On Improved Simpson‐Type Inequalities via Convexity and Generalized Fractional Operators

In this work, we develop novel Simpson‐type inequalities for mappings with convex properties by employing operators for tempered fractional integrals. These findings expand upon and refine classical results, including those linked to Riemann–Liouville fractional integrals. Using methodologies such as Hölder’s inequality, the power‐mean inequality, and convex function properties, we derive precise bounds for these inequalities. The main contributions include the derivation of Simpson‐type inequalities under various convexity conditions and their adaptations for specific cases, such as functions with bounded derivatives and Lipschitz continuity. Special cases, where these inequalities reduce to classical results involving standard integrals, are also explored. Explicitly, clear connections to classical integral inequalities are established for differentiable functions whose derivatives satisfy convexity, boundedness, and Lipschitz conditions. Additionally, future research directions are proposed, emphasizing the broad applicability of these results in fractional calculus and convex analysis.