analytic functions with a singular point on the real axis

This article analyzes the inhomogeneous Hilbert boundary value problem for an upper half-plane with the finite index and boundary condition on the real axis for one generalized Cauchy–Riemann equation with a singular point on the real axis. A structural formula was obtained for the general solution of this equation under restrictions leading to an infinite index of the logarithmic order of the accompanying Hilbert boundary value problem for analytic functions. This formula and the solvability results of the Hilbert problem in the theory of analytic functions were applied to solve the set boundary value problem.

CONSIDERATION OF GEOMETRIC NONLINEARITY IN MATHEMATICAL MODELING OF PROBLEMS OF THE THEORY OF SPRING

Solutions to many problems that are important for practice that arise in modern technology cannot always be obtained by traditional methods of the theory of analytical functions or by means of integral transformations. This applies, for example, to contact problems in which the finite dimensions of the region are taken into account in at least one direction, or media with curvilinear anisotropy are studied, etc. The means of the mathematical theory of elasticity are not very effective for the study of such problems. In this case, it is advisable to use the achievements of the potential theory. The use of asymptotic methods at the same time, even in complex cases, makes it possible to obtain well-founded approximate equations, clarify qualitative regularities, and obtain analytical solutions to problems. This paper presents a generalization of the perturbation method, which makes it possible to reduce the study of complex problems of geometrically nonlinear elasticity theory (in the plane and spatial formulation) to the consistent solution of simpler boundary value problems of the potential theory. The geometrically nonlinear theory of elasticity contains some features that make it different from the classical (linear) theory. The main difference is to take into account the difference between the geometry of the undeformed and deformed states of the body under study, when there are movements that cause significant changes in the geometry of the body. At the same time, the equilibrium equation must be drawn up taking into account changes in the shape and size of structures. Taking into account finite deformations, which when creating mathematical models leads to significant difficulties in solving problems, but at the same time brings the model closer to the real problem.

Analytic Functions in a Complete Reinhardt Domain Having Bounded L-Index in Joint Variables

The manuscript is an initiative to construct a full and exhaustive theory of analytical multivariate functions in any complete Reinhardt domain by introducing the concept of L-index in joint variables for these functions for a given continuous, non-negative, non-vanishing, vector-valued mapping L defined in an interior of the domain with some behavior restrictions. The complete Reinhardt domain is an example of a domain having a circular symmetry in each complex dimension. Our results are based on the results obtained for such classes of holomorphic functions: entire multivariate functions, as well as functions which are analytical in the unit ball, in the unit polydisc, and in the Cartesian product of the complex plane and the unit disc. For a full exhaustion of the domain, polydiscs with some radii and centers are used. Estimates of the maximum modulus for partial derivatives of the functions belonging to the class are presented. The maximum is evaluated at the skeleton of some polydiscs with any center and with some radii depending on the center and the function L and, at most, it equals a some constant multiplied by the partial derivative modulus at the center of the polydisc. Other obtained statements are similar to the described one.

The Riemann problem in a half-plane for generalized analytic functions with a supersingular point on the contour of the boundary condition

In this paper, we study an inhomogeneous Riemann boundary value problem with a finite index and a boundary condition on the real axis for a generalized equation Cauchy–Riemann with supersingular coefficients. To solve the problem, it was necessary to derive a structural formula for the general solution of this equation and to investigate the solvability of the Riemann boundary value problem of the theory of analytic functions with an infinite index due to the power-order vorticity point.

Countably Generated Algebras of Analytic Functions on Banach Spaces

In the paper, we study various countably generated algebras of entire analytic functions on complex Banach spaces and their homomorphisms. Countably generated algebras often appear as algebras of symmetric analytic functions on Banach spaces with respect to a group of symmetries, and are interesting for their possible applications. Some conditions of the existence of topological isomorphisms between such algebras are obtained. We construct a class of countably generated algebras, where all normalized algebraic bases are equivalent. On the other hand, we find non-isomorphic classes of such algebras. In addition, we establish the conditions of the hypercyclicity of derivations in countably generated algebras of entire analytic functions of the bounded type. We use methods from the theory of analytic functions of several variables, the theory of commutative Fréchet algebras, and the theory of linear dynamical systems.

Determination of some properties of starlike and close-to-convex functions according to subordinate conditions with convexity of a certain analytic function

UDC 517.5 Investigation of the theory of complex functions is one of the most fascinating aspects of theory of complex analytic functions of one variable. It has a huge impact on all areas of mathematics. Many mathematical concepts are explained when viewed through the theory of complex functions. Let f ( z ) ∈ A , f ( z ) = z + ∑ n ≥ 2 ∞ a n z n , be an analytic function in the open unit disc normalized by f ( 0 ) = 0 and f ' ( 0 ) = 1. For close-to-convex and starlike functions, new and different conditions are obtained by using subordination properties, where r is a positive integer of order 2 - r ( 0 < 2 - r ≤ 1 2 ) . By using subordination, we propose a criterion for f ( z ) ∈ S * [ a r , b r ] . The relations for starlike and close-to-convex functions are investigated under certain conditions according to their subordination properties. At the same time, we analyze the convexity of some analytic functions and study their regional transformations. In addition, the properties of convexity for f ( z ) ∈ A are examined.

New Applications of Fuzzy Set Concept in the Geometric Theory of Analytic Functions

Zadeh’s fuzzy set theory offers a logical, adaptable solution to the challenge of defining, assessing and contrasting various sustainability scenarios. The results presented in this paper use the fuzzy set concept embedded into the theories of differential subordination and superordination established and developed in geometric function theory. As an extension of the classical concept of differential subordination, fuzzy differential subordination was first introduced in geometric function theory in 2011. In order to generalize the idea of fuzzy differential superordination, the dual notion of fuzzy differential superordination was developed later, in 2017. The two dual concepts are applied in this article making use of the previously introduced operator defined as the convolution product of the generalized Sălgean operator and the Ruscheweyh derivative. Using this operator, a new subclass of functions, normalized analytic in U, is defined and investigated. It is proved that this class is convex, and new fuzzy differential subordinations are established by applying known lemmas and using the functions from the new class and the aforementioned operator. When possible, the fuzzy best dominants are also indicated for the fuzzy differential subordinations. Furthermore, dual results involving the theory of fuzzy differential superordinations and the convolution operator are established for which the best subordinants are also given. Certain corollaries obtained by using particular convex functions as fuzzy best dominants or fuzzy best subordinants in the proved theorems and the numerous examples constructed both for the fuzzy differential subordinations and for the fuzzy differential superordinations prove the applicability of the new theoretical results presented in this study.

Fuzzy Differential Inequalities for Convolution Product of Ruscheweyh Derivative and Multiplier Transformation

In this paper, the author combines the geometric theory of analytic function regarding differential superordination and subordination with fuzzy theory for the convolution product of Ruscheweyh derivative and multiplier transformation. Interesting fuzzy inequalities are obtained by the author.

Some effective solutions for Prandtl's type integro‐differential equation

Prandtl's type integro‐differential equations with different coefficients are investigated. Using the methods of the theory of analytic functions and integral transformations, the singular integro‐differential equation is reduced to boundary value problems of the theory of analytic functions. Effective solutions of this equation and the corresponding asymptotic estimates are obtained.

Hyper-power series and generalized real analytic functions

This article is a natural continuation of the paper Tiwari, D., Giordano, P., Hyperseries in the non-Archimedean ring of Colombeau generalized numbers in this journal. We study one variable hyper-power series by analyzing the notion of radius of convergence and proving classical results such as algebraic operations, composition and reciprocal of hyper-power series. We then define and study one variable generalized real analytic functions, considering their derivation, integration, a suitable formulation of the identity theorem and the characterization by local uniform upper bounds of derivatives. On the contrary with respect to the classical use of series in the theory of Colombeau real analytic functions, we can recover several classical examples in a non-infinitesimal set of convergence. The notion of generalized real analytic function reveals to be less rigid both with respect to the classical one and to Colombeau theory, e.g. including classical non-analytic smooth functions with flat points and several distributions such as the Dirac delta. On the other hand, each Colombeau real analytic function is also a generalized real analytic function.

Some Fundamental Results from Complex Analysis

This paper is going to introduce the most basic theory of analytic functions of one complex variable. It begins at the discussion of meaning of complex number and the historical development from the formula of cubic equation to the square root of negative number. In the middle section, which is divided in to four small parts. First part states the expression of complex number z, algebraic properties, and the relationship of each single complex number with whole complex plane. Second part concerns about several elementary functions of complex number. Next part relates to the derivative of complex number, such as the partial derivative and Cauchy-Riemann Equation to verdict that whether a function is analytic in the domain. Last part is the integral of complex functions, which concludes some important theorems with the proof, and examples. After the section of introduction of complex number, it comes to the final section which is talking about an identity involving complex number and the prove of it.

The adhesive contact problem for a piecewise-homogeneous orthotropic plate with an elastic patch

A piecewise-homogeneous elastic orthotropic plate, reinforced with a finite patch of the wedge-shaped, which meets the interface at a right angle and is loaded with tangential and normal forces is considered. Using methods of the theory of analytic functions, the problem is reduced to the system of singular integro-differential equations (SIDE) with fixed singularity. Under tension-compression of patch using an integral transformation a Riemann problem is obtained, the solution of which is presented in explicit form. The tangential contact stresses along the contact line are determined and their asymptotic behavior in the neighborhood of singular points is established.

Краевая задача Гильберта для одного класса обобщенных аналитических функций с сингулярной линией

In this paper, we study an inhomogeneous Hilbert boundary value problem with a finite index and a boundary condition on the real axis for one generalized equation Cauchy — Riemann with singular coefficient. A structural formula for the general solution of this equation is obtained under constraints leading to an infinite index of the accompanying Hilbert boundary value problem for analytic functions. The study of the solvability of the latter is the basis for solving the boundary value problem for generalized analytical functions. Note that here we have obtained a boundary value problem with an infinite index and two swirl points. Such a situation is not studied in the case of an unbounded domain, even for analytic functions. We have obtained a formula for the general solution of the indicated Hilbert problem in the theory of analytic functions. We carried out a complete study of the solvability of this problem in terms of the characteristics of the coefficient of the boundary condition. Note that here we have obtained a boundary value problem with an infinite index and two swirl points. Such a situation is not studied in the case of an unbounded domain, even for analytic functions. We have obtained a formula for the general solution of the indicated Hilbert problem in the theory of analytic functions. We carried out a complete study of the solvability of this problem in the class of bounded analytic functions. This study made it possible to obtain a formula for solving the Hilbert problem already for generalized analytic functions. We also formulated the conditions under which the problem has no solution, has a single solution or an infinite number of solutions.

Adhesive behavior between dissimilar materials subjected to thermo-elastic loadings with normal-tangential coupling effect

The temperature difference always exists between contacting objects in various thermal bonding technologies, where the adhesion strength is found to be very sensitive to the temperature. In this paper, the anisothermal adhesive contact between an elastic cylinder subjected to the external force with an inclined angle and an elastic substrate is investigated based on the non-slipping JKR model, where the coupling effect between the normal and tangential tractions within the contact region is taken into consideration. The stated problem is reduced to a system of coupled singular integral equations, which can be solved with the analytical function theory. The full analytical solutions of the oscillatory stress fields are obtained for the first time by taking the series expansion of the integrand induced by the thermoelastic effect, which is of substantial significance. The relationship between the contact size and the external loadings is established by virtue of the Griffith energy balance criterion. The difference between the oscillatory and non-oscillatory solutions of stress fields is discussed in detail. Numerical results demonstrate that the oscillatory solution cannot be well approximated by the corresponding non-oscillatory solution for the cases of large Dundurs’ index and external compressive force. Furthermore, it is found that both the thermoelastic effect and the inclined angle of the applied force have significant influences on the adhesive contact behaviors. The presence of temperature difference between contacting objects can either enhance or weaken the adhesion effect, which is related to the inclined angle of the external force. The results obtained from this paper may provide a new insight into the influence mechanism of the temperature on adhesion behaviors.

Hadron and light nucleus radii from electron scattering* *Supported by the National Natural Science Foundation of China (12135007); Natural Science Foundation of Jiangsu Province (BK20220122) and STRONG-2020 "The strong interaction at the frontier of knowledge: fundamental research and applications" which received funding from the European Union's Horizon 2020 research and innovation programme (824093)

Conceptually, radii are amongst the simplest Poincaré-invariant properties that can be associated with hadrons and light nuclei. Accurate values of these quantities are necessary so that one may judge the character of putative solutions to the strong interaction problem within the Standard Model. However, limiting their ability to serve in this role, recent measurements and new analyses of older data have revealed uncertainties and imprecisions in the radii of the proton, pion, kaon, and deuteron. In the context of radius measurement using electron+hadron elastic scattering, the past decade has shown that reliable extraction requires minimisation of bias associated with practitioner-dependent choices of data fitting functions. Different answers to that challenge have been offered; and this perspective describes the statistical Schlessinger point method (SPM), in unifying applications to proton, pion, kaon, and deuteron radii. Grounded in analytic function theory, independent of assumptions about underlying dynamics, free from practitioner-induced bias, and applicable in the same form to diverse systems and observables, the SPM returns an objective expression of the information contained in any data under consideration. Its robust nature and versatility make it suitable for use in many branches of experiment and theory.

Geometrical Theory of Analytic Functions

This Special Issue, devoted to the topic of the “Geometric Theory of Analytic Functions”, aims to bring together the newest research achievements of scholars studying the complex-valued functions of one variable [...]

A simple proof of the Fundamental Theorem of Algebra

Many proofs of the Fundamental Theorem of Algebra, including various proofs based on the theory of analytic functions of a complex variable, are known. To the best of our knowledge, this proof is different from the existing ones.