A Family of Holomorphic and m -Fold Symmetric Bi-Univalent Functions Endowed with Coefﬁcient Estimate Problems

: This paper presents a new general subfamily N u , v Σ m ( η , µ , γ , (cid:96) ) of the family Σ m that contains holomorphic normalized m -fold symmetric bi-univalent functions in the open unit disk D associated with the Ruscheweyh derivative operator. For functions belonging to the family introduced here, we ﬁnd estimates of the Taylor–Maclaurin coefﬁcients | a m + 1 | and | a 2 m + 1 | , and the consequences of the results are discussed. The current ﬁndings both extend and enhance certain recent studies in this ﬁeld, and in speciﬁc scenarios, they also establish several connections with known results.


Introduction
Let D = {z ∈ C : |z| < 1} be an open unit disk in the complex plane and A be a collection of functions which are holomorphic in D together with a normalization given by The Hadamard product f (z) * l(z) of f (z) and l(z) is defined by where the function l(z) = z + ∑ ∞ n=2 b n z n is also holomorphic in D.
Denote the sub-collection of A by S, consisting of univalent functions in D, and consider the sub-collection P of functions that are holomorphic in D and the real part, R(p(z)), is positive.
According to the Koebe 1/4 Theorem (see [2]), the image of D under any univalent function consists of a disk of radius 1/4.As a consequence, every function f ∈ S has an inverse f −1 such that The inverse of the function f (z) has a series expansion in some disk about the origin of the following form: A univalent function f (z) in the neighborhood of the origin and its inverse f −1 (w) satisfy the following condition: Using ( 1) and ( 3) in (4), we get If a function f ∈ A and its inverse are both univalent on D, then f is called a biunivalent function.Denote the family of all bi-univalent functions in D by Σ.
Lewin [3] conducted a study on the family Σ of bi-univalent functions and discovered that |a 2 | < 1.51 for the functions belonging to the family Σ.Later, Brannan and Clunie [4]  proposed the conjecture that |a 2 | ≤ √ 2. Subsequently, Netanyahu [5] demonstrated that max|a 2 | = 4  3 for f ∈ Σ.To explore various fascinating examples of f ∈ Σ, refer to the seminal work on this area by Srivastava et al. [6], which has revitalized the study of f ∈ Σ functions in recent years.
Srivastava et al. [6] showed that the family Σ is nonempty by providing some explicit examples, including the following function:  [6] actually revived the study of analytic and biunivalent functions in recent years.It was followed by a remarkably huge flood of sequels on the subject.Let 0 ≤ ϑ < 1. Brannan and Taha [7] introduced specific subfamilies of Σ, analogous to the well-known subfamilies, starlike functions S * (ϑ), and convex functions K(ϑ) of order θ.A function f ∈ Σ is in the family S * Σ (ϑ) of bi-starlike functions of order ϑ if both f and its inverse are starlike functions of order ϑ, or is in the family K Σ (ϑ) of bi-convex functions of order ϑ if both f and its inverse are convex functions of order ϑ.Moreover, for 0 < ϑ ≤ 1, the function f ∈ A is classified as a strongly bi-starlike function, S * Σ [ϑ] (see [7,8]), if it satisfies: where g = f −1 is defined by (5).
Recently, studying the family Σ and deriving non-sharp bounds on |a 2 | and |a 3 |, where a 2 and a 3 are the initial Taylor-Maclaurin coefficients, have become an active area of research.In particular, the pioneering work by Srivastava et al. [6] has crucially advanced the study of certain subfamilies within Σ and identified constraints on |a 2 | and |a 3 |.A substantial number of subsequent works have been published in the literature, building upon the groundbreaking research by Srivastava et al. [6] and focusing on coefficient problems for different subfamilies of Σ (see, for example, [9,10] and the above-cited works).However, the general coefficient estimate bounds on |a n |(n ∈ {4, 5, 6, . ..}) for functions f in the family Σ remain an unsolved problem.
For f ∈ S, the function is univalent and maps D into an m-fold symmetric region.A function f ∈ A is called m-fold symmetric (see [11]) if it is of the form: The family of all m-fold symmetric functions is denoted by A m .For a function f ∈ A m defined by (7), analogous to the Ruscheweyh derivative operator, the m-fold Ruscheweyh derivative D : A m → A m is defined as follows (see [12]): Let δ m denote the family of m-fold symmetric univalent functions in D normalized by (7).Then, the functions f ∈ S are one-fold symmetric.As stated by Koepf [11], the m-fold symmetric p in P has the form Recently, Srivastava et al. [13] defined the family of m-fold symmetric bi-univalent functions Σ m analogous to the family Σ, and the inverse of functions f given by ( 7) is specified as follows: For m = 1, the function in (9) coincides with (5) of the family Σ.Some examples of m-fold symmetric bi-univalent functions are given below: Recent research has been dedicated to analyzing the functions in the family Σ m and obtaining non-sharp bounds on |a m+1 | and |a 2m+1 |, where a m+1 and a m+2 are the initial Taylor-Maclaurin coefficients.In reality, Srivastava et al. [13] have greatly advanced the research on many subfamilies of the family Σ m and obtained restrictions on |a m+1 | and |a 2m+1 | in recent years.Later on, some scholars followed them (see, for example, [14,15]  and the above-cited works).
Motivated by the aforementioned works, the primary goal of this study is to propose a formula to determine the coefficients of the functions for the family Σ m utilizing the residue of calculus.As an example, we construct estimates of the coefficients |a m+1 | and |a 2m+1 | for functions belonging to a generic subfamily N u,v Σ m (η, µ, γ, ) of Σ m in D, and additional links to previously known results are made.Furthermore, by sufficiently specializing the parameters, some consequences of this family are demonstrated.

The Family N u,v
Σ m (λ, µ, γ, ) and Its Associated Coefficient Estimates In this section, the following general family N u,v Σ m (η, µ, γ, ) is introduced and investigated.

Definition 1.
A function f ∈ Σ m given by ( 7) belongs to the family if the following conditions are satisfied: and where z, w ∈ D, u and v, holomorphic in D, are defined by the expansion (8), and the function g = f −1 is defined by (9).
Many choices of the functions u and v can be used to create attractive subfamilies of the functions that are holomorphic in the family A m .
it can be seen that the functions u(z) and v(z) satisfy the conditions of Definition 1.
where the function g = f −1 is defined by (9).
and the family then the conditions of Definition 1 are satisfied for both functions u(z) and v(z).Thus, where the function g = f −1 is defined by (9).
).It can be seen that, for symmetric one-fold bi-univalent functions, by specializing η, µ, γ and , we get several known subfamilies of Σ recently investigated by various authors.Let us present some examples.
Now, we are able to express bounds for |a m+1 | and |a 2m+1 | for the subfamily N u,v Σ m (η, µ, γ, ) of the family Σ m .
Then, to obtain |a 2m+1 |, subtract (27) from ( 25), where ϕ 5 is given by (19).Now, putting the value of a 2 m+1 from (29) into (35), it follows that Therefore, taking the absolute value of (36) and using the relation given by (28), we deduce that By putting the value of a 2 m+1 from (30) into (35), we obtain Therefore, taking the absolute value of (38), we conclude the following bound Finally, from (37) and (39), we get the relevant estimate as asserted in (14).This completes the proof.

Corollaries and Consequences
If we put in Theorem 1, then Corollary 1 can be obtained.
If we set in Theorem 1, then Corollary 2 can be obtained.
By letting in Theorem 1 for the subfamily N u,v Σ (η, µ, γ, ) of the family Σ := Σ 1 that contains normalized holomorphic and bi-univalent functions, then Corollary 3 can be derived.Corollary 3. Let f (z) ∈ N Σ (η, µ, γ, ; α) be of the form (1).Then, and where in Theorem 1 for the subfamily N u,v Σ (η, µ, γ, ) of the family Σ that contains normalized holomorphic and bi-univalent functions, then Corollary 4 can be derived. and , where The following corollary follows from Theorem 1 for one-fold symmetric bi-univalent functions.
Σ (η, µ, γ, ) be of the form (1).Then, where and By specializing the parameters in Corollary 3, it can be seen that several estimate bounds for known subfamilies of Σ can be attained as special cases.
It worth noting that the Koebe function is not a member of Σ.Hence, Σ is a proper subfamily of A. In fact, this pioneering work of Srivastava et al.