In this paper, we intorduce a family of analytic functions in the open unit disk which is bi-univalent. By virtue Faber polynomial expansions, estimation n?th(n?3) Taylor–Maclaurin coefficients an obtained. Furthermore, bounds value first two such established.

In the contemporary paper, we introduce new subclasses of analytic and bi-univalent functions involving integral operator based upon Lucas polynomial. Furthermore, find estimates on first two Taylor-Maclaurin coefficients a2 a3 for in these obtain Fekete-Szegö problem subclasses.

Three subclasses of analytic and bi-univalent functions are introduced through the use q?Gegenbauer polynomials, which a generalization Gegenbauer polynomials. For falling within these subclasses, coefficient bounds a2 a3 as well Fekete–Szegö inequalities derived. Specializing parameters used in our main results leads to number new results.

The aim of this paper is to introduce some special families holomorphic and S\u{a}l\u{a}gean type bi-univalent functions by making use Horadam polynomials involving the modified sigmoid activation function $\phi(s)=\frac{2}{1+e^{-s} },\,s\geq0$ in open unit disc $\mathfrak{D}$. We investigate upper bounds on initial coefficients for form $g_{\phi}(z)=z+\sum\limits_{j=2}^{\infty}\phi(s)d_jz^j$, ...