Certain new subclasses of m-fold symmetric bi-pseudo-starlike functions using Q-derivative operator

in the open unit disk Ω = {z; z ∈ C and |z| < 1}. We denote by G the subclass of functions in A which are univalent in Ω (for more details see [1]). The Keobe-One Quarter Theorem [1] state that the image of Ω under all univalent function f ∈ A contains a disk of radius 4 . Hence all univalent function f ∈ A has an inverse f−1 satisfy f−1( f (z)) and f ( f−1(υ)) = υ (|υ| < r0( f ), r0( f ) ≥ 1 4 ), where g(υ) = f−1(υ) = υ− ρ2υ + (2ρ2 − ρ3)υ − (5ρ2 − 5ρ2ρ3 + ρ4)υ + · · · (2)

A domain Ψ is said to be m-fold symmetric if a rotation of Ψ about the origin through an angle 2π/m carries Ψ on itself. Therefore, a function f (z) holomorphic in Ω is said to be m-fold symmetric if A function is said to be m-fold symmetric if it has the following normalized form Let S m the class of m-fold symmetric univalent functions in Ω, that are normalized by (3), in which, the functions in the class S are one-fold symmetric. Similar to the concept of m-fold symmetric univalent functions, we introduced the concept of m-fold symmetric bi-univalent functions which is denoted by Σ m . Each of the function f ∈ Σ produces m-fold symmetric bi-univalent function for each integer m ∈ N .
In [29] Girgaonkar et al., introduced a new subclasses of holomorphic and bi-univalent functions as follows: is given by (2).

Definition 3. A function f (z) given by (1) is said to be in the class
is given by (2).
In this current research, we introduced two new subclasses denoted by M q,σ Σ,m (χ) and M q,σ Σ,m (ψ) of the function class Σ m and obtain estimates coefficient |ρ m+1 | and |ρ 2m+1 | for functions in these two new subclasses.
which was introduced and studied by Girgaonkar et al., [29].

Remark 2. We have the class lim
which was introduced and studied by Srivastava et al., [11].

Definition 5. A function f (z) given by (3) is said to be in the class
and where g(υ) is given by (2).

Conclusion
In this present paper, two new subclasses indicated by M q,σ Σ,m (χ) and M q,σ Σ,m (ψ) of function class of E m was obtained and worked on. Also, the estimates coefficients for |ρ m+1 | and |ρ 2m+1 | of functions in these classes are determined.
Conflicts of Interest: "The author declares no conflict of interest."