This study aims to construct some new Milne-type integral inequalities for functions whose modulus of the local fractional derivatives is convex on the fractal set. To that end, we develop a novel generalized integral identity involving first-order generalized derivatives. Finally, as applications, some error estimates for the Milne-type quadrature formula and new inequalities for the generalized arithmetic and p-Logarithmic means are derived. This paper’s findings represent a significant improvement over previously published results. The paper’s ideas and formidable tools may inspire and motivate further research in this worthy and fascinating field.

Building upon previous research in conformable fractional calculus, this study introduces a novel identity. Using this identity as a foundation, we derive a set of conformable fractional Milne-type inequalities specifically designed for differentiable convex functions. The obtained results recover some existing inequalities in the literature by fixing some parameters. These novel contributions aim to enrich the analytical tools available for studying convex functions within the realm of conformable fractional calculus. The derived inequalities reflect an inherent symmetry characteristic of the Milne formula, further illustrating the balanced and harmonious mathematical structure within these frameworks. We provide a thorough example with graphical representations to support our findings, offering both numerical insights and visual confirmation of the established inequalities.

This paper introduces new versions of Hermite–Hadamard, midpoint- and trapezoid-type inequalities involving fractional integral operators with exponential kernels. We explore these inequalities for differentiable convex functions and demonstrate their connections with classical integrals. This paper validates the derived inequalities through a numerical example with graphical representations and provides some practical applications, highlighting their relevance to special means. This study presents novel results, offering new insights into classical integrals as the fractional order $\beta$ approaches 1, in addition to the fractional integrals we examined.

In the present work, we introduce two new local fractional integral operators involving Mittag–Leffler kernel on Yang’s fractal sets. Then, we study the related generalized Hermite–Hadamard-type inequality using generalized $(E,h)$-convexity and obtain two identities pertaining to these operators, and the respective first- and second-order derivatives are given. In terms of applications, we provide some new generalized trapezoid-type inequalities for generalized ($E,h$)-convex functions. Finally, some special cases are deduced for different values of $\delta$, E, and h.

The present paper provides several corrected dual-Simpson-type inequalities for functions whose local fractional derivatives are generalized convex. To that end, we derive a new local fractional integral identity as an auxiliary result. Using this new identity along with generalized Hölder’s inequality and generalized power mean inequality, we establish some new variants of fractal corrected dual-Simpson-type integral inequalities. Furthermore, some applications for error estimates of quadrature formulas as well as some special means involving arithmetic and p-logarithmic mean are offered to demonstrate the efficacy of our findings.

This paper presents a novel framework for Katugampola fractional multiplicative integrals, advancing recent breakthroughs in fractional calculus through a synergistic integration of multiplicative analysis. Motivated by the growing interest in fractional calculus and its applications, we address the gap in generalized inequalities for multiplicative s-convex functions by deriving a Hermite–Hadamard-type inequality tailored to Katugampola fractional multiplicative integrals. A cornerstone of our work involves the derivation of two groundbreaking identities, which serve as the foundation for midpoint- and trapezoid-type inequalities designed explicitly for mappings whose multiplicative derivatives are multiplicative s-convex. These results extend classical integral inequalities to the multiplicative fractional calculus setting, offering enhanced precision in approximating nonlinear phenomena.