Initial Coefﬁcient Estimates and Fekete–Szegö Inequalities for New Families of Bi-Univalent Functions Governed by ( p − q ) -Wanas Operator

: The motivation of the present article is to deﬁne the ( p − q ) -Wanas operator in geometric function theory by the symmetric nature of quantum calculus. We also initiate and explore certain new families of holormorphic and bi-univalent functions A E ( λ , σ , δ , s , t , p , q ; ϑ ) and S E ( µ , γ , σ , δ , s , t , p , q ; ϑ ) which are deﬁned in the unit disk U associated with the ( p − q ) -Wanas operator. The upper bounds for the initial Taylor–Maclaurin coefﬁcients and Fekete–Szegö-type inequalities for the functions in these families are obtained. Furthermore, several consequences of our results are pointed out based on the various special choices of the involved parameters.


Introduction
Indicate by A the family of all holomorphicfunctions of the form in the open unit disk U = {z ∈ C : |z| < 1}. We also denote by T the subfamily of A consisting of functions which are also univalent in U.
The famous Koebe one-quarter theorem [1] ensures that the image of U under each univalent function f ∈ A contains a disk of radius 1 4 . Furthermore, each function f ∈ T has an inverse f −1 defined by f −1 ( f (z)) = z and A function f ∈ A is named bi-univalent in U if both f and f −1 are univalent in U. The family of all bi-univalent functions in U is denoted by E.
We notice that the family E is not empty. However, the Koebe function is not a member of E.
The problem to obtain the general coefficient bounds on the Taylor-Maclaurin coefficients |b n | (n ∈ N; n 3) for functions f ∈ E is still not completely addressed for many of the subfamilies of E.
The origin of the Fekete-Szegö functional b 3 − ηb 2 2 for f ∈ T was in the disproof [16] by Fekete and Szegö of the Littlewood-Paley conjecture that the coefficients of odd univalent functions are bounded by unity. Researchers in the Theory of Geometric Function have recently obtained remarkable results on this topic (see, for example, [17][18][19][20][21]).
With a view to recalling the principle of subordination between holomorphic functions, let the functions f and g be holomorphic in U. The function f is subordinate to g, if there exists a Schwarz function ω, which is analytic in U with ω(0) = 0 and |ω(z)| < 1 (z ∈ U), such that f (z) = g ω(z) .
The subordination is denoted by It is well known that (see [22]), if the function g is univalent in U, then f ≺ g (z ∈ U) ⇐⇒ f (0) = g(0) and f (U) ⊆ g(U).
For 0 < q < p ≤ 1, the (p, q)-derivative operator or (p, q)-difference operator for a function f is defined by and D p,q f (0) = f (0).
For more details on the concepts of (p, q)-calculus see [20,[23][24][25][26][27]. For function f ∈ A, we deduce that where the (p, q)-bracket number or twin-basic [k] p,q is given by which is a natural generalization of the q-number, that is, we have (see [28][29][30]) It is clear that the notation [k] p,q is symmetric, that is, In 2019, Wanas [31] introduced the following operator, which can also be called Wanas where s ∈ R, t ∈ R + 0 with s + t > 0, k − 1, σ ∈ N and δ ∈ N 0 . Now, for f ∈ A, we define the (p − q)-difference Wanas operator as given below Remark 1. The operator W σ,δ s,t,p,q is a generalization of several known operators studied in earlier investigations which are being recalled below.
For q −→ 1 − and p = σ = 1, the operator W σ,δ s,t,p,q reduces to the operator I δ s,t that was introduced and studied by Swamy [35]; 6.
For q −→ 1 − , p = σ = t = 1 and s > −1, the operator W σ,δ s,t,p,q reduces to the operator I δ s that was investigated by Cho and Srivastava [37];
We shall need the following Lemma in our investigation. Lemma 1 ([44], p. 41 and [45], p. 41). Let the function x ∈ P be given by the following series: The sharp estimate given by |x n | 2 (n ∈ N) holds true.

A Set of Main Results
Indicate by ϑ(z) the holomorphic function with positive real part in U such that and ϑ(U) is symmetric with respect to the real axis, which is of the type: where B 1 > 0. Using the (p − q)-Wanas operator, we now provide the following subfamilies of holomorphic and bi-univalent functions.
Proof. Let f ∈ A E (λ, σ, δ, s, t, p, q; ϑ) and g = f −1 . Then, there are holomorphic functions S, T : U −→ U with S(0) = T(0) = 0, which fulfill the following conditions: and (1 − λ) w W σ,δ s,t,p,q g(w) W σ,δ s,t,p,q g(w) Define the functions x and y by Then, x and y are analytic in U with x(0) = y(0) = 1. Since we have S, T : U −→ U, each of the functions x and y has a positive real part in U.
Furthermore, in order to find the bound on |b 3 |, we subtract (15) from (13) and also apply (16). We obtain x 2 1 = y 2 1 , hence, then, by substituting the value of b 2 2 from (17) into (20), gives In addition, substituting the value of b 2 2 from (18) into (20), we obtain p,q and we have p,q , which gives us the desired estimates of the coefficient |b 3 |.
In [8], the authors obtained Fekete-Szegö inequalities for classes of analytic and bi-univalent functions defined by (p, q)-derivative operator; 3.
In [23], the authors obtained Fekete-Szegö inequalities for subclasses of analytic and biunivalent functions defined by subordinations using the Sȃlȃgean operator; 4.
In [55], the authors obtained Fekete-Szegö inequalities and coefficients bounds for new classes of bi-univalent functions defined by the Sȃlȃgean integro-differential operator; 8.
In [56], the author obtained Fekete-Szegö inequalities for classes of bi-univalent functions defined in terms of subordinations.
Putting η = 1 in Theorem 3, we obtain the following result.