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In this article, we make use of convex analytic functions Ha(z) = [1/(1 - a)] log[(1 - az)/(1 - z)], a - ℝ, |a| ≤ 1, a ≠ 1 and starlike analytic functions Lb(z) = z/[(1 - bz)(1 - z)], b - ℝ, |b| ≤ 1, to construct univalent harmonic functions by means of a transformation on some normalized univalent analytic functions. Besides exploring mapping properties of harmonic functions so constructed, we establish sufficient conditions for their harmonic convolutions or Hadamard products to be locally univalent and sense preserving, univalent and convex in some direction. [ABSTRACT FROM AUTHOR]

In this effort, we present a new definition of the Steiner symmetrization by using special analytic functions in a complex domain (the open unit disk) with respect to the origin. This definition will be used to optimize the class of univalent analytic functions. Our method is based on the concept of differential subordination and the Caratheodory theory. Examples are illustrated in the sequel involving the modified Libera-Livingston-Bernardi integral operator over the open unit disk. The result gives that this integral satisfies the definition of bounded turning function (univalent analytic function). [ABSTRACT FROM AUTHOR]

As a continuation of the study of polynomial approximation and composition operators on Dirichlet spaces of unit disk, which has settled a problem posed by Cima in 1976, the present paper aims to consider the case of the unbounded domains, such as the half-plane. Specifically, we may obtain the rational approximations in the Dirichlet spaces and characterize the composition operators which have a dense range on the Dirichlet spaces over the half-plane. Moreover, this paper also considers the relationship between the Dirichlet spaces and Hardy spaces on half-plane. [ABSTRACT FROM AUTHOR]

The main purpose of this work is to study the third Hankel determinant for classes of Bernoulli lemniscate-related functions by introducing new subclasses of star-like functions represented by S L λ * and R L λ . In many geometric and physical applications of complex analysis, estimating sharp bounds for problems involving the coefficients of univalent functions is very important because these coefficients describe the fundamental properties of conformal maps. In the present study, we defined sharp bounds for function-coefficient problems belonging to the family of S L λ * and R L λ . Most of the computed bounds are sharp. This study will encourage further research on the sharp bounds of analytical functions related to new image domains. [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 interest in special complex functions and their wide-ranging implementations in geometric function theory (GFT) has developed tremendously. Recently, subordination theory has been instrumentally employed for special functions to explore their geometric properties. In this effort, by using a convolutional structure, we combine the geometric series, logarithm, and Hurwitz–Lerch zeta functions to formulate a new special function, namely, the logarithm-Hurwitz–Lerch zeta function (LHL-Z function). This investigation then contributes to the study of the LHL-Z function in terms of the geometric theory of holomorphic functions, based on the differential subordination methodology, to discuss and determine the univalence and convexity conditions of the LHL-Z function. Moreover, there are other subordination and superordination connections that may be visually represented using geometric methods. Functions often exhibit symmetry when subjected to conformal mappings. The investigation of the symmetries of these mappings may provide a clearer understanding of how subordination and superordination with the Hurwitz–Lerch zeta function behave under different transformations. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce a new subclass of normal- ized analytic and univalent functions in the open unit disk associated with Sigmoid function. Coeficient estimates, convolution conditions, convex- ity and some other geometric properties for functions in this class are investigated. Also, subordination and inclusion results are obtained. [ABSTRACT FROM AUTHOR]

The purpose of the present article to obtain some sufficient conditions for the hypergeometric matrix function belonging to certain classes of starlike and convex functions. Finally, we discuss an integral operator associated with this function. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

ABSTRACTThe authors introduce new classes of analytic function with respect [Formula: see text]-symmetric points subordinate to a domain that is not Carathéodory. To use the existing infrastructure or framework, usually, the study of analytic function have been limited to a differential characterization subordinate to functions which are Carathéodory. Here, we try to obtain various interesting properties of functions which are not Carathéodory. Integral representation, interesting conditions for starlikeness and inclusion relations for functions in these classes are obtained.

A TT-symmetric univalent function is a complex valued function that is conformally mapping the unit disk onto itself and satisfies the symmetry condition ϕ[T](ζ)=[ϕ(ζT)]1∕T{\phi }^{\left[T]}\left(\zeta )={\left[\phi \left({\zeta }^{T})]}^{1/T} for all ζ\zeta in the unit disk. In other words, it is a complex function that preserves the unit disk’s shape and orientation and is symmetric about the unit circle. They are used in the study of geometric function theory and the theory of univalent functions. In recent effort, we extend the class of fractional anomalous diffusion equations in a symmetric complex domain. we aim to present the analytic univalent solution for such a class using special functions technique. Our analysis and comparative findings are further supported by the geometric simulations for the univalent solution such as the convexity and starlikeness of the diffusion. As a consequence of illustration of a list of conditions yielding the univalent solutions (normalize analytic function in the open unit disk), the normalization of diffusion shape is achieved.

We establish that Grunsky norm of any normalized univalent function on a quasidisk is completely determined by the squares of holomorphic abelian differentials (in contrast to Teichmüller norm, which relates to all integrable holomorphic quadratic differentials). The result has important interesting applications in several fields of geometric function theory: it provides an explicit representation of Fredholm eigevalues of all quasiconformal curves, sheds a new light on the distortion theory for univalent functions with quasiconformal extension, etc. [ABSTRACT FROM AUTHOR]

Univalent functions with asymptotically conformal extension to the boundary form a subclass of functions with quasiconformal extension with rather special features. Such functions arise in various questions of geometric function theory and Teichmüller space theory and have important applications involving conformal and quasiconformal maps. The paper provides an approximative characterization of local conformality and its connection with univalent polynomials. Also, some other quantitative applications of this connection are given. [ABSTRACT FROM AUTHOR]

Geometric function theory has extensively explored the geometric characteristics of analytic functions within symmetric domains. This study analyzes the geometric properties of a specific class of analytic functions employing confluent hypergeometric functions and generalized Bessel functions of the first kind. Specific constraints are imposed on the parameters to ensure the inclusion of the confluent hypergeometric function within the analytic function class. The coefficient bound of the class is used to determine the inclusion properties of integral operators involving generalized Bessel functions of the first kind. Different results are observed for these operators, depending on the specific values of the parameters. The results presented here include some previously published findings as special cases. [ABSTRACT FROM AUTHOR]

If S denotes the class of all univalent functions in the open unit disk D : = z ∈ C : | z | < 1 with the form f (z) = z + ∑ n = 2 ∞ a n z n , then the logarithmic coefficients γ n of f ∈ S are defined by log f (z) z = 2 ∑ n = 1 ∞ γ n (f) z n , z ∈ D. The logarithmic coefficients were brought to the forefront by I.M. Milin in the 1960’s as a method of calculating the coefficients a n for f ∈ S . He concerned himself with logarithmic coefficients and their role in the theory of univalent functions, while in 1965 Bazilevič also pointed out that the logarithmic coefficients are crucial in problems concerning the coefficients of univalent functions. In this paper we estimate the bounds for the logarithmic coefficients | γ n (f) | when f belongs to the class B (α , β) of Bazilevič function of type (α , β) . [ABSTRACT FROM AUTHOR]

A new fractional integral operator is used to present a generalized class of analytic functions in a complex domain. The method of definition is based on a Hadamard product of analytic function, which is called convolution product. Then we formulate a convolution integral operator acting on the subclass of normalized analytic functions. Consequently, we investigate the suggested convolution operator geometrically. Differential subordination inequalities, taking the starlike formula are given. Some consequences of well known results are illustrated. [ABSTRACT FROM AUTHOR]

In this paper, we delve into the fascinating realm of analytic mappings within the open unit disc, focusing on functions that are standardly normalized. Our research is centered on determining the sharp upper bound of the third Hankel determinant for a kth-root transformation applied to a specific subclass of these analytic functions. [ABSTRACT FROM AUTHOR]

The focus of the present work is on the establishment and investigation of the coefficient estimates of two new subclasses of bi-close-to-convex functions and bi-concave functions; these are of an Ozaki type and involve a modified Caputo's fractional operator that is associated with three-leaf functions in the open unit disc. The classes are defined using the notion of subordination based on the previously established fractional integral operators and classes of starlike functions associated with a three-leaf function. For functions in these classes, the Fekete-Szegö inequalities and the initial coefficients, | a 2 | and | a 3 | , are discussed. Several new implications of the findings are also highlighted as corollaries. [ABSTRACT FROM AUTHOR]

Without qualms, studies show that quantum calculus has received great attention in recent times. This can be attributed to its wide range of applications in many science areas. In this exploration, we study a new qdifferential operator that generalized many known differential operators. The new q-operator and the concept of subordination were afterwards, used to define a new subclass of analytic-univalent functions that invariably consists of several known and new generalizations of starlike functions. Consequently, some geometric properties of the new class were investigated. The properties include coefficient inequality, growth, distortion and covering properties. In fact, we solved some radii problems for the class and also established its subordinating factor sequence property. Indeed, varying some of the involving parameters in our results led to some existing results. [ABSTRACT FROM AUTHOR]

In this paper we consider various subclasses of normalized, analytic functions defined in the open unit disk A = {z = C : z < 1} in the complex plane C and study the Hardy space of the functions in these subclasses. This study provides an analysis of the growth of these functions near the boundary of the open unit disk and the Taylor's coefficients of them. The study is carried out using the methods of integral means and subordination of analytic functions. Determination of explicit indices of the Hardy space and order of the growth rate of the Taylor coefficient of these functions are important results here. The novelty of the work here is an attempt to extend the study of the above mentioned features for functions in standard subclasses of analytic univalent functions which were not considered by researchers in the past. [ABSTRACT FROM AUTHOR]

In this present paper, we introduce two new subclasses HkΣ (α, δ, λ, µ, σ) and HkΣ (β, δ, λ, µ, σ) of normalized analytic bi-univalent functions defined in the open unit disk and associated with generalized differential operator. Further, we obtain bounds for the second and third Taylor- Maclaurin coefficients of the functions of these subclasses. Also, we obtain some consequences of results obtained. [ABSTRACT FROM AUTHOR]

In this effort, we present a new definition of the Steiner symmetrization by using special analytic functions in a complex domain (the open unit disk) with respect to the origin. This definition will be used to optimize the class of univalent analytic functions. Our method is based on the concept of differential subordination and the Carathéodory theory. Examples are illustrated in the sequel involving the modified Libera–Livingston–Bernardi integral operator over the open unit disk. The result gives that this integral satisfies the definition of bounded turning function (univalent analytic function).

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.

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.

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]

Establishing and evaluation of values of the basic curvelinear quasiinvariants of Jordan curves still remains an important problem of geometric and quasiconformal analysis, especially for applications. It is not solved completely even for polygonal domains. The most general known results were established for unbounded polygons with locally smooth boundaries containing the infinite point and having only a finite number of vertices. The present paper deals with convex polygonal domains having infinite (countable) number of vertices. It creates a new approach in this direction and establishes that quasiinvariants of such polygons are estimated by their geometric characteristics. [ABSTRACT FROM AUTHOR]

In this note we discuss some problems related to conformal slit-mappings. On the one hand, classical Loewner theory leads us to questions concerning the embedding of univalent functions into slit-like Loewner chains. On the other hand, a recent result from monotone probability theory motivates the study of univalent functions from a probabilistic perspective. [ABSTRACT FROM AUTHOR]

The field of geometric function theory has thoroughly investigated starlike functions concerning symmetric points. The main objective of this work is to derive certain geometric properties, such as the starlikeness of order δ , convexity of order δ , k-starlikeness, k-uniform convexity, lemniscate starlikeness and convexity, exponential starlikeness and convexity, and pre-starlikeness for the Galué type Struve function (GTSF). Furthermore, the conditions for GTSF belonging to the Hardy space are also derived. The results obtained in this work generalize several results available in the literature. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We consider a subclass S L ∗ (λ) of starlike functions f satisfying the subordination z f ′ (z) / f (z) ≺ φ (z) where the function φ defined by φ (z) = (1 + z) λ , 0 < λ ≤ 1 , maps the open unit disk in the complex plane to a domain symmetric with respect to real axis in the right-half plane. Applying the admissibility condition technique in the theory of differential subordination developed by Miller and Mocanu, we investigate various subordination for functions to belong this class. As applications, we present several sufficient conditions of normalized analytic functions to be in this class. [ABSTRACT FROM AUTHOR]

Exploring some characteristics of differential subordination and superordination of analytic univalent functions in an open unit disc is the aim of this work and additionally, we have the form's normalized Taylor-Maclaurin series: f(z) = z+ ∑n=2 ∞ anzn. It also aims to clarify the results of the sandwich. By utilizing the integral operator's properties to examine forth-order differential subordination and superordination of analytic univalent functions, some fascinating results are found and explore forth-order subordinations and superordinations in respect to the convolution. Ultimately, we acquired multiple outcomes concerning fourth-order sandwich theorems inside the open unit disk. [ABSTRACT FROM AUTHOR]

The m-fold symmetric in terms of a generalized distribution series has not been considered in the literature. In this study, however, the authors investigated the bi-univalency of m-fold symmetric functions for the generalized distribution of two subclasses of analytic functions. The initial few coefficient bounds a m S and a 2 m S are obtained for the two subclasses of functions defined and the results serve as a new generalization in this direction. [ABSTRACT FROM AUTHOR]

In this paper, we find sufficient conditions to be imposed on the parameters of a class of functions related to the Barnes–Mittag-Leffler function that allow us to conclude that it possesses certain geometric properties (such as starlikeness, uniformly starlike (convex), strongly starlike (convex), convexity, and close-to-convexity) in the unit disk. The key tools in some of our proofs are the monotonicity properties of a certain class of functions related to the gamma function. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to introduce a new subclass of harmonic univalent functions by applying q-calculus. Coefficient inequalities, extreme points, distortion bounds, covering results, convolution condition and convex combination are determined for this class. Finally, we discuss a class preserving integral operator for this class. [ABSTRACT FROM AUTHOR]

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In the present paper, we introduce two new subclasses of the m-fold symmetric, analytic and bi-univalent function class Σm defined in the open unit disk D1:= {z: z ∈ C and |z| < 1}. These two subclasses are denoted by SΣm (α) and S*Σm (β). For the functions f belong to both of these subclasses, we obtain estimates on the first two Taylor-Maclaurin coefficients |am+1| and |a²m+1|. Also, we obtain estimate on the Fekete-Szegö functional |a2m+1 - ka 2 m+1 |, k ∈ R. It is interesting to see that the geometrical similarities in these two subclasses also reflects in their coefficient estimates. Further, we pointed out interconnection of these results with some of the earlier known results. [ABSTRACT FROM AUTHOR]

This research introduces a new approach to elevate the precision of image edge detection through a new algorithm rooted in the coefficients derived from the subclass SCt,ρ (CSKP model). Our method employs convolution operations on input image pixels, utilizing the CSKP mask window in eight distinct directions, fostering a comprehensive and multi-directional analysis of edge features. To gauge the efficacy of our algorithm, image quality is assessed through perceptually significant metrics, including contrast, correlation, energy, homogeneity, and entropy. The study aims to contribute a valuable tool for diverse applications such as computer vision and medical imaging by presenting a robust and innovative solution to enhance image edge detection. The results demonstrate notable improvements, affirming the potential of the proposed algorithm to advance the current state-of-the-art in image processing.

Abstract Due to the major importance of the study of the logarithmic coefficients for univalent functions, in this paper we find the sharp upper bounds for some expressions associated with logarithmic coefficients of functions that belong to some well-known classes of analytic functions in the open unit disk of the complex plane.

In this paper, we studied some geometric properties of a class of analytic functions related to the generalized Mathieu type power series. Furthermore, we have identified interesting consequences and some examples accompanied by graphical representations to illustrate the results achieved.

The main purpose of this work is to study the third Hankel determinant for classes of Bernoulli lemniscate-related functions by introducing new subclasses of star-like functions represented by SLλ* and RLλ. In many geometric and physical applications of complex analysis, estimating sharp bounds for problems involving the coefficients of univalent functions is very important because these coefficients describe the fundamental properties of conformal maps. In the present study, we defined sharp bounds for function-coefficient problems belonging to the family of SLλ* and RLλ. Most of the computed bounds are sharp. This study will encourage further research on the sharp bounds of analytical functions related to new image domains.

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.

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 h 2 ,   h 3 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 f q , ∑   μ (ζ , n , ρ , σ , ϑ , γ , δ , φ) and the results of the subclass ℵ ∑ q , δ λ , ζ , n , ρ , σ , ϑ , φ . [ABSTRACT FROM AUTHOR]

Recently in relation to the theory of non-commutative probability, the notion of evolution family \{\omega _{s,t}\}_{s \le t} is generalized assuming only continuity in parameters, namely (s,t) \mapsto \omega _{s,t} is continuous with respect to locally uniform convergence on a planar domain. In this article we present various equivalence conditions to the continuous evolution families concerned with the left and right parameters. We also provide an example of a discontinuous evolution family in the last section. [ABSTRACT FROM AUTHOR]

The research presented in this paper concerns the notion of geometric function theory called fuzzy differential subordination. Using the technique associated with fuzzy differential subordination, a new subclass of analytic functions related with the strongly Janowski-type functions is defined. The class is introduced by using a new operator defined by the convolution of the generalized Sălăgean differential operator and Choi integral linear operator. Certain inclusion relations are proved for this class by using the notion of fuzzy differential subordination. In addition, new fuzzy differential subordinations are obtained that characterize this class. [ABSTRACT FROM AUTHOR]

The objective of this paper is to estimate the sharp bound of the third Hankel determinant for the kth-root transformation to the class of functions whose derivative has a positive real part satisfying the normalized conditions f(0) = 0 and f0(0) = 1 in the open unit disk D:= {z ∊ C: |z| < 1. [ABSTRACT FROM AUTHOR]

In this paper, we studied some geometric properties of a class of analytic functions related to the generalized Mathieu type power series. Furthermore, we have identified interesting consequences and some examples accompanied by graphical representations to illustrate the results achieved. [ABSTRACT FROM AUTHOR]

An important problem in applications of quasiconformal analysis and in its numerical aspect is to establish algorithms for explicit or approximate determination of the basic quasiinvariant curvelinear and analytic functionals intrinsically connected with conformal and quasiconformal maps, such as their Teichmüller and Grunsky norms, Fredholm eigenvalues and the quasireflection coefficients of associated quasicircles. We prove two general theorems of new type answering this question for univalent functions in arbitrary quasiconformal domains and provide its applications. The results are strengthened in the case of maps of the disk and give rise to extremal Beltrami coefficients of a new type. [ABSTRACT FROM AUTHOR]

Due to the major importance of the study of the logarithmic coefficients for univalent functions, in this paper we find the sharp upper bounds for some expressions associated with logarithmic coefficients of functions that belong to some well-known classes of analytic functions in the open unit disk of the complex plane. [ABSTRACT FROM AUTHOR]