Coeﬃcient Bounds and Fekete-Szegö inequality for a Certain Families of Bi-Prestarlike Functions Deﬁned by (M,N)-Lucas Polynomials

: In the current work, we use the (M,N)-Lucas Polynomials to introduce a new families of holomorphic and bi-Prestarlike functions deﬁned in the unit disk O and establish upper bounds for the second and third coeﬃcients of the Taylor-Maclaurin series expansions of functions belonging to these families. Also, we debate Fekete-Szegö problem for these families. Further, we point out several certain special cases for our results.


Introduction
Indicate by A the collection of functions U that are holomorphic in the unit disk O = {ξ ∈ C : |ξ| < 1} that have the shape: U(ξ) = ξ + ∞ n=2 a n ξ n . (1.1) Further, let S stands for the subfamily of the collection A consisting of functions in O satisfying (1.1) that are univalent in O. According to "the Koebe one-quarter theorem" (see [12]), each univalent function of this kind has an inverse U −1 that fulfills and U(U −1 (ζ)) = ζ, (|ζ| < r 0 (U), r 0 (U) ≥ 1 4 ), where U −1 (ζ) = ζ − a 2 ζ 2 + 2a 2 2 − a 3 ζ 3 − 5a 3 2 − 5a 2 a 3 + a 4 ζ 4 + · · · . (1.2) A function U ∈ A is said to be bi-univalent in O if both U and U −1 are univalent in O, let we name by the notation E the set of bi-univalent functions in O satisfying (1.1). In fact, Srivastava et al. [32] refreshed the study of holomorphic and biunivalent functions in recent years, it was followed by other works as those by Frasin and Aouf [15], Altinkaya
The problem to obtain the general coefficient bounds on the Taylor-Maclaurin coefficients |a n | (n ∈ N; n ≥ 4) for functions U ∈ E is still not completely addressed for many of the subfamilies of the bi-univalent function class E.
The Fekete-Szegö functional a 3 − µa 2 2 for U ∈ S is well known for its rich history in the field of Geometric Function Theory. Its origin was in the disproof by Fekete and Szegö [13] of the Littlewood-Paley conjecture that the coefficients of odd univalent functions are bounded by unity.
For U ∈ A given by (1.1) and J ∈ A defined by the "Hadamard product" of U and J is defined by Ruscheweyh [25] introduced and studied the family of "prestarlike functions" of order θ, that are the function U such as U * I θ is a starlike function of order θ, where The function I θ can be written in the form: We note that n (θ) is a decreasing function in θ and satisfies With a view to remembering the principle of subordination between holomorphic functions, let the functions U and such that

This subordination is indicated by
For two polynomials M (x) and N (x) that have real-valued coefficients, the following recurrence relation gives the (M,N)-Lucas Polynomials L M,N,k (x) (see [19]): The function that generates (M,N)-Lucas Polynomial L M,N,k (x) (see [20]) is given by We also note that the Lucas polynomials and other special polynomials plays an important role in a diversity of disciplines in the mathematical, statistical, physical and engineering sciences. More details associated with these polynomials can be found in [2,17,37,14,20,40].
Proof. Suppose that U ∈ WM E (τ, θ; x). Then there exists two holomorphic functions φ, ψ : In the light of (2.20) and (2.21), after simplifying, we find that If we add (2.23) to (2.25), we obtain By substitute the value of r 2 1 + s 2 1 from (2.27) in the right hand side of (2.28), we conclude that .
Proof. By making use of (2.16) and (2.17), we conclude that According to (1.3), we find that After some computations, we obtain .
Putting τ = 0 and θ = 1 2 in Theorem 2.4, we deduce the next outcome: Corollary 2.9. If U belongs to the family WM E (x), then Putting ρ = 1 in Theorem 2.4, we deduce the next outcome: .
Putting ρ = 1 in Corollary 2.9, we deduce the next outcome: Corollary 2.11. If U belongs to the family WM E (x), then