FRACTIONAL- AND INTEGER-ORDER ANALYSIS OF m-SUPERQUADRATIC STOCHASTIC PROCESSES WITH APPLICATIONS IN INFORMATION THEORY

In this study, we introduce and rigorously develop the concept of [Formula: see text]-superquadratic stochastic processes, a novel generalization and refinement of classical superquadratic stochastic processes. We systematically investigate their core structural properties and employ these to establish advanced forms of Jensen’s inequality and Hermite–Hadamard ([Formula: see text])-type inequalities within the framework of mean-square stochastic calculus. These inequalities are further extended to their fractional analogs via the stochastic Riemann–Liouville ([Formula: see text]) fractional integrals, providing a deeper analytical toolkit for fractional stochastic analysis. The theoretical results are substantiated through comprehensive graphical visualizations and detailed tabular representations, which are constructed from diverse illustrative examples. Additionally, we demonstrate the applicability of the proposed framework in information theory by formulating new classes of stochastic divergence measures. For reproducibility and computational transparency, we provide direct access to the commands used for generating all graphs and tables, along with the recorded execution times for each computation.