Class Of Analytic Functions Research Articles

Properties of a Class of Analytic Functions Influenced by Multiplicative Calculus

Motivated by the notion of multiplicative calculus, more precisely multiplicative derivatives, we used the concept of subordination to create a new class of starlike functions. Because we attempted to operate within the existing framework of the design of analytic functions, a number of restrictions, which are in fact strong constraints, have been placed. We redefined our new class of functions using the three-parameter Mittag–Leffler function (Srivastava–Tomovski generalization of the Mittag–Leffler function), in order to increase the study’s adaptability. Coefficient estimates and their Fekete-Szegő inequalities are our main results. We have included a couple of examples to show the closure and inclusion properties of our defined class. Further, interesting bounds of logarithmic coefficients and their corresponding Fekete–Szegő functionals have also been obtained.

A Class of Analytic Functions Defined by a Second Order Differential Inequality and Error Function

A Class of Analytic Functions Defined by a Second Order Differential Inequality and Error Function

On Some Interesting Classes of Analytic Functions Related to Univalent Functions

On Some Interesting Classes of Analytic Functions Related to Univalent Functions

Partial sums for a generalised class of analytic functions

Partial sums for a generalised class of analytic functions

A class of analytic functions defined using fractional Ruscheweyh–Goyal derivative and its majorization properties

A class of analytic functions defined using fractional Ruscheweyh–Goyal derivative and its majorization properties

On logarithmic coefficients for classes of analytic functions associated with convex functions

Let S denote the class of analytic and univalent functions in the unit disk D={z∈C:|z|<1} of the form f(z)=z+∑n=2∞anzn. For f∈S, the logarithmic coefficients defined by log⁡(f(z)/z)=2∑n=1∞γnzn,z∈D. In 1971, Milin [12] proposed a system of inequalities for the logarithmic coefficients of S. This is known as the Milin conjecture and implies the Robertson conjecture which implies the Bieberbach conjecture for the class S. Recently, the other interesting inequalities involving logarithmic coefficients for functions in S and some of its subfamilies have been studied by Roth [24], and Ponnusamy et al. [17]. In this article, we estimate the logarithmic coefficient inequalities for certain subfamilies of Ma-Minda family defined by a subordination relation. It is important to note that the inequalities presented in this study would generalize some of the earlier work.

Analytic Studies of a Class of Langevin Differential Equations Dominated by a Class of Julia Fractal Functions

In this investigation, we study a class of analytic functions of type Carathéodory style in the open unit disk connected with some fractal domains. This class of analytic functions is formulated based on a kind of Langevin differential equations (LDEs). We aim to study the analytic solvability of LDEs in the advantage of geometric function theory consuming the geometric properties of the Julia fractal (JF) and other fractal connected with the logarithmic function. The analytic solutions of the LDEs are obtainable by employing the subordination theory.

Approximation by double periodic functions in the class C^r_A

In this work we will consider approximation by polynomials in doubly periodic Weierstrass functions for functions that are analytic in a domain and continuous in its closure. This problem is closely related to the approximation by holomorphic polynomials in two variables of a function that is holomorphic in a domain on an elliptic curve. We assume that at the boundary of the region on the plane the length of the are is commensurable with length of the chord. This condition also applies to the region on the elliptic curve. The possibility of obtaining an approximation estimate that is consistent with the so-called the inverse theorem. i. e., with the restoration of the smoothness of the function by the speed of approximation, it was previously obtained for classed of function analytic in a domain for which in the closure of the domain the derivative of a given order has a modulus of continuity of the Holder type, with an order less than unity. The approximation method used earlier did not make it possible to study classes of analytic functions whose derivative of some order is limited. In this paper, we use a different approximation method to approximate by polynomials in doubly periodic Weierstrass functions the functions that are analytic in a domain and whose derivative of a given order is bounded in the domain.

Value of $n$-width of some classes of analytic functions in the Bergman space $B_{2}$

Value of $n$-width of some classes of analytic functions in the Bergman space $B_{2}$

Fractional operators on the bounded symmetric domains of the Bergman spaces

&lt;abstract&gt;&lt;p&gt;Mathematics has several uses for operators on bounded symmetric domains of Bergman spaces including complex geometry, functional analysis, harmonic analysis and operator theory. They offer instruments for examining the interaction between complex function theory and the underlying domain geometry. Here, we extend the Atangana-Baleanu fractional differential operator acting on a special type of class of analytic functions with the $ m $-fold symmetry characteristic in a bounded symmetric domain (we suggest the open unit disk). We explore the most significant geometric properties, including convexity and star-likeness. The boundedness in the weighted Bergman and the convex Bergman spaces associated with a bounded symmetric domain is investigated. A dual relations exist in these spaces. The subordination and superordination inequalities are presented. Our method is based on Young's convolution inequality.&lt;/p&gt;&lt;/abstract&gt;

Applications of $ q- $Ultraspherical polynomials to bi-univalent functions defined by $ q- $Saigo's fractional integral operators

&lt;abstract&gt;&lt;p&gt;This study established upper bounds for the second and third coefficients of analytical and bi-univalent functions belonging to a family of particular classes of analytic functions utilizing $ q- $Ultraspherical polynomials under $ q- $Saigo's fractional integral operator. We also discussed the Fekete-Szegö family function problem. As a result of the specialization of the parameters used in our main results, numerous novel outcomes were demonstrated.&lt;/p&gt;&lt;/abstract&gt;

Conformal Self-Mappings of the Complex Plane with Arbitrary Number of Fixed Points

There are known conformal self-mappings of the fundamental domains of analytic functions via Möbius transformations. When two adjacent fundamental domains have a straight line or an arc of a circle as a common boundary, the Schwarz symmetry principle can be applied for one of those mappings and what we obtain is a conformal self-mapping of the union of those domains in which each one of the domains is mapped onto itself. Repeating this operation until the whole plane is exhausted, we obtain a conformal self-mapping of the complex plane in which every fundamental domain is conformally mapped onto itself. We prove in this paper that this is true for any analytic function. Since the self-mappings of fundamental domains have each one at least one fixed point, ultimately, for the self-mapping of the complex plane, we obtain at least as many fixed points as is the number of fundamental domains. When dealing with a rational function, this number is finite, otherwise we obtain infinitely many fixed points. Computer experimentation allows the illustration of these concepts for most of the familiar classes of analytic functions. There are known applications of the Möbius transformations in physics via the Lorentz group. Relating those application to the present work may contribute to the advancement of the knowledge in that field.

On generalized close-to-convexity related with strongly Janowski functions

"Strongly Janowski functions are used to define certain classes of analytic functions which generalize the concepts of close-to-convexity and bounded boundary rotation. Coefficient results, a necessary condition, distortion bounds, Hankel determinant problem and several other interesting properties of these classes are studied. Some significant well-known results are derived as special cases. Keywords: Starlike, convex, bounded boundary rotation, Carathéodory and Janowski functions, subordination, convolution."

Radius problems for certain classes of analytic functions

"Radius constants for functions in three classes of analytic functions to be a starlike function of order α, parabolic starlike function, starlike function associated with lemniscate of Bernoulli, exponential function, cardioid, sine function, lune, a particular rational function, and reverse lemniscate are obtained. One of these classes are characterized by the condition Re g/(zez ) &gt; 0. The other two classes are defined by using the function g and they consist respectively of functions f satisfying Re f/g &gt; 0 and |f/g − 1| &lt; 1. Keywords: Starlike function, radius of starlikeness, exponential function."

Crouzeix's conjecture for classes of matrices

For a matrix A which satisfies Crouzeix's conjecture, we construct several classes of matrices from A for which the conjecture will also hold. We discover a new link between cyclicity and Crouzeix's conjecture, which shows that Crouzeix's Conjecture holds in full generality if and only if it holds for the differentiation operator on a class of analytic functions. We pose several open questions, which if proved, will prove Crouzeix's conjecture. We also begin an investigation into Crouzeix's conjecture for symmetric matrices and in the case of 3×3 matrices, we show Crouzeix's conjecture holds for symmetric matrices if and only if it holds for analytic truncated Toeplitz operators.

Logarithmic coefficients for a class of analytic functions defined by subordination

In this paper we consider a class of functions Mα(φ) defined by subordination, consisting of functions f ∈ A satisfying the condition (1 −α) zf′(z)/ f(z) + α (1 + zf′′(z)/ f′(z) )≺ φ(z), z ∈ U. In the study of univalent functions, estimates on the Taylor coefficients are usually given. Another significant problem deals with the estimates of logarithmic coefficients. For the class S of univalent functions no sharp bounds for the modulus of the individual logarithmic coefficients are known if n ≥ 3. For different subclasses of S the results are not better and in most cases only th e first three initial coefficients of log f(z)/z are considered. For the class Mα(φ) we obtain upper bounds for the logarithmic coefficients γn, n ∈ {1, 2, 3} and also for Γn, n ∈ {1, 2, 3}, the logarithmic coefficients of the inverse of Mα(φ). Connections with previous known results are pointed out.

Properties of Certain Subclasses of P-valent Functions Defined by Certain Integral Operator

In this paper, we introduce standard definitions for new subclasses of uniformly starlike, uniformly convex p-valent functions, and study properties some of uniformly certain classes of analytic functions defined by certain integral operator.

Certain Results on Subclasses of Analytic and Bi-Univalent Functions Associated with Coefficient Estimates and Quasi-Subordination

The purpose of the present paper is to introduce and investigate new subclasses of analytic function class of bi-univalent functions defined in open unit disks connected with a linear q-convolution operator, which are associated with quasi-subordination. We find coefficient estimates of h2, h3 for functions in these subclasses. Several known and new consequences of these results are also pointed out. There is symmetry between the results of the subclass fq,∑ μ(ζ,n,ρ,σ,ϑ,γ,δ,φ) and the results of the subclass ℵ∑q,δλ,ζ,n,ρ,σ,ϑ,φ.

Applications of Fuzzy Differential Subordination to the Subclass of Analytic Functions Involving Riemann–Liouville Fractional Integral Operator

In this research, we combine ideas from geometric function theory and fuzzy set theory. We define a new operator Dτ−λLα,ζm:A→A of analytic functions in the open unit disc Δ with the help of the Riemann–Liouville fractional integral operator, the linear combination of the Noor integral operator, and the generalized Sălăgean differential operator. Further, we use this newly defined operator Dτ−λLα,ζm together with a fuzzy set, and we next define a new class of analytic functions denoted by Rϝζ(m,α,δ). Several innovative results are found using the concept of fuzzy differential subordination for the functions belonging to this newly defined class, Rϝζ(m,α,δ). The study includes examples that demonstrate the application of the fundamental theorems and corollaries.

Certain Quantum Operator Related to Generalized Mittag–Leffler Function

In this paper, we present a novel class of analytic functions in the form h(z)=zp+∑k=p+1∞akzk 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 ℵq,pn(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.