Variable-Order Postquantum Fractional Integral Inequalities With Nonuniform Memory

Variable memory effects are essential for accurately modeling nonlocal phenomena in applied mathematics, physics, and engineering. Motivated by this observation, we develop a new class of variable-order postquantum fractional integral inequalities based on the Riemann–Liouville (RL)-type p,q-fractional integral operator with a q-shifting structure. Unlike the existing results restricted to constant fractional orders, the proposed framework allows the order to vary with the independent variable, enabling the description of nonuniform and evolving memory effects. A postquantum variable-order multiparameter fundamental identity on finite intervals is established, from which new generalized Hermite–Hadamard-, midpoint-, trapezoidal-, Simpson-, and Bullen-type inequalities are derived. The results are developed under several generalized convexity assumptions, including convex and α,m-convex functions, thereby extending the applicability of the theory. Many known constant-order postquantum inequalities are recovered as special cases. Numerical simulations, graphical illustrations, and an application to special means are presented to demonstrate the effectiveness of the proposed framework.