On Fractal–Fractional Simpson-Type Inequalities: New Insights and Refinements of Classical Results

Fractal–Fractional Simpson-Type Inequalities play a significant role in the fields of energy and industrial leadership by providing advanced mathematical tools to model and analyze complex systems. These inequalities extend traditional approaches to numerical integration and optimization, allowing for more accurate approximations in energy production, resource allocation, and industrial processes. By integrating fractal and fractional concepts, they can address irregularities and complexities often found in real-world data, enhancing decision-making, efficiency, and sustainability in industrial operations and energy management. Their application helps optimize resource utilization, improve predictive modeling, and foster innovation in technology development.
In this paper, we introduce a novel fractal–fractional identity, from which we derive new Simpson-type inequalities for functions whose first-order local fractional derivative exhibits generalized s-convexity in the second sense. This work introduces an approach that uses the first-order local fractional derivative, enabling the treatment of functions with lower regularity requirements compared to earlier studies. Additionally, we present two distinct methodological frameworks, one of which achieves greater precision by refining classical outcomes in the existing literature. The paper concludes with several practical applications that demonstrate the utility of our results.