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AbstractIn this paper we define a Bernardi type quantum integral operator. It transforms the starlike univalent in the unit disk into a starlike region in it. We show that the upper-bound of the third-order Hankel determinant for classes of q-starlike functions is connected with a q-analogue integral operator, defined by a modified q-Bernardi integral operator. The Fekete-Szegö inequality of these classes is also investigated. Numerous well-known specific instances, examples and graphics are listed in the paper. The computations are done by Mathematica 13.3. [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]

Orthogonal polynomials have been widely employed by renowned authors within the context of geometric function theory. This study is driven by prior research and aims to address the —Fekete-Szegö problem. Additionally, we provide bound estimates for the coefficients and an upper bound estimate for the second Hankel determinant for functions belonging to the category of analytical and bi-univalent functions. This investigation incorporates the utilization of Euler polynomials. [ABSTRACT FROM AUTHOR]

This work presents a novel investigation that utilizes the integral operator I p , λ n in the field of geometric function theory, with a specific focus on sandwich theorems. We obtained findings about the differential subordination and superordination of a novel formula for a generalized integral operator. Additionally, certain sandwich theorems were discovered. [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]

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 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]

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. [ABSTRACT FROM AUTHOR]

A new subclass T X q [ λ , A , B ] of analytic functions is introduced by making use of the q-derivative operator associated with the Pascal distribution. Certain properties of analytic functions in the subclass T X q [ λ , A , B ] are derived. Some known results are generalized. [ABSTRACT FROM AUTHOR]

We connect the pre-Schwarzian norm of logharmonic mappings to the pre-Schwarzian norm of an analytic function and establish some necessary and sufficient conditions under which locally univalent logharmonic mappings have a finite pre-Schwarzian norm. We also obtain a necessary and sufficient condition for a logharmonic function to be Bloch. Furthermore, we obtain the pre-Schwarzian norm and growth theorem for logharmonic Bloch mappings and their analytic and co-analytic parts. [ABSTRACT FROM AUTHOR]

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

This study presents a subclass S (β) of bi-univalent functions within the open unit disk region D . The objective of this class is to determine the bounds of the Hankel determinant of order 3, ( Ⱨ 3 (1) ). In this study, new constraints for the estimates of the third Hankel determinant for the class S (β) are presented, which are of considerable interest in various fields of mathematics, including complex analysis and geometric function theory. Here, we define these bi-univalent functions as S (β) and impose constraints on the coefficients │ a n │ . Our investigation provides the upper bounds for the bi-univalent functions in this newly developed subclass, specifically for n = 2, 3, 4, and 5. We then derive the third Hankel determinant for this particular class, which reveals several intriguing scenarios. These findings contribute to the broader understanding of bi-univalent functions and their potential applications in diverse mathematical contexts. Notably, the results obtained may serve as a foundation for future investigations into the properties and applications of bi-univalent functions and their subclasses. [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]

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 [6, 7], it was shown that mixed problems can be both ill-posed for Petrovsky well-conditioned equations and well-posed for ill-conditioned equations. In the present paper we study the existence and uniqueness of the solution of a mixed problem for a class of equations with complex-valued coefficients that behave as parabolic ones, despite the fact that they can change over "time" from parabolic type to Schrödinger type, or even to antiparabolic type. [ABSTRACT FROM AUTHOR]

In this paper, we present and investigate the notion of third-order strong differential subordinations, unveiling several intriguing properties within the context of specific classes of admissible functions. Furthermore, we extend certain definitions, presenting novel and fascinating results. We also derive several interesting properties of the results of third-order strong differential subordinations for analytic functions associated with the Srivastava–Attiya operator. [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]

A new linear integro-differential equation is considered on a closed curve located on the complex plane. The integrals in the equation are understood in the sense of a finite part, according to Hadamard. The order of the equation can be higher than one. The coefficients of the equation have a special structure. A characteristic of the coefficients is that they contain two arbitrary polynomials and their derivatives. Solvability conditions are explicitly stated. Whenever they are satisfied, an exact analytical solution is given. Generalized Sokhotsky formulas, the theory of Riemann boundary value problems, methods for solving linear differential equations, and the properties of analytic functions of a complex variable are used. An example is given. [ABSTRACT FROM AUTHOR]

We provide a complete description of zero sets of some new analytic area Nevanlinna type spaces in the unit disk. Our theorems extend recent sharp results of E. Rodikova to larger analytic function spaces of area Nevanlinna type. [ABSTRACT FROM AUTHOR]

In this paper, new integro-differential operators are introduced that defined by Salagean’s differential operator. The major object of the present study is to investigate convexity properties on new geometric subclasses included these new operators. [ABSTRACT FROM AUTHOR]

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]

One of the fundamental parts of Geometric Function Theory is the study of analytic functions in different domains with critical geometrical interpretations. This article defines a new generalized domain obtained based on the quotient of two analytic functions. We derive various properties of the new class of normalized analytic functions X defined in the new domain, including the sharp estimates for the coefficients a 2 , a 3 , and a 4 , and for three second-order and third-order Hankel determinants, H 2 , 1 X , H 2 , 2 X , and H 3 , 1 X . The optimality of each obtained estimate is given as well. [ABSTRACT FROM AUTHOR]

This paper presents a geometric approach to the problems in differential subordination theory. The necessary conditions for a function to be in various subfamilies of the class of starlike functions and the class of Carathéodory functions are studied, respectively. Further, several consequences of the findings are derived. [ABSTRACT FROM AUTHOR]

Using the Salagean q-differential operator, we investigate a novel subclass of analytic functions in the open unit disc, and we use the Hadamard product to provide some inclusion relations. Furthermore, the coefficient conditions, convolution properties, and applications of the q-fractional calculus operators are investigated for this class of functions. In addition, we extend the Miller and Mocanu inequality to the q-theory of analytic functions. [ABSTRACT FROM AUTHOR]

This article extends the study of q-versions of analytic functions by introducing and studying the association of starlike functions with trigonometric cosine functions, both defined in their q-versions. Certain coefficient inequalities like coefficient bounds, Zalcman inequalities, and both Hankel and Toeplitz determinants for the new version of starlike functions are investigated. It is worth mentioning that most of the determined inequalities are sharp with the support of relevant extremal functions. [ABSTRACT FROM AUTHOR]

In this study, a novel integral operator that extends the functionality of some existing integral operators is presented. Specifically, the integral operator acts as the inverse operator to the widely recognized Opoola differential operator. By making use of the integral operator, a certain subclass of analytic univalent functions defined in the unit disk is proposed and investigated. This new class encompasses some familiar subclasses, like the class of starlike and the class of convex functions, while some new ones are introduced. The investigation thereafter covers coefficient inequality, distortion, growth, covering, integral preserving, closure, subordinating factor sequence, and integral means properties. Furthermore, the radii problems associated with this class are successfully addressed. Additionally, a few remarks are provided, to show that the novel integral operator and the new class generalize some existing ones. [ABSTRACT FROM AUTHOR]

Non-local operators of differentiation are bestowed with capabilities of encompassing complex natural into mathematical equations. Symmetry as invariance under a specified group of transformations can allow for the concept to be applied extensively not only to spatial figures but also to abstract objects like mathematical expressions which can be said to be expressions of physical relevance, in particular dynamical equations. Derived from this point of view, it can be noted that the more complex physical problems are, the more complex mathematical operators of differentiation are required. Accordingly, the fractal–fractional operators (FFOs) are expanded into the complex plane in our research which revolves around a unique class of normalized analytic functions in the open unit disk. To bring FFOs (differential and integral) into the normalized class, the study aims to expand and modify them along with the investigation of the FFOs geometrically. The qualities of convexity and starlikeness are implicated in this study where the differential subordination technique serves as the foundation for the inquiry under consideration. Furthermore, a collection of differential FFO inequalities is taken into account, demonstrating that the normalized Fox–Wright function can contain all FFOs. Besides these steps, the concept of Grunsky factors is applied to investigate symmetry, while boundary value issues involving FFOs are probed. Consequently, the related properties and applications can be further developed, which requires the devotion to differential fractional problems and diverse complex problems in relation to viable applications, pointing out the room to modify and upgrade the existing methods for more optimal outcomes in challenging real-world problems. [ABSTRACT FROM AUTHOR]

The k -convoluted operators related to the k -Whittaker function, confluent hypergeometric function of the first kind, have been developed using the k -symbol calculus in which this sort of calculus presents a generalization of the gamma function. K -symbol fractional calculus is employed to generalize and extend many differential and integral operators of fractional calculus. Based on this premise, a new geometric formula for normalized functions in the symmetric domain known as the open unit disk using the conformable fractional differential operator has been presented in this study. Thus, our technique entails investigating the most well-known geometric properties of this new operator, such as the subordination features and coefficient bounds so that the theory of differential subordination can be adjusted accordingly. By means of this technique, numerical results have been investigated for the proposed method. To this end, a few prominent corollaries of our primary findings as standout instances have been pointed out based on the positivity of the solutions, computational and numerical analyses. [ABSTRACT FROM AUTHOR]

We introduce the class of analytic and univalent functions of the form f(z) = z + ∑ az ∞ j j=2 for z ∈ △:= {z: |z| < 1} that satisfies 1 + 1 μ {(Sλ tf(z)) ′ − 1} ≺ (1+Lz) (1+Mz), z∈△, such that L and M are fixed numbers where satisfy −1 ≤ M < L ≤ 1, T ≥ 0, λ > −1 and μ is arbitrary complex number. We denote this subclass by Qμ (τ, λ, L, M). In this paper, we determine coefficients estimates, distortion theorem and maximization theorem for the new subclass Qμ (τ, λ, L, M). [ABSTRACT FROM AUTHOR]

It is well known that there are two important classes of analytic functions of Ma-Minda type (MMT): Ma-Minda starlike and Ma-Minda convex functions. In this work, we suggest a new class of analytic functions, which is normalized in the open unit disk. The suggested class is generated by a roulette curve formula, which satisfies the symmetric behavior in the open unit disk. A roulette curve is shaped as the path outlined by the sum of two complex numbers, each affecting at a uniform rapidity in a circle. Special cases are illustrated involving special functions. Graphics of the curve are illustrated by using Mathematica 13.3. [ABSTRACT FROM AUTHOR]

In the current article, we utilize the concept of subordination to establish a new subclass of analytic functions associated with a bounded domain that is symmetric about the real axis. By applying the convolution technique, we derive the necessary and sufficient condition, the radius of convexity for this recently introduced class. Furthermore, we prove the sharp upper bounds for the second-order Hankel determinants | H 2 , 1 ξ | , | H 2 , 2 ξ | and third-order Hankel determinant | H 3 , 1 ξ | for the functions ξ belonging to the newly defined class. [ABSTRACT FROM AUTHOR]

Let S s * (e z) denote the class of starlike functions with respect to symmetric points subordinate to the exponential function, i.e., the functions which satisfy in the unit disk U the condition 2 z f ′ (z) f (z) − f (− z) ≺ e z (z ∈ U) . We obtained the sharp estimate of the second-order Hankel determinants H 2 , 3 (f) and improved the estimate of the third-order H 3 , 1 (f) for this functions class S s * (e z) . [ABSTRACT FROM AUTHOR]

A complex Layla and Majnun model system (CLMMS) is suggested in this study for a complex variable in the open-unit disk. Analytic solutions are discovered by using a technique of bounded turning functions. The set of necessary conditions is illustrated involving some special cases. [ABSTRACT FROM AUTHOR]

In 2012, new classes of analytic functions on U × U ¯ with coefficient holomorphic functions in U ¯ were defined to give a new approach to the concepts of strong differential subordination and strong differential superordination. Using those new classes, the extended Dziok–Srivastava operator is introduced in this paper and, applying fractional integral to the extended Dziok–Srivastava operator, we obtain a new operator D z − γ H m l α 1 , β 1 that was not previously studied using the new approach on strong differential subordinations and superordinations. In the present article, the fractional integral applied to the extended Dziok–Srivastava operator is investigated by applying means of strong differential subordination and superordination theory using the same new classes of analytic functions on U × U ¯. Several strong differential subordinations and superordinations concerning the operator D z − γ H m l α 1 , β 1 are established, and the best dominant and best subordinant are given for each strong differential subordination and strong differential superordination, respectively. This operator may have symmetric or asymmetric properties. [ABSTRACT FROM AUTHOR]

Fuzzy set theory, introduced by Zadeh, gives an adaptable and logical solution to the provocation of introducing, evaluating, and opposing numerous sustainability scenarios. The results described in this article use the fuzzy set concept embedded into the theories of differential subordination and superordination from the geometric function theory. In 2011, fuzzy differential subordination was defined as an extension of the classical notion of differential subordination, and in 2017, the dual concept of fuzzy differential superordination appeared. These dual notions are applied in this paper regarding the fractional integral applied to Dziok–Srivastava operator. New fuzzy differential subordinations are proved using known lemmas, and the fuzzy best dominants are established for the obtained fuzzy differential subordinations. Dual results regarding fuzzy differential superordinations are proved for which the fuzzy best subordinates are shown. These are the first results that link the fractional integral applied to Dziok–Srivastava operator to fuzzy theory. [ABSTRACT FROM AUTHOR]

Abstract The Lupaş q-transform has first appeared in the study of the Lupaş q-analogue of the Bernstein operator. Given 0 < q < 1 and f ∈ C [0, 1], the Lupaş q-transform is defined by During the last decades, this transform has been investigated from a variety of angles, including its analytical, geometric features, and properties of its block functions along with their sums. As opposed to the available studies dealing with a fixed value of q, the present work is focused on the injectivity of Λ q with respect to parameter q. More precisely, the conditions on f such that equality Λ q (f ; x) = Λ r (f ; x), x ⩾ 0 implies q = r have been established. [ABSTRACT FROM AUTHOR]

In this paper, we introduce some new subclasses of analytic functions in the open unit disk U with negative coefficients defined by generalized Alexander integral operator. The aim of the present paper is to determine coefficient inequalities, inclusion relations, neighborhoods, partial sums and integral means properties for functions f belonging to these subclasses. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to present classical second-order differential subordination knowledge extended in this study to include new results regarding third-order differential subordinations. The focus of this study is on the main problems examined by differential subordination theory. Hence, the new results obtained here reveal techniques for identifying dominants and the best dominant of certain third-order differential subordinations. Another aspect of novelty is the new application of the Gaussian hypergeometric function. Novel third-order differential subordination results are obtained using the best dominant provided by the theorems and the operator previously defined as Gaussian hypergeometric function's fractional integral. The research investigation is concluded by giving an example of how the results can be implemented. [ABSTRACT FROM AUTHOR]

The paper introduces a new family of analytic bi-univalent functions that are injective and possess analytic inverses, by employing a q-analogue of the derivative operator. Moreover, the article establishes the upper bounds of the Taylor–Maclaurin coefficients of these functions, which can aid in approximating the accuracy of approximations using a finite number of terms. The upper bounds are obtained by approximating analytic functions using Faber polynomial expansions. These bounds apply to both the initial few coefficients and all coefficients in the series, making them general and early, respectively. [ABSTRACT FROM AUTHOR]

A new subclass of bi-univalent functions associated with the Hohlov operator is introduced. Certain properties such as the coefficient bounds, Fekete-Szegö inequality and the second Hankel determinant for functions in the subclass are obtained. In particular, several known results are generalized. [ABSTRACT FROM AUTHOR]

This article comprises the study of strongly starlike functions which are defined by using the concept of subordination. The function φ defined by φ (ζ) = (1 + ζ) λ , 0 < λ < 1 maps the open unit disk in the complex plane to a domain symmetric with respect to the real axis in the right-half plane. Using this mapping, we obtain some radius results for a family of starlike functions. It is worth noting that all the presented results are sharp. [ABSTRACT FROM AUTHOR]

For the analytic functions ϕ (ζ) = ζ + ∑ k = n ∞ ϕ k ζ k in the unit disk O , we calculate the values of n and α , where the condition ℜ 1 + ζ ϕ ″ (ζ) / ϕ ′ (ζ) > − α or ℜ 1 + ζ ϕ ″ (ζ) / ϕ ′ (ζ) < 1 + α / 2 yields univalence and starlikeness. Conditions imply ϕ in the class where all normalized analytic functions U , with ζ / ϕ (ζ) 2 ϕ ′ (ζ) − 1 < 1 are obtained. Recent findings are gained, and unique cases are demonstrated. The generalization of the Jack lemma serves the proof of the main result and that our methodology is based on the idea of subordination. [ABSTRACT FROM AUTHOR]

Some well-known authors have extensively used orthogonal polynomials in the framework of geometric function theory. We are motivated by the previous research that has been conducted and, in this study, we solve the Fekete–Szegö problem as well as give bound estimates for the coefficients and an upper bound estimate for the second Hankel determinant for functions in the class  G Σ (v , σ)  of analytical and bi-univalent functions, implicating the Euler polynomials. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to introduce and study some new subclasses of p-valent functions by making use of a linear q-differential Borel operator.We also deduce some properties, such as inclusion relationships of the newly introduced classes and the integral operator J μ , p . [ABSTRACT FROM AUTHOR]

The problem of completely describing the approximation of the number by the elements of the sequence , , is considered. To this end, the function , which is analytic in the complex plane with a cut along the half-line of the real line, is studied in detail. We prove that the power series , where all are positive, which represents this function on the unit disk, envelops it in the open right half-plane. This gives a series of double inequalities for the deviation on the positive half-line, which are asymptotically sharp as . Integral representations of the function and of the coefficients are obtained. They play an important role in the study. A two-term asymptotics of the coeffients as is found. We show that the coefficients form a logarithmically convex completely monotone sequence. We also obtain integral expressions for the derivatives of all orders of the function . It turns out that is completely monotone on the half-line . Applications and development of the results are discussed. [ABSTRACT FROM AUTHOR]

Gegenbauer polynomials constitute a numerical tool that has attracted the interest of many function theorists in recent times mainly due to their real-life applications in many areas of the sciences and engineering. Their applications in geometric function theory (GFT) have also been considered by many researchers. In this paper, this powerful tool is associated with the prolific concepts of convolution and subordination. The main purpose of the research contained in this paper is to introduce and study a new subclass of analytic functions. This subclass is presented using an operator defined as the convolution of the generalized distribution and the error function and applying the principle of subordination. Investigations into this subclass are considered in connection to Carathéodory functions, the modified sigmoid function and Bell numbers to obtain coefficient estimates for the contained functions. [ABSTRACT FROM AUTHOR]

In this research, a novel linear operator involving the Borel distribution and Mittag-Leffler functions is introduced using Hadamard products or convolutions. This operator is utilized to develop new subfamilies of bi-univalent functions via the principle of subordination with Gegenbauer orthogonal polynomials. The investigation also focuses on the estimation of the coefficients | a ℓ | (ℓ = 2 , 3) and the Fekete–Szegö inequality for functions belonging to these subfamilies of bi-univalent functions. Several corollaries and implications of the findings are discussed. Overall, this study presents a new approach for constructing bi-univalent functions and provides valuable insights for further research in this area. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to study certain geometric properties (like univalence, starlikeness, convexity, close-to-convexity) for the normalized Miller-Ross function. The various results, which we have established in the present investigation, are believed to be new, and their importance is illustrated by several interesting consequences and examples. Furthermore, some of the main results improve the corresponding results available in the literature [15]. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to derive the sharp lower and upper bounds on Hermitian Toeplitz determinants for starlike and convex functions with respect to symmetric points. Some of the results provide improvements (or corrections) to several recent results. [ABSTRACT FROM AUTHOR]

Fractional differential operators have recently been linked with numerous other areas of science, technology and engineering studies. For a real variable, the class of fractional differential and integral operators is evaluated. In this study, we look into the Prabhakar fractional differential operator, which is the most applicable fractional differential operator in a complex domain. In terms of observing a group of normalized analytical functions, we express the operator. In the open unit disc, we deal with its geometric performance. Applying the Prabhakar fractional differential operator d c θ α , β γ , ω to a subclass of analytic univalent function results in the creation of a new subclass of mathematical functions: W ( γ , ω , α , β , θ ,m,c,z,p,q). We obtain the characteristic, neighborhood and convolution properties for this class. Some of these properties are extensions of defined results. [ABSTRACT FROM AUTHOR]