Bridging Pre-Invex Mappings and Fractional Integrals: A Pathway to Iterative Schemes via Error Boundaries of Maclaurin’s Rule

In this paper, we aim to investigate corrected Euler–Maclaurin inequalities involving pre-invex mappings within the framework of fractional calculus. We want to find a number of important results for differentiable pre-invex mappings and Riemann–Liouville (RL) fractional integrals so that we can make more accurate error estimates. Additionally, we present examples with graphical illustrations to substantiate our major findings and deduce several special cases under certain conditions. Afterwards, we introduce applications such as the linear combination of means, composite corrected Maclaurin’s rule, modified Bessel mappings, and novel iterative methods for solving nonlinear equations.