Probabilistic Mean-Square Extensions of Fractional Hermite–Hadamard–Mercer Inequalities with Applications to Distortion Models and Special Functions

This paper develops probabilistic mean-square extensions of fractional Hermite–Hadamard–Mercer (HHM) type inequalities for convex stochastic processes. By employing generalized mean-square stochastic fractional integral operators, we establish new fractional inequalities that extend deterministic HHM estimates to a stochastic framework. The stochastic inequalities are interpreted in the almost sure sense, while the associated fractional operators are considered within the mean-square setting for second-order stochastic processes. Several special cases are also discussed, showing that the obtained results reduce to known fractional and stochastic inequalities for suitable choices of the parameters. As analytical consequences, the proposed results are applied to two-variable means, modified Bessel functions and the k-Digamma function. To strengthen the applied interpretation, we also present a stochastic nonlinear conductivity distortion model in which the effective conductivity is represented by a positive convex stochastic process. The corresponding heat-conduction setting, heat-flux interpretation and Monte Carlo illustrations show how the derived bounds can be used to estimate fractional stochastic averaging errors under nonlinear conductivity distortion. The numerical plots are presented as illustrative demonstrations of the behaviour of the theoretical bounds under admissible parameters.