The fractional integral of the generalized Marcum Q-function and its geometric properties

In this paper, we investigate geometric function theoretic properties of a normalized fractional integral operator constructed from the Riemann–Liouville fractional integral combined with the generalized Marcum Q-function. In the first part, we establish sufficient conditions under which the operator is starlike or convex of order \(\alpha \) in the unit disk. In the second part, we study convolution properties of the operator, providing conditions under which it maps functions in the class \(\mathcal {R}(\beta )\) into \(\mathcal {R}(\sigma )\) and belongs to the Hardy space \(\mathcal {H}^{\infty }(\mathbb {D})\) . The analysis relies on techniques from differential subordination, admissible functions, and classical tools from geometric function theory, highlighting the rich interplay between fractional calculus and the generalized Marcum Q-function.