Error estimates for perturbed Milne-type inequalities by twice-differentiable functions using conformable fractional integrals

Abstract

Abstract Milne’s inequality provides an upper bound for the error in definite integral approximations using Milne’s rule, making it a useful tool for evaluating the rule’s precision. For this reason, this inequality is widely applied in engineering, physics, and applied mathematics. Additionally, conformable fractional integral operators establish a stronger relationship between classical and fractional calculus, enhancing the modeling, analysis, and resolution of complex problems. Therefore, we focus on the study of conformable fractional integral operators and Milne-type inequalities, which have significant applications in various fields. In this study, we first obtain an integral identity involving conformable fractional integral operators and twice-differentiable functions. Building on this new identity, we develop various perturbed Milne-type integral inequalities for twice-differentiable convex functions. We also validate them numerically through examples, computational analysis, and visual representations. In conclusion, it is evident that our findings significantly enhance and expand upon prior findings regarding integral inequalities. In addition to improving the scope of previous discoveries, the obtained results offer meaningful approaches and methods for tackling mathematical and scientific issues.

Keywords

• Milne-type integral inequalities
• Convex functions
• Riemann–Liouville fractional integrals
• Conformable fractional integrals