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

Abstract

Using the Al-Oboudi type operator, we present and investigate two special families of bi-univalent functions in $\mathfrak{D}$, an open unit disc, based on $\phi(s)=\frac{2}{1+e^{-s} },\,s\geq0$, a modified sigmoid activation function (MSAF) and Horadam polynomials. We estimate the initial coefficients bounds for functions of the type $g_{\phi}(z)=z+\sum\limits_{j=2}^{\infty}\phi(s)d_jz^j$ 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.

Bi-univalent Function Subclasses Subordinate to Horadam Polynomials

“…Srivastava et al [23] have proposed certain subfamilies of σ subordinate to Horadam polynomials, Frasin et al [12] examined a comprehensive subfamily of σ associated with k-Fibonacci numbers, a comprehensive subfamily of σ subordinate to Horadam polynomials was investigated by Shammaky et al in [21], Swamy and Wanas [29] have introduced a subfamily of σ subordinate to (m, n)-Lucas polynomials, Swamy and Yalçin [30], initiated two subclasses of σ linked with Gegenbauer polynomials, Wanas et al [33] explored a comprehensive subfamily of σ making use of Gegenbauer polynomials, Horadam polynomials were used by Wanas and Lupas [34] to define Bazilevic bi-univalent function class, Frasin et al [11] investigated coefficient bounds for a subfamily of σ defined by Horadam polynomials and so on. Coefficient related investigations for elements of certain subclasses of σ linked with any of the aforementioned polynomials and a modified sigmoid function appeared like the ones published in [2,26,27].…”

Coefficient bounds for regular and bi-univalent functions linked with Gegenbauer polynomials

“…The recent research trends are the outcomes of the study of functions ∈ Σ linked with any of the above mentioned polynomials, can be seen in [4], [20], [24], [29], [30], [33] and [32]. Generally interest was shown to estimate the initial Taylor-Maclaurin coefficients and the celebrated inequality of Fekete-Szegö for the special subfamilies of Σ.…”