Coefficient Estimates for Certain Families of Analytic Functions Associated with Faber Polynomial

In this paper, we use the Faber polynomial expansion to obtain bounds for the general coe ﬃ cients j a n j of bi-univalent functions in the family of analytic functions in the open unit disk. Estimation of the bound value for the initial coe ﬃ cients of the functions in these classes is also established


Introduction
Let A denote the class of all functions of the form which are analytic in the open unit disc U = fz : z ∈ ℂ and jzj < 1g.Also, let S be the class of all functions in A which are univalent in U.For f ∈ A, Airault and Bouali ( [1], page 184) used Faber polynomial to show that where F j−1 ða 2 , a 3 , ⋯, a j Þ is the Faber polynomial defined by The first terms of the Faber polynomial F j−1 , j ≥ 2, are given by (e.g., see [2], page 52) The Koebe one-quarter theorem ( [3], page 31) ensured that the range of each function of the class S contains the disc fw : jwj < 1/4g.Therefore, the univalent function f ∈ S has an inverse f −1 , which is defined by The inverse g ≔ f −1 of the function f ∈S has Taylor expansion given by (see [1], page 185) where the coefficients K p n ða 2 , a 3 , ⋯, a n Þ are given by and V j is homogeneous polynomial of degree j in the variables a 3 , ⋯, a n (see [4], page 349 and [1], pages 183 and 205).
Let f and g be analytic functions in U; we say that the function f is subordinate to g, written as follows: if there exists a Schwarz function w, which (by definition) is analytic in U with wð0Þ = 0 and jwðzÞj < 1ðz ∈ UÞ, such that f ðzÞ = gðwðzÞÞ for all z ∈ U.In particular, if the function g ∈ S in U, then we have the following equivalence relation (cf., e.g., [5,6]; see also [3]): Let ϕ be analytic function with positive real part in U, satisfying ϕð0Þ = 1, ϕ ′ ð0Þ > 0: Also, let ϕðUÞ be symmetric with respect to the real axis.Such a function has a Taylor series of the form with β, γ ∈ ℂ, Re ðαÞ > max f0, Re ðkÞ − 1g and Re ðkÞ > 0. Also, Re ðαÞ = 0 when Re ðkÞ = 1; β ≠ 0: Here, μ γ,k α,β is the generalized Mittag-Leffler function defined by [8] (see also [7]), and the symbol ð * Þ denotes the Hadamard product or convolution.
A single-valued function f analytic in a domain D ⊂ ℂ is said to be univalent, if it never takes the same value twice in We denote by Σ the class of bi-univalent functions in U given by (1).The class of analytic bi-univalent functions was introduced and studied by Lewin [13] and showed that ja 2 j < 1:51.Recently, many authors found nonsharp estimates on the first two Taylor-Maclaurin's coefficients ja 2 j and ja 3 j.
In this paper, we use the Faber polynomial expansion to obtain bounds for the general coefficients ja n j of bi-univalent functions in S Σ ðα, β, γ, k, ϕÞ and T Σ ðα, β, γ, k, ϕÞ as well as we estimate the bounds of the initial coefficients of the functions in these classes.

Coefficient Estimates of S
where B 1 is defined in (11).
Putting p = n in Theorem 5, we have the following: Corollary 6.Let the function f ∈ Σ given by (1) be in the class where B 1 is defined in (11).
Using the same technique used in Theorem 5, we get the following theorem: Theorem 7. Let the function f ∈ Σ given by (1) be in the class T Σ ðα, β, γ, k, ϕÞ: Also, let a m = 0 and a p ≠ 0 for 2 ≤ m, p ≤ n, where ðp − 1Þ is a divisor of ðn − 1Þ: Then, where B 1 is defined in (11).
Putting p = n in Theorem 7, we have the following: Corollary 8. Let the function f ∈ Σ given by (1) be in the class where B 1 is defined in (11).
To prove our next theorem, we shall need the following lemma.

Conclusion
By using the Faber polynomial expansion, we obtain bounds for the general coefficients ja n j of bi-univalent functions for functions in the classes S Σ ðα, β, γ, k, ϕÞ and T Σ ðα, β, γ, k, ϕÞ in U = fz : z ∈ ℂ and jzj < 1g; also, estimation of the bound value for the initial coefficients of the functions in the classes S Σ ðα, β, γ, k, ϕÞ and T Σ ðα, β, γ, k, ϕÞ is established.