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In the present work, the authors introduce a new linear differential operator associated with $ q $-calculus which uses the $ q $-derivative operator. For the new class involving the linear differential operator, interesting properties like coefficient inequality, growth, distortion, convolution and radii of starlikeness are established. Various connections of the present class with that of the earlier works are pointed out. [ABSTRACT FROM AUTHOR]

A fractal–fractional calculus is presented in term of a generalized gamma function (ℓ−gamma function: Γ ℓ (.) ). The suggested operators are given in the symmetric complex domain (the open unit disk). A novel arrangement of the operators shows the normalization associated with every operator. We investigate a number of significant geometric features thanks to this. Additionally, some integrals, such the Alexander and Libra integral operators, are associated with these operators. Simple power functions are among the illustrations that are provided. Additionally, the formulation of the discrete ℓ − fractal–fractional operators is conducted. We demonstrate that well-known examples are involved in the extended operators. [ABSTRACT FROM AUTHOR]

This paper investigates the third Hankel determinant, denoted H 3 (1) , for functions within the subclass R S ∑ * (λ) of bi-univalent functions associated with crescent-shaped regions φ ⦅ z = z + 1 + z 2 . The primary aim of this study is to establish upper bounds for H 3 (1) . By analyzing functions within this specific geometric context, we derive precise constraints on the determinant, thereby enhancing our understanding of its behavior. Our results and examples provide valuable insights into the properties of bi-univalent functions in crescent-shaped domains and contribute to the broader theory of analytic functions. [ABSTRACT FROM AUTHOR]

The q-Durrmeyer polynomials are one of the popular q-versions of the classical operators of approximation theory. They have been studied from different points of view by a number of researchers. The aim of this work is to estimate the rate of convergence for the sequence of the q-Durrmeyer polynomials in the case 0 < q < 1. It is proved that for any compact set 퓓 ⊂ ℂ, the rate of convergence is O(qn) as n → ∞. The sharpness of the obtained result is demonstrated. [ABSTRACT FROM AUTHOR]

In this article, we study the growth of solutions of the homogeneous complex linear differential equation f(k) + Ak-1(z)f(k-1) + ··· + A1(z)f' + A0(z)f = 0, where the coefficients Aj(z) (j = 0, 1,..., k-1) are analytic or meromorphic functions in C\{z0}. Under the sufficient condition that there exists one dominant coefficient by its logarithmic lower order or by its logarithmic lower type, we extend some precedent results due to Liu, Long, and Zeng and others. [ABSTRACT FROM AUTHOR]

Our aim was to develop a new class of bi starlike functions by utilizing the concept of subordination, driven by the idea of multiplicative calculus, specifically multiplicative derivatives. Several restrictions were imposed, which were indeed strict constraints, because we have tried to work within the current framework or the design of analytic functions. To make the study more versatile, we redefined our new class of function with Miller-Ross Poisson distribution (MRPD), in order to increase the study's adaptability. We derived the first coefficient estimates and Fekete-Szegő inequalities for functions in this new class. To demonstrate the characteristics, we have provided a few examples. [ABSTRACT FROM AUTHOR]

A special function is a function with a particular use in mathematical physics or another branch of mathematics and is often named after an early scientist who researched its characteristics. A few noteworthy instances exist, such as the hypergeometric function and its distinct species. By using k-calculus, this sort of special function is made more generic. K-symbol calculus is utilized in this study to develop the k-convoluted operators associated with the k-Whittaker function (conuent hypergeometric function of the first kind). Through the use of this recently created operator, we propose a new geometric formula of normalized functions in the unit disk. Our strategy is to modify the theory of differential subordination, thus we geometrically investigate the most well-known characteristics of this new operator, including subordination features and coefficient bounds. We draw attention to some notable corollaries of our main conclusions as exceptional examples. [ABSTRACT FROM AUTHOR]

In 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]

A sequence of n trials from a finite population with no replacement is described by the hypergeometric distribution as the number of successes. Calculating the likelihood that factory-produced items would be defective is one of the most popular uses of the hypergeometric distribution in industrial quality control. Very recently, several researchers have applied this distribution on certain families of analytic functions. In this study, we provide certain adequate criteria for the generalized hypergeometric distribution series to be in two families of analytic functions defined in the open unit disk. Furthermore, we consider an integral operator for the hypergeometric distribution. Some corollaries will be implied from our main results. [ABSTRACT FROM AUTHOR]

The objective of this paper is to present some geometric properties of the close to convex function f when f is an analytic, univalent self-conformal mapping defined in the open unit disk D={z∈C: |z|<1},and on the boundary of D. One of the goals of this work was determining the sharp bound for the function of form f(z)=z/(1-z)2δ, 0<δ<1, when R(f¹/g¹ (w_1))≥-2+δ/w_1, for a some w_1∈∂D And another, if f has angular limit at p∈∂D . Then the inequality |(f¹ (z))/(g¹ (z) )|≥(-(1-2δ))/2(1+ δ); is sharp with extremal functions f(z)=z/(1-z)2δ;, and g(z)=z/(1+z), where 0<δ<1. Finally, if f is extended continuously to the boundary of D, then | (f¹/g¹ )(p)|≥;|1-2δ|(|1-2δ|-2)/|p-δ |; 0<δ<1. [ABSTRACT FROM AUTHOR]

In this paper, we defined a new family of meromorphic functions whose analytic characterization was motivated by the definition of the multiplicative derivative. Replacing the ordinary derivative with a multiplicative derivative in the subclass of starlike meromorphic functions made the class redundant; thus, major deviation or adaptation was required in defining a class of meromorphic functions influenced by the multiplicative derivative. In addition, we redefined the subclass of meromorphic functions analogous to the class of the functions with respect to symmetric points. Initial coefficient estimates and Fekete–Szegö inequalities were obtained for the defined function classes. Some examples along with graphs have been used to establish the inclusion and closure properties. [ABSTRACT FROM AUTHOR]

In our current study, we apply differential subordination and quantum calculus to introduce and investigate a new class of analytic functions associated with the q-differential operator and the symmetric balloon-shaped domain. We obtain sharp results concerning the Maclaurin coefficients the second and third-order Hankel determinants, the Zalcman conjecture, and its generalized conjecture for this newly defined class of q-starlike functions with respect to symmetric points. [ABSTRACT FROM AUTHOR]

The manuscript is an attempt to consider all methods which are applicable to investigation a directional index for composition of an analytic function in some domain and an entire function. The approaches are applied to find sufficient conditions of the L-index boundedness in a direction b 2 Cn n f0g, where the continuous function L satisfies some growth condition and the condition of positivity in the unit polydisc. The investigation is based on a counterpart of the Hayman Theorem for the class of analytic functions in the polydisc and a counterpart of logarithmic criterion describing local conduct of logarithmic derivative modulus outside some neighborhoods of zeros. The established results are new advances for the functions analytic in the polydisc and in multidimensional value distribution theory. [ABSTRACT FROM AUTHOR]

To obtain the main result of the present paper, we use the technique of differential superordination. As special cases of our main result, we obtain sufficient conditions for f 2 A to be Φ-like, parabolic Φ-like, starlike, parabolic starlike, close-to-convex and uniform close-to-convex. We also obtain sandwich-type results regarding these functions. For demonstration of the results, we have plotted the images of open unit disk under certain functions using Mathematica 7.0. [ABSTRACT FROM AUTHOR]

The Bessel function and its various generalizations have extensively been studied in various branches of applied mathematics and theoretical physics, including the Geometric Function Theory. In this paper, we study basic characteristics of Bessel functions of order μ and degree ν . Among the results that we investigate are the results giving the characteristic properties of univalence, convexity and starlikeness. We further investigate the conditions under which the function L μ , ν are strongly convex and strongly starlike. Several corollaries are also mentioned depicting the usefulness of the main results, one of the Corollary providing improvement in a result for normalized Bessel function. [ABSTRACT FROM AUTHOR]

Our aim was to develop a new class of bi starlike functions by utilizing the concept of subordination, driven by the idea of multiplicative calculus, specifically multiplicative derivatives. Several restrictions were imposed, which were indeed strict constraints, because we have tried to work within the current framework or the design of analytic functions. To make the study more versatile, we redefined our new class of function with Miller-Ross Poisson distribution (MRPD), in order to increase the study's adaptability. We derived the first coefficient estimates and Fekete-Szegő inequalities for functions in this new class. To demonstrate the characteristics, we have provided a few examples.

Special functions and special polynomials have been used and studied widely in the context of Geometric function theory of Complex Analysis by the many authors. Here, in our present investigations, making use of the convolution, we first introduce a new subclass of Bazilevivč bi-univalent functions involving the Bernoulli Polynomials. We then find the first two Taylor-Maclaurin coefficient [Formula: see text] and [Formula: see text] for our defined functions class. We then calculate the bounds for the Fekete Szegö inequality for the defined function class. Also some known consequences of our main results are highlighted in the form of Corollaries.

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In the present paper with the aid of subordination, the authors introduce two subclasses of analytic functions denoted by S α , β (λ) (α , β , λ ∈ R , α < 1 , β > 1 , λ ≥ 0) and G (λ) defined in the open unit disk D : = { z ∈ C : | z | < 1 } . These subclasses are defined through a certain univalent function S α , β and the generating function of the Gregory coefficients G (λ) . We determine upper bounds of the initial coefficients, Fekete–Szeg o ¨ functional, Hankel determinant of second order, logarithmic coefficients and inverse coefficients of the functions belongs to these subclasses. Some of the corollaries of the main results are also pointed out. [ABSTRACT FROM AUTHOR]

In this paper, we derive several fuzzy differential subordination and fuzzy differential superordination results for analytic functions $ \mathcal{M}_{\xi, \beta}^{s, \gamma} $, which involve the extended Mittag-Leffler function and the Pascal distribution series. We also investigate and introduce a class $ \mathcal{MB}_{\xi, \beta}^{F, s, \gamma}(\rho) $ of analytic and univalent functions in the open unit disc $ \mathcal{D} $ by employing the newly defined operator $ \mathcal{M}_{\xi, \beta}^{s, \gamma} $. We determine a specific relationship of inclusion for this class. Further, we establish prerequisites for a function role in serving as both the fuzzy dominant and fuzzy subordinant of the fuzzy differential subordination and superordination, respectively. Some novel results that are sandwich-type can be found here.

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]

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 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]

In this paper, we derive several fuzzy differential subordination and fuzzy differential superordination results for analytic functions Ms,γξ,β, which involve the extended Mittag-Leffler function and the Pascal distribution series. We also investigate and introduce a class MBF,s,γξ,β (ρ) of analytic and univalent functions in the open unit disc D by employing the newly defined operator Ms,γξ,β. We determine a specific relationship of inclusion for this class. Further, we establish prerequisites for a function role in serving as both the fuzzy dominant and fuzzy subordinant of the fuzzy differential subordination and superordination, respectively. Some novel results that are sandwich-type can be found here. [ABSTRACT FROM AUTHOR]

This paper introduces a new subclass of analytic functions employing the operator that was recently defined by the authors. The coefficients estimate |as| (s = 2, 3) of the Taylor-Maclaurin series in this new class, as well as the Fekete-Szegö functional problems, have been derived. Furthermore, we obtained the sharp upper bound for the functional |a2a4 -- a3²| for functions belonging to this new subclass. [ABSTRACT FROM AUTHOR]

This paper contains the solution to the problem of a semi-infinite moving crack situated in an orthotropic strip bonded between two identical strips. The crack moves with a constant velocity, and the surface is under shear wave disturbance. We first examine the equations of elasticity, which include equilibrium equations and stress and displacement constitutive relations with the model-specific continuity and boundary conditions. Using the Fourier integral transform, the standard form for the Wiener–Hopf (W-H) equation is obtained, which is solved using the W-H method. The analytical expressions for the considered crack problem have been obtained for stress intensity factor (SIF), normalized stress intensity factor (NSIF), and stress magnification factor (SMF). The behavior of NSIF has been graphically presented for particular cases of composite materials for different crack propagation velocities and various depth ratios of the strips. The novelty of this paper lies in the analytic solutions using the W-H method for the semi-infinite moving crack problem under the influence of anti-plane shear waves. The pictorial presentations of normalized SIF clearly show their dependency on strip depths, crack propagation velocities and elastic constants. [ABSTRACT FROM AUTHOR]

Using the results of 3rd order differential subordination, we introduce certain families of admissible functions and discuss some applications of 3rd order differential subordination for normalized analytic functions associated with novel fractional operator namely Zeta-Riemann Fractional differential operator. Some new results on differential subordination and superordination with some sandwich theorems are obtained. Moreover, several particular cases are also noted. [ABSTRACT FROM AUTHOR]

The results of this work belong to the field of geometric function theory, being based on differential subordination methods. Using the idea of the q-calculus operators, we define the q-analogue of the multiplier- Ruscheweyh operator of a specific family of linear operators, Isq,µ(λ,l). Our major goal is to build and investigate some analytic function subclasses using Isq,µ(λ, l). Also, some differential subordination and superordination results are obtained. Moreover, based on the new theoretical results, several examples are constructed. For every differential superordination under investigation, the best subordinant is provided. [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]

In this paper, we investigate analytic solutions of higher order linear non-homogeneous directional differential equations whose coefficients are analytic functions in the unit ball. We use methods of theory of analytic functions in the unit ball having bounded L-index in direction, where L : Bn → R+ is a continuous function such that L(z) > β|b| /1−|z| for all z ∈ Bn, b ∈ Cn \ {0} be a fixed direction, β > 1 is some constant. Our proofs are based on application of inequalities from analog of Hayman’s theorem for analytic functions in the unit ball. There are presented growth estimates of their solutions which contains parameters depending on the coefficients of the equations. Also we obtained sufficient conditions that every analytic solution of the equation has bounded L-index in the direction. The deduced results are also new in one-dimensional case, i.e. for functions analytic in the unit disc. [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 consider the Banach space of analytic functions in a given bounded convex domain of a multidimensional complex space with a restriction on derivatives for each logarithmically convex sequence. On the union of such Banach spaces we define the linear topology of the canonical inductive limit and prove the completeness of the system of all complex exponentials. The main result is a description of the space of Laplace transforms of linear continuous functionals in the algebraic sense under certain conditions on sequences and their family. In the resulting weight space, we define the topology of the projective limit so that the Laplace transform establishes an isomorphism between the strongly conjugate space to the original space and the resulting projective limit of Banach spaces of entire functions. Previously, similar problems were considered for functions of a single variable. This work is the final part of the researches of I.Kh. Musin, R.S. Yulmukhametov, and the author for functions of many variables. [ABSTRACT FROM AUTHOR]

The paper considers a family of Banach spaces of functions analytic in a bounded convex domain. In the first paragraph, a continuously parameterized family including the classical Smirnov and Bergman spaces is constructed. In the second section, based on this family, we construct a normal scale of Banach spaces, which also includes Smirnov and Bergman spaces. The authors suggest that the constructed scale will be an effective tool in the study of problems of approximation of analytical functions by exponentials. [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]

This paper presents a study on the univalence extension of a specific class of functions defined by an integral operator. The research provides sufficient conditions for determining this univalence extension, which depend on two arbitrary analytic functions defined in the unit disk, alongside three complex numbers. The main result of the study includes these functions and parameters, and by specializing them, it leads to various well-known univalent conditions. The paper showcases examples to demonstrate how these conditions can be applied. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce and investigate a new subcategory H_Σ (q,β;ς) of analytic and bi-univalent functions in the open unit disk D based on the ideas of q-derivative operator and subordination. For the functions belong to the subcategory H_Σ (q,β;ς) , upper bounds for the second and third coefficients are found and some special outcomes of the main result are also presented.

The results of this work belong to the field of geometric function theory, being based on differential subordination methods. Using the idea of the $ \mathfrak{q} $-calculus operators, we define the $ \mathfrak{q} $-analogue of the multiplier- Ruscheweyh operator of a specific family of linear operators, $ I_{\mathfrak{q}, \mu }^{s}(\lambda, \ell). $ Our major goal is to build and investigate some analytic function subclasses using $ I_{\mathfrak{q}, \mu }^{s}(\lambda, \ell) $. Also, some differential subordination and superordination results are obtained. Moreover, based on the new theoretical results, several examples are constructed. For every differential superordination under investigation, the best subordinant is provided.

Abstract The Ruscheweyh derivative operator is used in this paper to introduce and investigate interesting general subclasses of the function class Σ m $\Sigma_{m}$ of m-fold symmetric bi-univalent analytic functions. Estimates of the initial Taylor-Maclaurin coefficients | a m + 1 | $\vert a_{m+1} \vert $ and | a 2 m + 1 | $\vert a_{2 m+1} \vert $ 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.

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.

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]

In this paper we extend some previously known results in the unit ball and in the unit disk on Bergman projections in bounded strongly pseudoconvex domains and in tubular domains over symmetric cones. It should be also added that on the other hand all our new results are also related with analytic spaces in product domains. Below first we give some definitions and basic lemmas and facts on bounded strongly pseudoconvex domains and then in tubular domains over symmetric cones. [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]

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]

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]

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 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]