Fractional Gauss-Lobatto error bounds for multiplicatively twice-differentiable functions via multiplicative (mu,Mu)-Riemann-Liouville Integrals

This paper initiates the study by establishing Gauss-Lobatto-type identity for multiplicatively twice-differentiable [Formula: see text]-convex functions within the framework of multiplicative [Formula: see text]-Riemann-Liouville fractional integrals. Subsequently, we develop Gauss-Lobatto-type inequalities under the following two conditions: (i) [Formula: see text] is multiplicatively [Formula: see text]-convex, or (ii) [Formula: see text] is [Formula: see text]-convex [Formula: see text]. To substantiate the theoretical findings, we further provide numerical simulations and graphical representations that elucidate the practical implications of our analysis. Lastly, we highlight the utility of the established inequalities in the context of quadrature formulas and special means.