Some novelty integral inequalities on fractal sets using generalized convex functions and their applications

Abstract

Classical Hermite-Hadamard inequalities, which provide bounds for the integral means of convex functions, encounter limitations when applied to fractal domains due to the non-differentiable and irregular nature of such spaces. This work addresses these limitations by extending Hermite-Hadamard-type inequalities using Yang’s local fractional calculus in conjunction with the generalized Beta function. We derive new bounds for integral means via fractal-specific integration techniques and convex function theory, presenting trapezoidal and midpoint inequalities that generalize classical H\"{o}lder and Young inequalities within a fractal framework. Notably, the proposed results reduce to their classical counterparts when the fractal parameter $\varpi = 1$, ensuring consistency with standard calculus. Applications to special means and Mittag-Leffler functions highlight the effectiveness of the results in modeling anomalous diffusion, material heterogeneity, and complex dynamical behavior. Graphical illustrations confirm the adaptability and robustness of the proposed inequalities under varying fractal dimensions, showcasing practical implications for modeling non-smooth phenomena. Furthermore, the integration of fractal calculus and convex analysis offers a robust mathematical foundation for analyzing irregular systems across disciplines such as physics, engineering, and data science.