Lemniscate Starlikeness and Convexity for the Generalized Marcum Q-Function

In this paper, we investigate new geometric properties of normalized analytic functions associated with the generalized Marcum Q-function. In particular, we focus on two analytic forms derived from a normalized derivative of a representation involving the Marcum Q-function, and its Alexander transform. For these functions, we establish sufficient conditions ensuring membership in the classes of lemniscate starlike and lemniscate convex functions. Special attention is given to the case $\nu =1$, where explicit admissible parameter ranges for b are derived. We further examine inclusion relations between these normalized analytic forms and lemniscate subclasses, complemented by several corollaries, illustrative examples, and graphical visualizations. These results extend and enrich the geometric function theory of special functions related to the generalized Marcum Q-function.