EXPLORING NEW GENERALIZED INEQUALITIES OF FRACTAL-FRACTIONAL INTEGRAL OPERATORS BY MEANS OF GENERALIZED MITTAG-LEFFLER KERNELS AND APPLICATION

Mathematical inequalities and fractional calculus are two important topics of mathematics that have numerous implications. The aim of this research is to introduce newly generalized inequalities, including Pólya–Szegö, Chěbyshěv, Grüss and various types of inequalities in terms of fractal-fractional operators containing generalized Mittag-Leffler kernels. With the aid of the fractal-fractional operator, we derive several other new results through the use of Jensen inequality for convex functions, Lah-Ribarič, Young inequality and various other useful generalizations. Moreover, we obtain the resulting inequalities based on the following three criteria: (i) fixing fractional-order [Formula: see text], we attain new results for fractal integrals; (ii) fixing fractal-dimension [Formula: see text] we attain new results for Atangana–Baleanu–Riemann fractional integral operator; (iii) fixing fractional-order and fractal-dimension [Formula: see text], we attain the results in the existing literature. Additionally, the applications and justifications for the provided outcomes are concisely reviewed, which generates novel estimates for bounding mappings that will be significant for further research in fractal theory and numerical analysis. Ultimately, the interaction of mathematical inequalities and fractal-fractional operators offers a rich foundation for analysis and application in many domains. Understanding these connections can lead to more deeper concepts and advances in both theoretical and applied mathematics.