Geometric Properties of New Subclasses of the (λ,q)-Fractional Differential Operator of Analytic Functions Associated with Euler Generating Function

In this paper, we introduce and investigate a new subclass of analytic functions in the open unit disk by combining the (λ,q)-fractional differintegral operator with the Euler generating function through the technique of subordination. This class, denoted by Kq,λ(τ), unifies and extends several families of q-starlike and q-convex functions as special cases corresponding to particular choices of the parameter λ. For this class, coefficient estimates for the initial Taylor coefficients and a Fekete–Szego inequality are established. Furthermore it is shown that the transformed function Dqλf is q-starlike in the unit disk under suitable assumptions on the parameters. In addition, first-order distortion estimates are obtained for the associated q-starlike and q-convex subclasses. Several consequences are derived for the special cases λ=0 and λ=1, corresponding, respectively, to q-starlike and q-convex function classes. The classical limit q→1− is also investigated and establish a new connection between Euler generating functions, fractional q-operators, and subclasses of analytic and univalent functions.