LAUSR

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 $$\phi (s)= \frac{2}{1+e^{-s} },\,s\ge 0$$ ϕ ( s ) = 2 1 + e - s , s ≥ 0 in the open unit disc $${\mathfrak {D}}$$ D . We investigate the upper bounds on initial coefficients for functions of the form $$g_{\phi }(z)=z+\sum \nolimits _{j=2}^{\infty }\phi (s)d_jz^j$$ g ϕ ( z ) = z + ∑ j = 2 ∞ ϕ ( s ) d j z 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.