(P,Q)-Lucas polynomial coefficient inequalities of the bi-univalent function class

Recently, Lucas polynomials and other special polynomials gained importance in the field of geometric function theory. In this study, by connecting these polynomials, subordination, and the Al-Oboudi differential operator, we introduce a new class of bi-univalent functions and obtain coefficient estimates and Fekete–Szegö inequalities for this new class.


Introduction
Let A denote the class of functions of the form u(z) = z + ∞ n=2 a n z n , (1) which are analytic in the open unit disk U = {z : |z| < 1}, and let S = {u ∈ A : u is univalent in U } .
The Koebe one-quarter theorem [3] states that the range of every function u ∈ S contains the disc of radius w : |w| < 1 4 . Thus, every such function u ∈ S has an inverse u −1 , which satisfies u −1 (u (z)) = z (z ∈ U ) and u u −1 (w) = w |w| < r 0 (u) , r 0 (u) ≥ 1 4 , If both u and u −1 are univalent in U, then a function u ∈ A is said to be bi-univalent in U. We say that u is in the class Σ for such functions.
For analytic functions u and v, u is said to be subordinate to v , denoted if there is an analytic function w such that For a function u (z) ∈ A, Al-Oboudi [1] defined the following differential operator, named the Al-Oboudi differential operator: If u is given by (1), then from (4) and (5) we see that with D n δ u(0) = 0. When δ = 1 , we get Salagean's differential operator [9].
Definition 1 [6] Let P(x) and Q(x) be polynomials with real coefficients. The (P, Q)-Lucas polynomials L P,Q,n (x) are defined by the reccurence relation from which the first few Lucas polynomials can be found as follows: L P,Q,0 (x) = 2, Definition 2 [6] Let G {Ln(x)} (z) be the generating function of the (P, Q) -Lucas polynomial sequence L P,Q,n (x). Then (10)

The class Q Σ,δ (ζ, n; x) and the Fekete-Szegö inequality
We begin this section by defining the class Q Σ,δ (ζ, n; x) and by finding the estimates on the coefficients |a 2 | and |a 3 | for functions in this class.

Definition 3
The function u is said to be in the class Q Σ,δ (ζ, n; x) if the following conditions are satisfied: where the function D n δ is the Al-Oboudi differential operator and v = u −1 is given by (2).