Coefﬁcient Bounds for Certain Classes of Analytic Functions Associated with Faber Polynomial

: In this paper, we intorduce a family of analytic functions in the open unit disk which is bi-univalent. By the virtue of the Faber polynomial expansions, the estimation of n − th ( n ≥ 3 ) Taylor–Maclaurin coefﬁcients | a n | is obtained. Furthermore, the bounds value of the ﬁrst two coefﬁcients of such functions is established.


Introduction
Faber polynomials, which were introduced by Faber in 1903 [1], play an important role in the theory of functions of a complex variable and different areas of mathematics and there is a rich literature [2][3][4][5][6][7] describing their properties and their applications. Given a function h(z) of the form h(z) = z + b 0 + b 1 z −1 + b 2 z −2 + . . . , consider the expansion valid for all ζ in some neighborhood of ∞. The function Ψ n (w) = w n + n ∑ k=1 a nk w n−k is a polynomial of degree n, called the n-th Faber polynomial with respect to the function h(z).
2. M(1, β, γ, A single-valued function f analytic in a domain D ⊂ C is said to be univalent there if it never take the same value twice; that is, if f (z 1 ) = f (z 2 ) for all points z 1 and z 2 in D with z 1 = z 2 (see [8], p. 26). A function f ∈ A is said to be bi-univalent in U if f and its inverse map f −1 are univalent in U. Let σ denote the class of bi-univalent functions in U given by (1). The class of analytic bi-univalent functions was first introduced and studied by Lewin [21] and showed that |a 2 | < 1.51. Recently, many authors found non-sharp estimates on the first two Taylor-Maclaurin coefficients |a 2 | and |a 3 | for various subclasses of bi-univalent functions, see for example, ( ). For other related topics see also, ( [44][45][46][47]).
In this paper, we use the Faber polynomial expansions to obtain bounds for the general coefficients |a n | of bi-univalent functions in M σ (λ, β, γ, φ) and S σ (λ, β, γ, φ) as well as we provide estimates for the initial coefficients of these functions.
To prove our next theorem, we shall need the following lemma .

Remark 1.
If we take β = 0 in Theorem 2 we obtain that the bounds on a 3 − a 2 2 given by Deniz et al. [52] when γ = 1.
if the following conditions are satisfied: where g = f −1 .
Using the parameter setting of Definition 5 in Theorems 1 and 2, respectively, we get the following corollaries.
Using the parameter setting of Definition 6 in Theorems 1 and 2, respectively, we get the following corollaries.
Using the parameter setting of Definition 7 in Theorems 1 and 2, respectively, we get the following corollaries.

Definition 8. A function f ∈ σ given by
where g = f −1 .
Using the parameter setting of Definition 8 in Theorems 1 and 2, respectively, we get the following corollaries.

Definition 9.
A function f ∈ σ given by (1) is said to be in the class M ∆ σ (λ, β, γ) if the following conditions are satisfied: where g = f −1 .
Using the parameter setting of Definition 9 in Theorems 1 and 2, respectively, we get the following corollaries.
Using the parameter setting of Definition 10 in Theorems 1 and 2, respectively, we get the following corollaries.
Definition 11. A function f ∈ σ given by (1) is said to be in the class M σe (λ, β, γ) if the following conditions are satisfied: and where g = f −1 .
Using the parameter setting of Definition 11 in Theorems 1 and 2, respectively, we get the following corollaries.
By using (9), we conclude that Remark 2. If we take β = 0 in Theorem 3, then we have the results which were given by Zireh et al. [51] when ϕ(z) = 1.
Using the parameter setting of Definition 12 in Theorems 3 and 4, respectively, we get the following corollaries.
Using the parameter setting of Definition 13 in Theorems 3 and 4, respectively, we get the following corollaries.
Using the parameter setting of Definition 15 in Theorems 3 and 4, respectively, we get the following corollaries.

Definition 16.
A function f ∈ σ given by (1) is said to be in the class S ∆ σ (λ, β, γ) if the following conditions are satisfied: where g = f −1 .
Using the parameter setting of Definition 9 in Theorems 3 and 4, respectively, we get the following corollaries.
Using the parameter setting of Definition 17 in Theorems 3 and 4, respectively, we get the following corollaries.
Using the parameter setting of Definition 18 in Theorems 3 and 4, respectively, we get the following corollaries.
Author Contributions: All authors contributed equally to the writing of this paper. All authors have read and agreed to the published version of the manuscript.