Fekete–Szegö Inequality for Bi-Univalent Functions Subordinate to Horadam Polynomials

Making use of Horadam polynomials, we propose a special family of regular functions of the type g(z) � z + 􏽐j�2 djz j which are bi-univalent (or bi-schlicht) in the disc z ∈ C: |z|< 1 { }. We find estimates on the coefficients |d2| and |d3| and the functional of Fekete–Szegö for functions in this subfamily. Relevant connections to existing results and new observations of the main result are also presented.


Preliminaries
Let R and N :� 1, 2, 3, . . . { } � N 0 \ 0 { } be the sets of real numbers and positive integers, respectively. Let C be the set of all complex numbers, and let D denote the disc z ∈ C: |z| < 1 { }. We denote by A, the set of all regular functions in D that has the series of the form and S be the set of all members of A that are univalent in D.
A function g of A is said to be bi-univalent (or bischlicht) in D if both g and g − 1 are univalent in D. Let Σ stand for the set of bi-univalent functions having form (1). Lewin [2] investigated the family Σ and proved that |d 2 | < 1.51. Brannan and Clunie [3] claimed that |d 2 | < � 2 √ . Later, Tan [4] obtained initial coefficient estimates for biunivalent functions. Subsequently, Brannan and Taha [5] examined certain well-known subfamilies of Σ in D. e momentum on the study of bi-univalent function family was gained recently, which is due to the work of Srivastava et al. [6]. is article has revived the topic apparently, and many researchers have investigated several interesting special families of Σ (see [7][8][9][10]).
Recently, Hörçum and Koçer [11] (see also Horadam and Mahon [12]) examined the Horadam polynomials H j (x) (or H j (x, a, b; p, q)), which is defined by the recurrence relation where j ∈ N 1, 2 { }, x ∈ R, p, q, a, and b are real constants. It is seen from (3) that H 3 (x) � pbx 2 + qa. e generating function of the sequence H j (x), j ∈ N, is as follows (see [11]): where z ∈ C is such that R(z) ≠ x, x ∈ R. Few particular cases of H j (x, a, b; p, q) are The Pell polynomials, P j (x) � H j (x, 1, 2; 2, 1) e estimates on |d 2 | and |d 3 | and the very popular Fekete-Szegö functional were determined for bi-univalent functions linked with certain polynomials like Lucas polynomials, Fibonacci polynomials, Chebyshev polynomials, Horadam polynomials, and Gegenbauer polynomials. It is well-known that these polynomials play a potentially important role in architecture, approximation theory, physics, statistics, mathematical, and engineering sciences. e recent research trend is the study of functions in Σ linked with any of the abovementioned polynomials. Generally, interest was shown to obtain the initial coefficient bounds and the celebrated inequality of Fekete-Szegö for the special subfamilies of Σ. Recently, the Horadam polynomial was used by Abirami et al. [13] to find coefficient estimates for the families of bi-Bazilevic and λ-bistarlike function, Frasin et al. [14] obtained coefficient estimates and Fekete-Szegö inequalities for certain subfamilies of Al-Oboudi-type bi-univalent functions related to k-Fibonacci numbers involving modified activation function, initial coefficient bounds for certain subsets of biunivalent functions family subordinate to Horadam polynomials were obtained in [15,16], Shaba and Wanas [17] obtained coefficient bounds which are sharp, for a family of bi-univalent functions using (U, V)-Lucas polynomials, Srivastava et al. [18] have proposed a methodology to estimate coefficient bounds and Fekete-Szegö problem for certain subsets of bi-univalent function family linked with Horadam polynomials, and Swamy [19] and Swamy et al. [20,21] have initiated the study of some subfamilies of bi-univalent function family subordinate to Horadam polynomials involving modified activation function. Swamy and Sailaja [22] have used Horadam polynomials to investigate coefficient estimates for two families of bi-univalent functions, Swamy et al. [23] have introduced some subfamilies of Sȃlȃgean type biunivalent functions subordinate to (m, n)-Lucas polynomials and found initial coefficients, and Wanas and Alina [24] have fixed the Fekete-Szegö problem for Bazilevic biunivalent function class linked with Horadam polynomials.
For functions g and f holomorphic in D, g is said to subordinate f, if there is a Schwarz function ψ in D, such that ψ(0) � 0, |ψ(z)| < 1, and g(z) � f(ψ(z)), z ∈ D. is subordination is indicated as g≺f. In particular, if f ∈ S, then g(z)≺f(z) is equivalent to g(0) � f(0) and g(D) ⊂ f(D).
Inspired by the article [25] and the recent trends on functions in Σ, we present a comprehensive family of Σ associated with Horadam polynomials H j (x) as in (3) having the generating function (4). roughout this paper, the inverse function g − 1 (ω) � f(ω) is as in (2) and G(x, z) is as in (4).
Journal of Function Spaces

Bi-Univalent Function Class SS τ Σ (x, γ, μ)
We determine the initial coefficients bounds and the inequality of Fekete-Szegö for functions in SS τ Σ (x, c, μ), in the following theorem.