On $q$-differential operators acting on bounded turning functions associated with epicycloidal curves

The $q$-calculus has found numerous applications in engineering, particularly in mass and heat transfer problems, as well as in the study of nonlinear and fuzzy differential equations arising in science and technology. The $q$-difference operator ${\mathfrak{D}}_q$ and related $q$-calculus operators play an important role in the development of this theory. In this paper, we introduce and investigate a new class of analytic functions associated with epicycloidal domains by means of subordination theory and a $q$-differential operator. Our primary objective is to characterize this class and establish several of its geometric properties in the unit disk $\mathbb{U}$. In particular, we study coefficient estimates, the Zalcman conjecture and its generalized form, the second and third Hankel determinants, the Krushkal inequality, and the Fekete–Szegö functional.