In the current work, we introduce a special family of the function family of analytic and m-fold symmetric bi-univalent functions and obtain estimates of the Taylor–Maclaurin coefficients $|d_{m+1}|$ and $|d_{2m+1}|$ for functions in the special family. For $\delta$ a real number, Fekete–Szegö functional $|d_{2m+1}-\delta d_{m+1}^2|$ for functions in the special family is also estimated. We indicate several cases of the defined family and connections to existing results are also discussed.

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

In this paper we consider two subclasses of bi-univalent functions defined by the Horadam polynomials. Further, we obtain coefficient estimates for the defined classes.

Making use of Horadam polynomials, we propose a special family of regular functions of the type which are bi-univalent (or bi-schlicht) in the disc . We find estimates on the coefficients and 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.

By using -Fibonacci numbers, we present a comprehensive family of regular and biunivalent functions of the type in the open unit disc . We estimate the upper bounds on initial coefficients and also the functional of Fekete-Szegö for functions in this family. We also discuss few interesting observations and provide relevant connections of the result investigated.