New Generalizations of Fractional Iyengar-Type Inequalities Involving Multi-Point Quadratures and $L_p$ Norms

Abstract

In this paper, we establish several new generalizations of Iyengar-type integral inequalities within the setting of Riemann-Liouville fractional calculus. By employing a fractional version of the Montgomery identity, we derive sharp estimates for the deviation between the fractional integral of a function and its discrete weighted averages at $n$ arbitrary nodes. We extend these results to functions whose derivatives belong to the $L_r[a,b]$ spaces by utilizing H\"{o}lder's inequality. The established inequalities provide a unified setting that recovers the classical Iyengar-type results as special cases when the fractional order $\alpha=1$. Moreover, the influence of the node distribution on the associated error bounds is examined, and several corollaries corresponding to midpoint and trapezoidal-type rules are derived.