Coefﬁcient Estimates and Fekete–Szegö Functional Inequalities for a Certain Subclass of Analytic and Bi-Univalent Functions

: The present paper introduces a new class of bi-univalent functions deﬁned on a symmetric domain using Gegenbauer polynomials. For functions in this class, we have derived the estimates of the Taylor–Maclaurin coefﬁcients, | a 2 | and | a 3 | , and the Fekete-Szegö functional. Several new results follow upon specializing the parameters involved in our main results.


Definitions and Preliminaries
Let A denote the class of all analytic functions f defined in the open unit disk U = {ξ ∈ C : |ξ| < 1} and normalized by the conditions f (0) = 0 and f (0) = 1. Thus, each f ∈ A has a Taylor-Maclaurin series expansion of the form f (ξ) = ξ + ∞ ∑ n=2 a n ξ n , (ξ ∈ U). (1) Further, let S denote the class of all functions f ∈ A which are univalent in U.
Let the functions f and g be analytic in U. We say that the function f is subordinate to g, written as f ≺ g, if there exists a Schwarz function w, which is analytic in U with w(0) = 0 and |w(ξ)| < 1 (ξ ∈ U) such that f (ξ) = g(w(ξ)).
In addition, if the function g is univalent in U, then the following equivalence holds if and only if f (0) = g(0) and f (U) ⊂ g(U).
It is well known that every function f ∈ S has an inverse f −1 , defined by A function is said to be bi-univalent in U if both f (ξ) and f −1 (ξ) are univalent in U. Let Σ denote the class of bi-univalent functions in U given by (1). Example of functions in the class Σ are However, the familiar Koebe function is not a member of Σ.
Other common examples of functions in U such as 2ξ − ξ 2 2 and ξ 1 − ξ 2 are also not members of Σ.
Lewin [1] investigated the bi-univalent function class Σ and showed that |a 2 | < 1.51. Subsequently, Brannan and Clunie [2] conjectured that |a 2 | < √ 2. Netanyahu [3], on the other hand, showed that max f ∈Σ The coefficient estimate problem for each of the Taylor-Maclaurin coefficients |a n | (n ≥ 3; n ∈ N) is presumably still an open problem.
In 1933, Fekete and Szegö [18] obtained a sharp bound of the functional ηa 2 2 − a 3 , with real η (0 ≤ η ≤ 1) for a univalent function f . Since then, the problem of finding the sharp bounds for this functional of any compact family of functions f ∈ A with any complex η is known as the classical Fekete-Szegö problem or inequality.
Orthogonal polynomials have been studied extensively as early as they were discovered by Legendre in 1784 [19]. In the mathematical treatment of model problems, orthogonal polynomials often arise to find solutions of ordinary differential equations under certain conditions imposed by the model.
The importance of orthogonal polynomials for contemporary mathematics and a wide range of their applications in physics and engineering is beyond any doubt. It is wellknown that these polynomials play an essential role in problems of the approximation theory. They occur in the theory of differential and integral equations and mathematical statistics. Their applications in quantum mechanics, scattering theory, automatic control, signal analysis, and axially symmetric potential theory are also known [20,21].
Very recently, Amourah et al. [22] considered the Gegenbauer polynomials, whose generating function H α (x, ξ) is given by where x ∈ [−1, 1] and ξ ∈ U. For a fixed x, the function H α is analytic in U, so it can be expanded in a Taylor series as where C α n (x) is a Gegenbauer polynomial of degree n.
Obviously, H α generates nothing when α = 0. Therefore, the generating function of the Gegenbauer polynomial is set to be for α = 0. Moreover, it is worth to mention that a normalization of α to be greater than −1/2 is desirable [21,23]. The following recurrence relations can also define Gegenbauer polynomials: with the initial values Many researchers have recently explored bi-univalent functions associated with orthogonal polynomials, refs. [24][25][26][27][28] to mention a few. For a Gegenbauer polynomial, as far as we know, there is little work associated with bi-univalent functions in the literature.
Motivated essentially by the work of Amourah et al. [22,29,30], we introduce here a new subclass of bi-univalent functions subordinate to Gegenbauer polynomials and obtain bounds for the Taylor-Maclaurin coefficients |a 2 | and |a 3 | and Fekete-Szegö functional problems for functions in this new class.
First, we give the coefficient estimates for the class G α Σ (x, λ, µ, δ) given in Definition 1.
Making use of the values of a 2 2 and a 3 , we prove the following Fekete-Szegö inequality for functions in the class G α Σ (x, λ, µ, δ).

Corollaries and Consequences
In this section, we apply our main results to deduce each of the following new corollaries and consequences.

Remark 2.
By taking α = 1, one can deduce the above results for the various subclasses studied by Yousef et al. [26].

Concluding Remark
In this present investigation, we introduced and studied the coefficient problems associated with each of the new subclasses G α Σ (x), G α Σ (x, λ), G α Σ (x, λ, µ) and G α Σ (x, λ, µ, δ) of the class of bi-univalent functions in the open unit disk U. We derived estimates of the Taylor-Maclaurin coefficients |a 2 | and |a 3 | and Fekete-Szegö functional problems for functions belonging to these new subclasses.