Abstract

The aim of the present paper is to introduce some special families of holomorphic and Al-Oboudi type bi-univalent functions associated with $(m,n)$-Lucas polynomials involving 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-Szeg\"o problem. Some interesting consequences of the results established here are also indicated.

Keywords and Phrases

Holomorphic function, Bi-univalent function, Fekete - Szeg\"o inequality, $(m,n)$-Lucas polynomials, Modified sigmoid function.

A.M.S. subject classification

Primary : 30C45, 30C50 Secondary: 11B39.

.....