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Fractal Fract
Volume 9
Issue 8
10.3390/fractalfract9080516
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Article

Exploring Advanced Weighted Integral Inequalities via Extended Fractional Calculus Approaches

by
Areej A. Almoneef
1,*,
Abd-Allah Hyder
2,*,
Hüseyin Budak
3,4 and
Mohamed A. Barakat
5
1
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P. O. Box 84428, Riyadh 11671, Saudi Arabia
2
Department of Mathematics, College of Science, King Khalid University, P.O. Box 9004, Abha 61413, Saudi Arabia
3
Department of Mathematics, Saveetha School of Engineering, SIMATS, Saveetha University, Chennai 602105, India
4
Department of Mathematics, Faculty of Science and Arts, Kocaeli University, Kocaeli 41001, Türkiye
5
Department of Basic Science, University College of Alwajh, University of Tabuk, Tabuk 71491, Saudi Arabia
*
Authors to whom correspondence should be addressed.
Fractal Fract. 2025, 9(8), 516; https://doi.org/10.3390/fractalfract9080516 (registering DOI)
Submission received: 3 June 2025 / Revised: 30 July 2025 / Accepted: 2 August 2025 / Published: 7 August 2025
(This article belongs to the Special Issue Mathematical Inequalities in Fractional Calculus and Applications, 2nd Edition)
Versions Notes

Abstract

This paper investigates weighted Milne-type (M−t) inequalities within the context of Riemann–Liouville (R−L) fractional integrals. We establish multiple versions of these inequalities, applicable to different function categories, such as convex functions with differentiability properties, bounded functions, functions satisfying Lipschitz conditions, and those exhibiting bounded variation behavior. In particular, we present integral equalities that are essential to establish the main results, using non-negative weighted functions. The findings contribute to the extension of existing inequalities in the literature and provide a deeper understanding of their applications in fractional calculus. This work highlights the advantage of the established inequalities in extending classical results by accommodating a broader class of functions and yielding sharper bounds. It also explores potential directions for future research inspired by these findings.
Keywords: weighted integral inequalities; convex analysis; boundedness conditions; Lipschitzian functions; functions of bounded variation weighted integral inequalities; convex analysis; boundedness conditions; Lipschitzian functions; functions of bounded variation

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MDPI and ACS Style

Almoneef, A.A.; Hyder, A.-A.; Budak, H.; Barakat, M.A. Exploring Advanced Weighted Integral Inequalities via Extended Fractional Calculus Approaches. Fractal Fract. 2025, 9, 516. https://doi.org/10.3390/fractalfract9080516

AMA Style

Almoneef AA, Hyder A-A, Budak H, Barakat MA. Exploring Advanced Weighted Integral Inequalities via Extended Fractional Calculus Approaches. Fractal and Fractional. 2025; 9(8):516. https://doi.org/10.3390/fractalfract9080516

Chicago/Turabian Style

Almoneef, Areej A., Abd-Allah Hyder, Hüseyin Budak, and Mohamed A. Barakat. 2025. "Exploring Advanced Weighted Integral Inequalities via Extended Fractional Calculus Approaches" Fractal and Fractional 9, no. 8: 516. https://doi.org/10.3390/fractalfract9080516

APA Style

Almoneef, A. A., Hyder, A.-A., Budak, H., & Barakat, M. A. (2025). Exploring Advanced Weighted Integral Inequalities via Extended Fractional Calculus Approaches. Fractal and Fractional, 9(8), 516. https://doi.org/10.3390/fractalfract9080516

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