Some special families of holomorphic and Al-Oboudi type bi-univalent functions related to k-Fibonacci numbers involving modified Sigmoid activation function

The aim of the present paper is to introduce some special families of holomorphic and Al-Oboudi type bi-univalent functions related to k-Fibonacci numbers involving modified Sigmoid activation function $$\phi (s)= \frac{2}{1+e^{-s} },\,s\ge 0$$ in the open unit disc $${\mathfrak {D}}$$ . We investigate the upper bounds on initial coefficients for functions of the form $$g_{\phi }(z)=z+\sum \nolimits _{j=2}^{\infty }\phi (s)d_jz^j$$ , in these newly introduced special families and also discuss the Fekete–Szego problem. Some interesting consequences of the results established here are also indicated.