A New Class of Center-Radius Order (P, 𝔪)-Superquadratic Interval-Valued Stochastic Processes with Fractional Integral Inequalities and Applications

In this study, we formulate and rigorously establish, for the first time, a novel and analytically enriched class of center-radius ordered [Formula: see text]-superquadratic interval-valued stochastic processes, referred to as cr-[Formula: see text]- superquadratic [Formula: see text]s, thereby providing a substantial advancement in the theoretical framework of interval stochastic analysis and contemporary fractional inequality theory, and present a comprehensive investigation of their fundamental structural properties. Within this newly developed framework, we derive re ned Jensen- and Hermite-Hadamard (H,H)-type inequalities under the center-radius total order relation. Furthermore, we derive their fractional analogues by employing mean- square stochastic Riemann-Liouville (R,L) fractional integral operators, thereby extending classical inequalities to a broader stochastic interval-valued setting. The sharpness and ap- plicability of the proposed findings are substantiated via carefully constructed tabular and graphical illustrations. In the application part, we introduce novel notions of stochastic divergence, stochastic H,H divergence, and R,L fractional stochastic H,H divergence within the framework of cr-[Formula: see text]-superquadratic [Formula: see text]s. In light of these formulations, we establish a series of new information-theoretic inequalities that underscore the theoretical significance and applicability of the proposed concepts, thereby engendering far-reaching generalizations and pronounced refinements of a broad spectrum of pre-existing results documented in the literature. The results presented herein not only enrich the theory of (interval-valued) [Formula: see text] stochastic processes but also provide a robust foundation for further developments in fractional stochastic analysis and information theory.