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

Saima Rashid (), Zeemal Zia (), Muhammad Kashif Iqbal () and Miguel Vivas-Cortez
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Saima Rashid: Department of Mathematics, Government College University, Faisalabad 38000, Pakistan
Zeemal Zia: Department of Mathematics, Government College University, Faisalabad 38000, Pakistan
Muhammad Kashif Iqbal: Department of Mathematics, Government College University, Faisalabad 38000, Pakistan
Miguel Vivas-Cortez: ��Escuela de Ciencias Fisicas y Matematicas, Facultad de Ciencias Exactas y Naturales, Pontificia Universidad Catolica del Ecuador, Av. 12 de, Octubre 1076, Apartado, Quito 17-01-2184, Ecuador

FRACTALS (fractals), 2025, vol. 33, issue 06, 1-31

Abstract: 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, ChÄ›byshÄ›v, 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-RibariÄ , Young inequality and various other useful generalizations. Moreover, we obtain the resulting inequalities based on the following three criteria: (i) fixing fractional-order δ = 1, we attain new results for fractal integrals; (ii) fixing fractal-dimension λ = 1, we attain new results for Atangana–Baleanu–Riemann fractional integral operator; (iii) fixing fractional-order and fractal-dimension δ = λ = 1, 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.

Keywords: Proportional Delay Partial Differential Equations; Atangana–Baleanu Fractional Derivative in the Sense of Caputo; q̂-homotopy Analysis Method; Jafari Transform (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1142/S0218348X25401188

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