For functions of the form 𝜙(𝜉)=𝜉+∑∞𝑛=2𝑐𝑛𝜉𝑛 , 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.