On k ‐Riemann–Liouville Maclaurin‐Type Inequalities for s ‐Convex Stochastic Processes

Source abstract

ABSTRACT In this paper, we introduce a new class of stochastic fractional integrals, namely, the stochastic mean‐square ‐Riemann–Liouville fractional integrals, by combining elements of fractional calculus and stochastic analysis. We first establish the necessary theoretical framework by recalling fundamental concepts from both domains. A novel integral identity involving these operators is then derived, which serves as a key tool for our main results. Using this identity, we prove several ‐fractional Maclaurin‐type inequalities for differentiable ‐convex stochastic processes. These inequalities extend classical deterministic results to the stochastic setting and generalize them via ‐fractional operators, offering enhanced flexibility in modeling uncertainty and memory effects. The obtained results contribute to the growing theory of stochastic fractional analysis and provide new tools for the study of probabilistic bounds and convexity in random environments.