Fractional q-Starlikeness in Mittag–Leffler and Limaçon Domains

Abstract

In this paper, we use a normalized fractional q-derivative operator, constructed by combining Jackson’s q-difference operator with a fractional q-integral, to introduce two subclasses of normalized analytic functions in the open unit disk whose fractional transforms are starlike and univalent. The first subclass, Sq,λ*(ML), is defined by subordination to a normalized q-Mittag–Leffler function, whereas the second, Sq,λ*(L), is associated with a generalized limaçon domain. For both classes, we investigate sharp initial coefficient bounds and sharp estimates for the Fekete–Szegö functional. We also study inclusion and radius relations between the two classes. This comparative approach clarifies how an analytically defined Mittag–Leffler domain and a geometrically defined limaçon domain affect coefficient estimates when the underlying operator is held fixed. Relevant classical and quantum calculus results are recovered through suitable parameter choices and in the limiting case q→1−.