For functions of the form ϕ(ξ)=ξ+∑n=2∞cnξn, we identified two new subclasses of bi-starlike functions and bi-convex functions by using Mathieu-type series defined in the disc Δ={ξ∈C:|ξ|1}. We derived constraints for |c2| and |c3|, and the subclasses are connected to the shell-shaped area. The Fekete–Szegö functional properties for the aforementioned function subclasses were also investigated. Additionally, a number of related corollaries are shown.
