Some $m$-Fold Symmetric Bi-Univalent Function Classes and Their Associated Taylor-Maclaurin Coefficient Bounds

The Ruscheweyh derivative operator is used in this paper to introduce and investigate interesting general subclasses of the function class $\Sigma_{\mathrm{m}}$ of $m$-fold symmetric bi-univalent analytic functions. Estimates of the initial Taylor-Maclaurin coefficients $\left|a_{m+1}\right|$ and $\left|a_{2 m+1}\right|$ are obtained for functions of the subclasses introduced in this study, and the consequences of the results are discussed. The results presented would generalize and improve on some recent works by many earlier authors. In some cases, our estimates are better than the existing coefficient bounds. Furthermore, within the engineering domain, this paper delves into a series of complex issues related to analytic functions, $m$-fold symmetric univalent functions, and the utilization of the Ruscheweyh derivative operator. These problems encompass a broad spectrum of engineering applications, including the optimization of optical system designs, signal processing for antenna arrays, image compression techniques, and filter design for control systems. The paper underscores the crucial role of these mathematical concepts in addressing practical engineering dilemmas and fine-tuning the performance of various engineering systems. It emphasizes the potential for innovative solutions that can significantly enhance the reliability and effectiveness of engineering applications.


Introduction
Let A denote the class of the functions f that are analytic in the open unit disk U = {z ∈ C : |z| < 1}, normalized by the conditions f (0) = f (0) -1 = 0 of the Taylor-Maclaurin series expansion Assume that S is the subclass of A that contains all univalent functions in U of the form (1.1), and P is the subclass of all functions h(z) of the form which is analytic in the open unit disk U and Re(h(z)) > 0, z ∈ U.
The Koebe 1/4-theorem [2] asserts that every univalent function f ∈ S has an inverse f -1 defined by The inverse function g = f -1 has the form g(w) = f -1 (w) = wa 2 w 2 + 2a 2 2a 3 w 3 -5a 3  2 -5a 2 a 3 + a 4 w 4 + • • • . (1.3) A function f ∈ A is said to be bi-univalent if both f and f -1 are univalent.The class of bi-univalent functions in U is denoted by .The following are some examples of functions in the class : with the corresponding inverse functions: e w and e 2w -1 e 2w + 1 , respectively.
Estimates on the bounds of the Taylor-Maclaurin coefficients |a n | are an important concern problem in geometric function theory because they provides information about the geometric properties of these functions.Lewin [3] studied the class of bi-univalent functions and discovered that |a 2 | < 1.51 for the functions belonging to the class .Later on, Brannan and Clunie [4] conjectured that |a 2 | √ 2. Subsequently, Netanyahu [5] showed that max |a 2 | = 4/3 for f ∈ .Recently, many works have appeared devoted to studying the bi-univalent functions class and obtaining non-sharp bounds on the Taylor-Maclaurin coefficients |a 2 | and |a 3 |.In fact, in their pioneering work, Srivastava et al. [6] have revived and significantly improved the study of the analytic and bi-univalent function class in recent years.They also discovered bounds on |a 2 | and |a 3 | and were followed by such authors (see, for example, [7][8][9][10][11][12][13][14] and references therein).The coefficient estimates on the bounds of |a n | (n ∈ {4, 5, 6, . ..}) for a function f ∈ defined by (1.1) remains an unsolved problem.In fact, for coefficients greater than three, there is no natural way to obtain an upper bound.There are a few articles where the Faber polynomial techniques were used to find upper bounds for higher-order coefficients (see, for example, [15][16][17][18]).
For each function f ∈ S, the function is univalent and maps the unit disk into a region with m-fold symmetry.A function f is said to be m-fold symmetric (see [19]) and is denoted by A m if it has the following normalized form: Assume that S m denotes the class of m-fold symmetric univalent functions in U that are normalized by the series expansion (1.5).In fact, the functions in class S are 1-fold symmetric.According to Koepf [19], the m-fold symmetric function h ∈ P has the form Analogous to the concept of m-fold symmetric univalent functions, Srivastava et al. [20] defined the concept of m-fold symmetric bi-univalent function in a direct way.Each function f ∈ generates an m-fold symmetric bi-univalent function for each m ∈ N. The normalized form of f is given as (1.5), and the extension g = f -1 is given by as follows: We denote the class of m-fold symmetric bi-univalent functions in U by m .For m = 1, the series (1.7) coincides with the series expansion (1.3) of the class .Following are some examples of m-fold symmetric bi-univalent functions: , with the corresponding inverse functions: For a function f ∈ A m defined by (1.5), one can think of the m-fold Ruscheweyh derivative operator R δ : A m → A m , which is analogous to the Ruscheweyh derivative R δ : A → A and can define as follows: In engineering, optimizing optical systems and designing effective control systems pose enormous challenges.Describing complex wavefronts necessitates the use of analytic and univalent functions tailored to specific optical constraints, while in signal processing for antenna arrays, employing m-fold symmetric univalent functions is crucial for beamforming amidst electromagnetic wave complexities, demanding innovation and precision.Control systems engineering utilizes univalent functions for filter design, where achieving the desired frequency response must align with system stability and minimal phase distortion, posing a continual challenge.Additionally, modeling complex mechanical systems requires leveraging the Ruscheweyh derivative operator to analyze functions representing system dynamics, facilitating critical parameter identification for system performance optimization.In robotics, univalent functions aid in controlling manipulators while navigating constraints related to joint angles and velocities.Moreover, in image compression and transmission for communication systems, the use of m-fold symmetric bi-univalent functions offers the potential for optimizing compression ratios while preserving image quality, representing an ongoing engineering challenge (see, for example, [25,26]).
This paper aims to introduce new general subclasses of m-fold symmetric bi-univalent functions in U applying the m-fold Ruscheweyh derivative operator, obtain estimates on initial coefficients |a m+1 | and |a 2m+1 | for functions in subclasses Q m (τ , λ, γ , δ; α) and m (τ , λ, γ , δ; β), and improve many recent works.Moreover, we have derived the Fekete-Szegö inequalities for these classes.To derive our main results, we need to use the following lemmas that will be useful in proving the basic theorems in Sects. 2 and 3.
Lemma 1 [2] If h ∈ P with h(z) given by (1.2), then Lemma 2 [27] If h ∈ P with h(z) given by (1.2) and μ is a complex number, then

Coefficient bounds for the function class
In this section, we assume that For a function h ∈ P given by (1.2).If K(z) is any complex-valued function such that satisfies the following conditions: and where z, w ∈ U and the function g = f -1 is given by (1.7).

.24)
Finally, taking the absolute value of (2.24) and applying Lemma 1 once again for the coefficients p m , p 2m , q m , and q 2m , we deduce that This completes the proof.

Coefficient bounds for the function class m (τ , λ, γ , δ; β)
In this section, we assume that Definition 2 A function f ∈ m given by (1.5) is called in the class m (τ , λ, γ , δ; β) if it satisfies the following conditions: and where z, w ∈ U and the function g = f -1 is given by (1.7).
Proof It follows from (3.1) and (3.2) that and where p(z) and q(w) have the forms (2.9) and (2.10), respectively.Clearly, we have and Equating the corresponding coefficients of (3.5) and (3.6) yields (1 + 2m(λ + γ ) + λγ ((2m + 1) 2 + 1) and In view of (3.9) and (3.11), we find that and Adding (3.10) to (3.12), we obtain Hence, we find from (3.14) and (3.15) that and respectively.By taking the absolute value of (3.16) and (3.17) and applying Lemma 1 for the coefficients p m , p 2m , q m , and q 2m , we deduce that and 2 , respectively.To determine the bound on |a 2m+1 |, by subtracting (3.12) from (3.10), we get Upon substituting the value of a 2 m+1 from (3.16) and (3.17) into (3.18),we conclude that and Now, taking the absolute value of (3.19) and (3.20) and applying Lemma 1 once again for the coefficients p m , p 2m , q m , and q 2m , we deduce that and respectively.This completes the proof.

Corollaries and consequences
This section is devoted to demonstrating of some special cases of the definitions and theorems.These results are given in the form of remarks and corollaries.
For 1-fold symmetric bi-univalent functions, Theorem 1 reduces to the following corollary: .
By taking δ = 0 in Corollary 6, we have the following result.