Coeﬃcient Bounds for Al-Oboudi Type Bi-univalent Functions based on a Modiﬁed Sigmoid Activation Function and Horadam Polynomials

Using the Al-Oboudi type operator, we present and investigate two special families of bi-univalent functions in D , an open unit disc, based on φ ( s ) = 2 1+ e − s , s ≥ 0, a modiﬁed sigmoid activation function (MSAF) and Horadam polynomials. We estimate the initial coeﬃcients bounds for functions of the type g φ ( z ) = z + ∞ (cid:80) j =2 φ ( s ) d j z j in these families. Continuing the study on the initial cosﬃcients of these families, we obtain the functional of Fekete-Szeg¨o for each of the two families. Furthermore, we present few interesting observations of the results investigated. functional, regular function, MSAF, bi-univalent function, Horodam polynomials.


Preliminaries
Let the set of complex numbers be denoted by C and the set of normalized regular functions in D = {z ∈ C : |z| < 1} that have the power series of the form be indicated by A and the set of all functions of A that are univalent in D is symbolized by S. The famous Koebe theorem (see [12]) ensures that any function They are named as Fibonacci polynomials, second type Chebyshev polynomials, first type Chebyshev polynomials, Lucas polynomials, Pell-Lucas polynomials and Pell polynomials, respectively.
The recent research trend is the study of bi-univalent functions linked with any one of the above mentioned polynomials using well-known operators, which can be seen in the research papers [4], [13], [25], [28], [34], [36], [37] and [39]. Generally interest was shown to estimate the initial Taylor-Maclaurin coefficients and the celebrated inequality of Fekete-Szegö for the special families of that are being introduced using known operators.
In this work, we present two special sets of using Al-Oboudi type operator which was precisely defined in the paper [19]. We determine the initial coefficient bounds and also obtain the relevant connection to the celebrated Fekete-Szegö functional for functions in the defined families.
For regular functions g and f in D, g is said to subordinate to 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. This subordination is indicated as Inspired by the articles [6], [33] and the trends on functions ∈ , we present two special families of by using Al-Oboudi type operator, which is as in Definition 1.1 and Horadam polynomials H j (x) as in the relation (1.3) having the generating function (1.4).
Throughout this paper, f φ (ω) = g −1 φ (ω) is an extension of g −1 to D given by (1.2), p, q, a and b are as in (1.3) and G is as in (1.4), unless and otherwise mentioned.
In view of (1.3), we conclude that , which gets (2.3) with J as in (2.4). This evidently ends the proof of Theorem 2.1.

Conclusion
Two special families of holomorphic and bi-univalent (or bi-schlicht) functions are introduced by using Al-Oboudi type operator involving a modified sigmoid activation function associated with Horadam polynomials. Bounds of the first two coefficients |d 2 |, |d 3 | and the celebrated Fekete-Szegö functional have been fixed for each of the two families. Through corollaries of our main results, we have highlighted many interesting new consequences.
The special families examined in this research paper using Al-Oboudi type operator could inspire further research related to other aspects such as families using q-derivative operator [22], [35], meromorphic bi-univalent function families associated with Al-Oboudi differential operator [30] and families using integro-differential operators [27].