Error Bound of Hermite–Hadamard–Mercer Inequalities via ψ−Conformable Fractional Integral Operators on the Coordinates

This paper develops a family of Hermite–Hadamard–Mercer-type inequalities within the framework of generalized conformable fractional calculus. By interpreting generalized conformable fractional integrals as weighted integral means depending on the order parameter, we derive refined two-sided bounds for coordinated convex functions that incorporate Mercer-type improvements inspired by Jensen–Mercer structures. The proposed estimates reduce to the classical Hermite–Hadamard–Mercer inequalities when the fractional order equals one, and they recover several known Hermite–Hadamard-type bounds under suitable parameter choices and assumptions. In addition, we present weighted (Fejér-type) variants by introducing admissible weight functions, which enhances applicability in settings where nonuniform densities naturally arise. Overall, the conformable fractional Hermite–Hadamard–Mercer framework provides a unified and flexible approach for integral estimation problems in coordinated convex analysis.