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The aim of this study is to present novel collections of bi-univalent functions, which are characterized using the Bell Distribution. These collections are delineated through the application of Jacobi polynomials. We have established bounds for the Taylor-Maclaurin coefficients, particularly |a2| and |a3|. Additionally, we have investigated the Fekete-Szegö functional issues pertinent to functions within these subclasses. By concentrating on particular parameters in our principal findings, we have identified numerous new insights. [ABSTRACT FROM AUTHOR]

This study defines three novel classes of bi‐univalent functions connected to generalized telephone numbers for the first time. We produced assessments about the Taylor–Maclaurin coefficients |a2| and |a3| and Fekete–Szegö functional problems for functions involving these novel subclasses for functions in every one regarding these three bi‐univalent function classes. Additional findings are anticipated to emerge after expertise the factors included in our significant outcomes, which have not yet been investigated by subordinating with generalized telephone numbers. [ABSTRACT FROM AUTHOR]

In this investigation, we present a new collection of analytic functions that includes Touchard polynomials. We then aim to calculate the Maclaurin coefficients |a2| and |a3| and address the Fekete-Szegö functional problem within this specific subfamily. Additionally, we demonstrate several new outcomes by specifying the parameters used in our main findings. [ABSTRACT FROM AUTHOR]

Making use of the definition of subordination, we introduce certain subclasses of spiral-like and Robertson functions in the open unit disk and study some important results such as convolution results, coefficients estimate, subordination properties and Fekete-Szego problems for these subclasses. Further, some known and new outcomes which follow as special cases of our outcomes are also mentioned. [ABSTRACT FROM AUTHOR]

The study of q-calculus is becoming increasingly prominent in the field of geometric function theory, reflecting a growing interest in its applications. In this research work, we first develop a new type of modified Sigmoid-Salagean q-differential operator in the open unit disk D, utilizing the concepts of quantum calculus and the Sigmoid activation function. Using this newly defined q-analogous differential operator and Horadam polynomials, we introduce new subclasses of bi-univalent functions in D. We determine upper bounds on initial coefficients, as well as the Fekete-Szegö problems, for functions belonging to these special families. Additionally, we discuss several interesting consequences related to the findings presented in this study. [ABSTRACT FROM AUTHOR]

In this paper, we present a novel class of Yamakawa-type bi-starlike functions. These functions are defined using Gegenbauer polynomials associated with q-calculus. We have derived estimates for the Maclaurin coefficients | a2 | and | a3 | for functions in the Yamakawa-type bi-starlike function subclass. Additionally, we have solved the Fekete-Szegö problems for functions in this new subclass. By specializing the parameters in our main results, we have obtained several new findings. [ABSTRACT FROM AUTHOR]

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The aim of this article is to establish two new and qualitative subfamilies F(ε, κ, ℵ) and G(ε, κ, ℵ) of biunivalent functions. For functions in these subfamilies, we determine the first two Maclaurin coefficient estimations |C2| and |C3|, and address the Fekete-Szegö problem. Additionally, we mention some corollaries related to the main results. [ABSTRACT FROM AUTHOR]

Our research introduces new subclasses of analytical functions that are defined by Euler polynomials. We then proceed to estimate the Fekete-Szegö functional problem and the Maclaurin coefficients for this specific subfamily, denoted as |a2| and |a3|. Furthermore, we demonstrate several new results that emerge when we specialize the parameters used in our main findings. [ABSTRACT FROM AUTHOR]

Bi-univalent functions associated with the leaf-like domain within the open unit disk are investigated and a new subclass is introduced and studied in the research presented here. This is achieved by applying the subordination principle for analytic functions in conjunction with q -calculus. The class is proved to be not empty. By proving its existence, generalizations can be given to other sets of functions. In addition, coefficient bounds are examined with a particular focus on | α 2 | and | α 3 | coefficients, and Fekete–Szegö inequalities are estimated for the functions in this new class. To support the conclusions, previous works are cited for confirmation. [ABSTRACT FROM AUTHOR]

In this paper we present new classes of univalent functions, or more generally, a new univalence criteria for normalized analytic functions, using an operator defined concerning to the idea behind a Marx–Strohhäcker type theorem. Also, we determine coefficient estimates and study Fekete–Szegö problem for these classes. [ABSTRACT FROM AUTHOR]

Bi-univalent functions associated with the leaf-like domain within open unit disks are investigated, and a new subclass is introduced and studied in the research presented here. This is achieved by applying the subordination principle for analytic functions in conjunction with q -calculus. The class is proved to not be empty. By proving its existence, generalizations can be given to other sets of functions. In addition, coefficient bounds are examined with a particular focus on | α 2 | and | α 3 | coefficients, and Fekete–Szegö inequalities are estimated for the functions in this new class. To support the conclusions, previous works are cited for confirmation. [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]

The aim of this work is to introduce two families, BΣ(P;ϑ) and OΣ(X;ϑ), of holomorphic and bi-univalent functions involving the Bazilevič functions and the Ozaki-close-to-convex functions, by using generalized telephone numbers. We determinate upper bounds on the Fekete-Szegö type inequalities and the initial Taylor-Maclaurin coefficients for functions in these families. We also highlight certain edge cases and implications for our findings. [ABSTRACT FROM AUTHOR]

In the current work, we alter the Opoola derivative operator using quantum calculus. We examine a novel specific subfamily of analytical functions with complex order connected to the subordination principal by using the modified q -Opoola derivative operator. Next, we examine the Fekete–Szegö inequality for the generated subclass in the open unit disk. This paper sheds light the significant connections between the results introduced in this work and previous results. [ABSTRACT FROM AUTHOR]

The aim of this work is to introduce two families, $ \mathcal{B}_{\Sigma}(\wp; \vartheta) $ and $ \mathcal{O}_{\Sigma}(\varkappa; \vartheta) $, of holomorphic and bi-univalent functions involving the Bazilevič functions and the Ozaki-close-to-convex functions, by using generalized telephone numbers. We determinate upper bounds on the Fekete-Szegö type inequalities and the initial Taylor-Maclaurin coefficients for functions in these families. We also highlight certain edge cases and implications for our findings.

The Mittag-Leffler-type Borel distribution is widely recognized and utilized as a beneficial and pertinent model across numerous applications. This study presents a new subclass of normalized analytic bi-univalent functions that combines Gegenbauer polynomials and the Mittag-Leffler-type Borel distribution. Employing this subclass enables us to derive novel approximations for Taylor-Maclaurin coefficients, |a2| and |a3|, as well as delve into the investigation of the Fekete-Szegö functional. Additionally, we explore a variety of new findings that arise through the specialization of parameters in our primary results.

In this study, we introduce a new class of normalized analytic and bi-univalent functions denoted by D Σ (δ , η , λ , t , r) . These functions are connected to the Bazilevič functions and the λ -pseudo-starlike functions. We employ Sakaguchi Type Functions and Horadam polynomials in our survey. We establish the Fekete-Szegö inequality for the functions in D Σ (δ , η , λ , t , r) and derive upper bounds for the initial Taylor–Maclaurin coefficients | a 2 | and | a 3 | . Additionally, we establish connections between our results and previous research papers on this topic. [ABSTRACT FROM AUTHOR]

Let $ \hat {\mathcal {S}}_{(\alpha,\beta)} $ S ^ (α , β) be the familiar class of almost spirallike functions of type β and order α in the unit disk (see Definition 1.1). In this paper, first, we prove that for a function $ f(z)=z+\sum _{n=2}^\infty a_nz^n $ f (z) = z + ∑ n = 2 ∞ a n z n in the class $ \hat {\mathcal {S}}_{(\alpha,\beta)} $ S ^ (α , β) , then $$\begin{align*} & |a_{3}-\lambda a_{2}^2|\\ & \quad \leq (1-\alpha)\cos{\beta}\max\{1,|1-4(1-\alpha)(1-\lambda)\cos{\beta}\,{\rm e}^{{\rm i}\beta}|\},\quad \lambda\in \mathbb{C}. \end{align*} $$ | a 3 − λ a 2 2 | ≤ (1 − α) cos ⁡ β max { 1 , | 1 − 4 (1 − α) (1 − λ) cos ⁡ β e i β | } , λ ∈ C. The above estimation is sharp. Second, we extend this result to the bounded starlike circular domain in $ \mathbb {C}^{n} $ C n and obtain the sharp estimates. The results presented here would provide extensions of those given by Xu et al. [The Fekete and Szegö problem on the bounded starlike circular domain in $ \mathbb {C}^n $ C n . Pure Appl Math Q. 2016;12:621–638] and Xiong [Sharp coefficients bounds for class of almost starlike mappings of order α in $ \mathbb {C}^n $ C n . J Math Inequalities. 2020;14:853–865]. Finally, a certain conjecture is also formulated. [ABSTRACT FROM AUTHOR]

In the present article, we introduce two families ΒΣ(δ,α,λ,m,n,t) and FΣ(δ,λ,γ,m,n,t) of holomorphic and bi-univalent functions associating λ-pseudo-starlike and convex functions with Sakaguchi type functions defined by Gegenbauer polynomials. We derive the initial Maclaurin coefficients estimates and determinate the Fekete-Szegö problem of functions in these families. [ABSTRACT FROM AUTHOR]

The paper is devoted to two problems pivotal in the generation theory. The first concerns verifying whether a normalized holomorphic function is an infinitesimal generator. It is also important to trace the influence of properties of a generator on the dynamic behavior of the generated semigroup. Recently, these problems have been studied using the so-called filtrations of generators. In this paper we study three filtrations. The study of the first began earlier, the second is a new one, while the third consists of functions studied previously with no connection to complex dynamics. For all filtration classes studied in the paper, dynamic properties of the corresponding semigroups are established and estimates for the Fekete-Szegö functional are given. [ABSTRACT FROM AUTHOR]

In this paper we establish upper bounds for the second and third coefficients of holomorphic and bi-univalent functions in a new family which involve the Bazilevič functions and β-pseudo-starlike functions under a new operator joining Poisson distribution with Ruscheweyh derivative operator. Also, we discuss Fekete-Szegö problem of functions in this family. [ABSTRACT FROM AUTHOR]

Let $ \hat {\mathcal {S}}_{(\alpha,\beta)} $ S ^ (α , β) be the familiar class of almost spirallike functions of type β and order α in the unit disk (see Definition 1.1). In this paper, first, we prove that for a function $ f(z)=z+\sum _{n=2}^\infty a_nz^n $ f (z) = z + ∑ n = 2 ∞ a n z n in the class $ \hat {\mathcal {S}}_{(\alpha,\beta)} $ S ^ (α , β) , then $$\begin{align*} & |a_{3}-\lambda a_{2}^2|\\ & \quad \leq (1-\alpha)\cos{\beta}\max\{1,|1-4(1-\alpha)(1-\lambda)\cos{\beta}\,{\rm e}^{{\rm i}\beta}|\},\quad \lambda\in \mathbb{C}. \end{align*} $$ | a 3 − λ a 2 2 | ≤ (1 − α) cos ⁡ β max { 1 , | 1 − 4 (1 − α) (1 − λ) cos ⁡ β e i β | } , λ ∈ C. The above estimation is sharp. Second, we extend this result to the bounded starlike circular domain in $ \mathbb {C}^{n} $ C n and obtain the sharp estimates. The results presented here would provide extensions of those given by Xu et al. [The Fekete and Szegö problem on the bounded starlike circular domain in $ \mathbb {C}^n $ C n . Pure Appl Math Q. 2016;12:621–638] and Xiong [Sharp coefficients bounds for class of almost starlike mappings of order α in $ \mathbb {C}^n $ C n . J Math Inequalities. 2020;14:853–865]. Finally, a certain conjecture is also formulated. [ABSTRACT FROM AUTHOR]

Considering the interesting results obtained recently by studying Rabotnov function, a new investigation is presented in this paper related to the topic of introducing new classes of bi-univalent functions. Using the normalized Rabotnov function and the concept of subordination, a new class of bi-univalent functions ℋ Σ (α , β , δ ; ψ) is defined and studied regarding coefficient estimates. Bounds of the Taylor–Maclaurin coefficients a 2 and a 3 are given and using those estimates, Fekete–Szegö problem is also investigated for the newly introduced class. For particular values of the parameters involved in the definition of the class ℋ Σ (α , β , δ ; ψ) , certain particular classes are obtained and investigated in the corollaries associated to each result proved for the class ℋ Σ (α , β , δ ; ψ). [ABSTRACT FROM AUTHOR]

The purpose of this article is to introduce and study certain families of normalized certain functions with symmetric points connected to Gegenbauer polynomials. Moreover, we determine the upper bounds for the initial Taylor–Maclaurin coefficients | a 2 | and | a 3 | and resolve the Fekete–Szegöproblem for these functions. In addition, we establish links to a few of the earlier discovered outcomes. [ABSTRACT FROM AUTHOR]

This research article introduces a novel operator termed q-convolution, strategically integrated with foundational principles of q-calculus. Leveraging this innovative operator alongside q-Bernoulli polynomials, a distinctive class of functions emerges, characterized by both analyticity and bi-univalence. The determination of initial coefficients within the Taylor-Maclaurin series for this function class is accomplished, showcasing precise bounds. Additionally, explicit computation of the second Hankel determinant is provided. These pivotal findings, accompanied by their corollaries and implications, not only enrich but also extend previously published results. [ABSTRACT FROM AUTHOR]

Over the last few years, Geometric Function Theory (GFT) as one of the most prime branch of complex analysis has gained a considerable and an impressive attention from many researchers, largely because it deals with the study of the geometric properties of analytic functions and their numerous applications in various fields of mathematics such as in special functions, probability distributions, and fractional calculus. The investigations in this paper are on two new classes of analytic functions defined in the unit disk ε = {z ∈ C : |z| < 1} and denoted by XSq (b,K) and XTq (b,K). Function f in the classes satisfy the conditions f(0) = f ′(0) -- 1 = 0, hence can be of series type f(z) = z + a2z² + a3z³ + …, z ∈ ε. The definition of the two new classes of analytic functions embed some well-known special functions such as the Galuê-type Struve function, modified error function and a starlike function whose coefficients are Bell's numbers while some involving mathematical principles are the q-derivative, inequalities, convolution and subordination. The main results from these classes are however, the upper estimates of some initial bounds such as |an| (n = 2, 3, 4) and the Fekete-Szegö functional |a3 - øa2/2| (ø ∈ C) of functions f ∈ XSq (b,K) and f ∈ XTq (b,K). [ABSTRACT FROM AUTHOR]

In this paper, making use of certain linear operators, we consider Fekete-Szegö problem for some subclasses of analytic functions. [ABSTRACT FROM AUTHOR]

Abstract. In this article, we introduce and investigate a new family of analytic and bi-prestarlike functions by using the Horadam polynomials defined in the open unit disk U. We determine upper bounds for the first two coefficients /a2/ and /a3/ and solve Fekete-Szegő problem of functions that belong to this family. Also, we point out several certain special cases for our results. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce three new classes of bi-univalent functions defined by means of Gegenbauer polynomials. For functions in each of these three bi-univalent function classes, we have derived the estimates of the Taylor–Maclaurin coefficients a 2 and a 3 and Fekete–Szegö functional problems for functions belonging to these new subclasses. A number of new results are shown to follow upon specializing the parameters involved in our main results. [ABSTRACT FROM AUTHOR]

One of the important problems regarding coefficients of analytical functions (i.e., Fekete–Szegö inequality) was raised by Fekete and Szegö in 1933. The results of this research are dedicated to determine upper coefficient estimates and the Fekete–Szegö problem in the class W Σ (δ , λ ; ϑ) , which is defined by generalized telephone numbers. We also indicate some specific conditions and consequences of results found by us. [ABSTRACT FROM AUTHOR]

We consider the class of λ -pseudo starlike functions f (z) = z + a 2 z 2 + a 3 z 3 + ⋯ such that z (f ′ (z)) λ / f (z) maps the open unit disk | z | < 1 onto a strip domain w with α − π 2 sin α < ℜ (w (z)) < α 2 sin α for some α , π / 2 ≤ α < π. We estimate | a 2 | , | a 3 | and solve the Fekete–Szegö problem for functions in this class. [ABSTRACT FROM AUTHOR]

We introduce a new subclass J Σ η , δ , μ (p ˜) of bi-univalent functions, defined by shell-like curves connected with Fibonacci numbers. Our main results in this paper include estimates of the Taylor–Maclaurin coefficients a 2 and a 3 for functions in this subclass, as well as solutions to Fekete–Szegö functional problems. We also show novel outcomes resulting from the specialization of the parameters used in our main results. [ABSTRACT FROM AUTHOR]

In this article we introduce three new subclasses of the class of bi-univalent functions Σ , namely HG Σ , GM Σ (μ) and G Σ (λ) , by using the subordinations with the functions whose coefficients are Gregory numbers. First, we evidence that these classes are not empty, i.e., they contain other functions besides the identity one. For functions in each of these three bi-univalent function classes, we investigate the estimates a 2 and a 3 of the Taylor–Maclaurin coefficients and Fekete–Szegő functional problems. The main results are followed by some particular cases, and the novelty of the characterizations and the proofs may lead to further studies of such types of similarly defined subclasses of analytic bi-univalent functions. [ABSTRACT FROM AUTHOR]

This paper is concerned with sharp solutions to majorization and Fekete-Szegő problems for several relatively new normalized subfamilies of starlike functions. Regarding starlike functions which are subordinated to Sine and Cosine functions, a number of new results are established. The results provide improvements to some recent results. As applications, some interesting deductions are also listed. The findings of this study are thoroughly supported by examples wherever necessary. [ABSTRACT FROM AUTHOR]

In the first part of this paper, we will give the Fekete–Szegö inequality for various subfamilies of spirallike mappings of type β on the unit ball of a complex Banach space. Our results give extensions of those given by Lai and Xu (Results Math 76(4), Paper No. 191, 2021) and Elin and Jacobzon (Results Math 77(3), Paper No. 137, 2022). We next give the Fekete–Szegö inequality for close-to-quasi-convex mappings of type B on the unit ball of a complex Banach space. Our results give extensions of that given by Xu et al. (Complex Var Elliptic Equ. ). [ABSTRACT FROM AUTHOR]

In the present paper, we obtain the upper bounds for the second Hankel determinant and Fekete-Szegö inequalities for a new subclass of analytic functions in the open unit disk defined by the hypergeometric functions. Moreover, several interesting applications of the results presented here are also discussed. [ABSTRACT FROM AUTHOR]

In current manuscript, using Laguerre polynomials and (p − q) -Wanas operator, we identify upper bounds a 2 and a 3 which are first two Taylor-Maclaurin coefficients for a specific bi-univalent functions classes W ∑ (η , δ , λ , σ , θ , α , β , p , q ; h) and K ∑ (ξ , ρ , σ , θ , α , β , p , q ; h) which cover the convex and starlike functions. Also, we discuss Fekete-Szegö type inequality for defined class. [ABSTRACT FROM AUTHOR]

In the real world, there are many applications that find the Pascal distribution to be a useful and relevant model. One of these is the normal distribution. In this work, we develop a new subclass of analytic bi-univalent functions by making use of the q-Pascal distribution series as a construction. These functions involve the q-Gegenbauer polynomials, and we use them to establish our new subclass. Moreover, we solve the Fekete–Szegö functional problem and analyze various different estimates of the Maclaurin coefficients for functions that belong to the new subclass. [ABSTRACT FROM AUTHOR]

In the real world there are many applications that find the Bell distribution to be a useful and relevant model. One of these is the normal distribution. In this paper, we develop a new subclass of analytic bi-univalent functions by making use of the Bell distribution as a building block. These functions involve the Gegenbauer polynomials, and we use them to establish our new subclass. In this study, we solve the Fekete–Szegö functional problem and analyse various different estimates of the Maclaurin coefficients D 2 and D 3 for functions that belong to the built class. [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]

Several different subclasses of the bi-univalent function class Σ were introduced and studied by many authors using distribution series like Pascal distribution, Poisson distribution, Borel distribution, the Mittag-Leffler-type Borel distribution, Miller–Ross-Type Poisson Distribution. In the present paper, by making use of the Bell distribution, we introduce and investigate a new family G Σ t (x , p , q , λ , β , γ) of normalized bi-univalent functions in the open unit disk U , which are associated with the Horadam polynomials and estimate the second and the third coefficients in the Taylor-Maclaurin expansions of functions belonging to this class. Furthermore, we establish the Fekete–Szegö inequality for functions in the family G Σ t (x , p , q , λ , β , γ). After specializing the parameters used in our main results, a number of new results are demonstrated to follow. [ABSTRACT FROM AUTHOR]

The main goal of this investigation is to obtain sharp upper bounds for Fekete-Szegö functional and the third Hankel determinant for a certain subclass SL ∗ u , v , α of holomorphic functions defined by the Carlson-Shaffer operator in the unit disk. Finally, for some special values of parameters, several corollaries were presented. [ABSTRACT FROM AUTHOR]

For the first time, we attempted to define two new sub-classes of bi-univalent functions in the open unit disc of the complex order involving Mathieu-type series, associated with generalized telephone numbers. The initial coefficients of functions in these classes were obtained. Moreover, we also determined the Fekete–Szegö inequalities for function in these and several related corollaries. [ABSTRACT FROM AUTHOR]

In the current study, we provide a novel qualitative new subclass of analytical and bi-univalent functions in the symmetry domain U defined by the use of Gegenbauer polynomials. We derive estimates for the Fekete–Szegö functional problems and the Taylor–Maclaurin coefficients a 2 and a 3 for the functions that belong to each of these new subclasses of the bi-univalent function classes. Some more results are revealed after concentrating on the parameters employed in our main results. [ABSTRACT FROM AUTHOR]

In this article, we introduce and investigate two new subclasses of λ -pseudo bi-univalent functions with respect to symmetric points in the open unit disk U defined by means of Gegenbauer polynomial. Coefficient bounds and Fekete–Szegö inequalities for functions belong to these subclasses are obtained. Furthermore, several new special cases upon specializing the parameters involved of our results are pointed out which are new and not discussed sofar. [ABSTRACT FROM AUTHOR]