Sharp bounds of HH-Mercer inequalities for quadrature formulas in fractional calculus as part of numerical analysis

In this new study, sharp bounds for Hermite–Hadamard–Mercer inequalities are established in the setting of Riemann–Liouville fractional integrals. A modified and sharper version of the Jensen–Mercer inequality is used to establish three distinct error bounds for Hermite–Mercer inequalities. This modified version of the Jensen–Mercer inequality includes a correction factor on the right hand side, which is why it is sharper than the classical Jensen–Mercer inequality. Examples illustrate that the established error bounds for fractional Hermite–Hadamard inequalities, derived using the modified Jensen–Mercer inequality, are sharper than existing ones. This study examines for the first time the influence of the fractional parameter on the integral component of the Hermite–Hadamard inequality, specifically in terms of its convergence of lower and upper bounds. To demonstrate the application of these new results, sharp error bounds for midpoint and trapezoidal inequalities for single time differentiable convex functions are established within the framework of fractional calculus. The error bounds for midpoint and trapezoidal formulas are derived using the modified Jensen–Mercer inequality, providing sharper bounds due to the correction factor on the right hand side. Numerical examples are provided to validate exactness and efficiency of the new results. Finally, applications of the new results are presented for quadrature formulas.