New stochastic fractional integral and related inequalities of Jensen–Mercer and Hermite–Hadamard–Mercer type for convex stochastic processes

In this investigation, we unfold the Jensen–Mercer (J− M) inequality for convex stochastic processes via a new fractional integral operator. The incorporation of convex stochastic processes, the J− M inequality and a fractional integral operator having an exponential kernel brings a new direction to the theory of inequalities. With this in mind, estimations of Hermite–Hadamard–Mercer (H− H− M)-type fractional inequalities involving convex stochastic processes are presented. In the context of the new fractional integral operator, we also investigate a novel identity for differentiable mappings. Then, a new related H− H− M-type inequality is presented using this identity as an auxiliary result. Applications to special means and matrices are also presented. These findings are particularly appealing from the perspective of optimization, as they provide a larger context to analyze optimization and mathematical programming problems.