Fekete–Szegö Inequalities for a New Subclass of Bi-Univalent Functions Associated with Gegenbauer Polynomials

: We introduce and investigate in this paper a new subclass of bi-univalent functions associated with the Gegenbauer polynomials which satisfy subordination conditions deﬁned in a symmetric domain, which is the open unit disc. For this new subclass, we obtain estimates for the Taylor–Maclaurin coefﬁcients | a 2 | , | a 3 | and the Fekete–Szegö


Introduction
Let A represent the class of functions whose members are of the form f (z) = z + ∞ ∑ n=2 a n z n , (z ∈ ∆), (1) which are analytic in ∆ = {z ∈ C : |z| < 1}. A subclass of A with members that are univalent in ∆ is indicated by the symbol S. The Koebe one-quarter theorem [1] guarantees that a disk with a radius of 1/4 exists in the image of ∆ for every univalent function f ∈ A. As a result, each univalent function f has a satisfied inverse function f −1 f −1 ( f (z)) = z, (z ∈ ∆) and f f −1 (ω) = ω, (|ω| < r 0 ( f ), r 0 ( f ) ≥ 1 4 ).
Let f and g be the analytic functions in ∆. We say that f is subordinate to g and denoted by if there exists a Schwarz function w, which is analytic in ∆ with w(0) = 0 and |w(z If g is a univalent function in ∆, then In [6], by means of Loewner's method, the Fekete-Szegö inequality for the coefficients of f ∈ S is that on the normalized analytic functions f in the open unit disk ∆ plays an important role in geometric function theory. The problem of maximizing the absolute value of the functional F µ ( f ) is called the Fekete-Szegö problem. The Fekete-Szegö inequalities introduced in 1933, see [12], preoccupied researchers regarding different classes of univalent functions [13][14][15][16]; hence, it is obvious that such inequalities were obtained regarding bi-univalent functions too and very recently published papers can be cited to support the assertion that the topic still provides interesting results [17][18][19].
In recent years orthogonal polynomials have been explored from a variety of angles. We know their relevance in mathematical physics, mathematical statistics, probability theory and engineering. The classical orthogonal polynomials are the most typically encountered orthogonal polynomials in applications (Laguerre polynomials, Jacobi polynomials and Hermite polynomials). For more details about the classical orthogonal polynomials we mention the papers: [17,18,[20][21][22][23][24].
The Gegenbauer polynomials [25] are defined in terms of the Jacobi polynomials P From (3), it follows that B λ n (l) is a polynomial of degree n with real coefficients and B λ n (1) = n − 1 + 2λ n , while the leading coefficient of B λ n (l) is 2 n n + λ − 1 n . According to Jacobi polynomial theory, for µ = v = λ − 1 2 , with λ > − 1 2 , and λ = 0, we have In [25,26], the Gegenbauer polynomials' generating function is provided by and this equivalence may be deduced from the Jacobi polynomial generating function. From (4), we obtain (5) and the proof is given in [6,23,25]. When λ = 1, the relation 5 yields the ordinary generating function for the Chebyshev polynomials, and when λ = 1 2 , we obtain the ordinary generating function for the Legendre polynomials (see [27]).
First, we define a new subclass of bi-univalent functions associated with Gegenbauer polynomials.
Upon allocating the parameters τ and ϑ, one can obtain several new subclasses of Σ, as illustrated in the following two examples.
In Theorem 1, we obtain the following result for τ = ϑ = 1.
For λ = 1 2 and l = 1 in Corollary 2, we obtain the following corollary.

The Fekete-Szegö Problem for the Function Class M Σ τ, ϑ; φ λ l
Due to the Zaprawa result, which is discussed in [19], we obtain the Fekete-Szegö inequality for the class M Σ τ, ϑ; φ λ l .
In Theorem 2, we have the following result for τ = ϑ = 1.

Conclusions
In this paper, we introduced and investigated a new subclass of bi-univalent functions in the open unit disk defined by Gegenbauer polynomials which satisfy subordination conditions. Furthermore, we obtain upper bounds for |a 2 |, |a 3 | and the Fekete-Szegö inequality a 3 − µa 2 2 for functions in this subclass. In addition, the approach presented here has been extended to establish new subfamilies of bi-univalent functions with other special functions. The related outcomes may be left to researchers for practice.