The Study of the New Classes of m-Fold Symmetric bi-Univalent Functions

: In this paper, we introduce three new subclasses of m-fold symmetric holomorphic functions in the open unit disk U , where the functions f and f − 1 are m-fold symmetric holomorphic functions in the open unit disk. We denote these classes of functions by FS p , q , s Σ , m ( d ) , FS p , q , s Σ , m ( e ) and FS p , q , s , h , r Σ , m . As the Fekete-Szegö problem for different classes of functions is a topic of great interest, we study the Fekete-Szegö functional and we obtain estimates on coefﬁcients for the new function classes.


Introduction and Preliminary Results
Let A denote the family of functions of the form f (z) = z + ∞ ∑ k=2 a k z k (1) which are analytic in the open unit disk U = {z ∈ C : |z| < 1} and normalized by the conditions f (0) = 0, f (0) = 1. Let S ⊂ A denote the subclass of all functions in A which are univalent in U (see [1]). In [1], the Koebe one-quarter theorem ensures that the image of the unit disk under every f ∈ S function contains a disk of radius 1/4.
It is well known that every function f ∈ S has an inverse f −1 , which is defined by A function f ∈ A is said to be bi-univalent in U if both f and f −1 are univalent in U. Let Σ denote the class of all bi-univalent functions in U given by (1).
The class of bi-univalent functions was first introduced and studied by Lewin [2] and it was shown that |a 2 | < 1.51.
The domain D is m-fold symmetric if a rotation of D about the origin through an angle 2π/m carries D on itself.
We said that the holomorphic function f in the domain D is m-fold symmetric if the following condition is true: f (e 2πi m z) = e 2πi m f (z).
A function is said to be m-fold symmetric if it has the following normalized form: The normalized form of f is given as in (3) and the series expansion for f −1 (z) is given below (see [3]): We can give examples of m-fold symmetric bi-univalent functions: The important results about the m-fold symmetric analytic bi-univalent functions are given in [3][4][5][6][7].
Definition 1. Let f ∈ A be given by (1) and 0 < q < p ≤ 1. Then, the (p, q)-derivative operator for the function f of the form (1) is defined by and (D p,q f )(0) = f (0) (6) and it follows that the function f is differentiable at 0.
We deduce from (2) that where the (p, q)-bracket number is given by which is a natural generalization of the q-number. [26,27]. . Let the function w ∈ P be given by the following series: w(z) = 1 + w 1 z + w 2 z 2 + . . . , z ∈ U, where we denote by P the class of Carathéodory functions analytic in the open disk U, P = {w ∈ A|w(0) = 1, Re(w(z)) > 0, z ∈ U}.
The sharp estimate given by |w n | ≤ 2, n ∈ N * holds true.

Main Results
Definition 3. The function f given by (3) is in the function class FS p,q,s and where g is the function given by (4).

Remark 1.
In the case when m = 1 (one-fold case) and s = 1, we obtain the class defined in [29].
We obtain coefficient bounds for the functions class FS p,q,s Σ,m (d) in the next theorem.

Theorem 1. Let f given by (3) be in the class FS
and Proof. If we use the relations (8) and (9), we obtain and where the functions α(z) and β(w) are in P and are given by and It is obvious that . . If we compare the coefficients in the relations (12) and (13), we have We obtain from the relations (16) and (18) and Now, from the relations (17), (19) and (21), we obtain that If we apply Lemma 1 for the coefficients α 2m and β 2m , we have If we use the relations (17) and (19), we obtain the next relation It follows from (20), (21) and (23) that If we apply Lemma 1 for the coefficients α m , α 2m , β m , β 2m , we obtain Remark 3. For one-fold case m = 1 and s = 1 in Theorem 1, we obtain the results obtained in [29].

Definition 4. The function f given by
where the function g is defined by Relation (4).

Remark 5.
For m = 1 (one-fold case) and s = 1, we obtain the class of functions obtained in [29].
In the next theorem, we obtain coefficient bounds for the function class FS p,q,s Σ,m (e).
From the relations (29) and (30), if we compare the coefficients, we obtain the following relations: We obtain from Relations (31) and (33) and We obtain now from Relations (32) and (34) the following relation: From Lemma 1 for the coefficients α m , α 2m , β m , β 2m , we obtain that If we use Relations (32) and (34) to find the bound on |a 2m+1 |, we obtain the following relation: or equivalently From Relation (36), if we substitute the value of a 2 m+1 , we obtain Remark 7. For one fold case (m = 1) and s = 1 in Theorem 2, we obtain the results given in [29].
Proof. We want to calculate a 2m+1 − σa 2 m+1 . For this, from Relations (22) and (24), where we know the values of the coefficients a 2 m+1 and a 2m+1 : According to Lemma 1 and after some computations, we obtain Theorem 4. Let f be a function of the form (3) in the class FS p,q,s Σ,m (e). Then, where Proof. We will compute a 2m+1 − σa 2 m+1 , using the values of the coefficients a 2 m+1 and a 2m+1 given in Relations (37) and (39).
A function f given by (3) is said to be in the class FS p,q,s,h,r Σ,m , where s ≥ 1, 0 < q < p ≤ 1, m ∈ N if the conditions are satisfied: and where the function g is given by (4).
We obtain coefficient bounds for the functions class FS p,q,s,h,r Σ,m in the following theorem.

Theorem 5.
Let the function f given by (3) Proof. In Relations (43) and (44), the equivalent forms of the argument inequalities are and (D p,q g(w)) s = r(w), where h(z) and r(w) satisfy the conditions from Definition 5, and have the following Taylor-Maclaurin series expansions: If we substitute (49) and (50) into (47) and (48)

Conclusions
As future research directions, the symmetry properties of this operator, the (p, q)derivative operator, can be studied.