Weighted Milne-type inequalities through Riemann-Liouville fractional integrals and diverse function classes

Areej A Almoneef; Abd-Allah Hyder; Hüseyin Budak; Areej A Almoneef; Abd-Allah Hyder; Hüseyin Budak

doi:10.3934/math.2024898

AIMS Mathematics

2024, Volume 9, Issue 7: 18417-18439. doi: 10.3934/math.2024898

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Research article

Weighted Milne-type inequalities through Riemann-Liouville fractional integrals and diverse function classes

1.
Department of Mathematical Sciences, College of Science, Princess Nourah Bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabi
2.
Department of Mathematics, College of Science, King Khalid University, P.O. Box 9004, Abha 61413, Saudi Arabia
3.
Department of Mathematics, Faculty of Science and Arts, Düzce University, Düzce, Türkiye

Received: 13 March 2024 Revised: 17 April 2024 Accepted: 08 May 2024 Published: 03 June 2024
MSC : 26D07, 26D10, 26D15

This research paper investigated weighted Milne-type inequalities utilizing Riemann-Liouville fractional integrals across diverse function classes. A key contribution lies in the establishment of a fundamental integral equality, facilitated by the use of a nonnegative weighted function, which is pivotal for deriving the main results. The paper systematically proved weighted Milne-type inequalities for various function classes, including differentiable convex functions, bounded functions, Lipschitzian functions, and functions of bounded variation. The obtained results not only contribute to the understanding of Milne-type inequalities but also offer insights that pave the way for potential future research in the considered topics. Furthermore, it is evident that the results obtained encompass numerous findings that were previously presented in various studies as special cases.
- weighted Milne-type inequalities,
- Riemann-Liouville fractional integrals,
- nonnegative weighted function,
- differentiable convex functions
Citation: Areej A Almoneef, Abd-Allah Hyder, Hüseyin Budak. Weighted Milne-type inequalities through Riemann-Liouville fractional integrals and diverse function classes[J]. AIMS Mathematics, 2024, 9(7): 18417-18439. doi: 10.3934/math.2024898

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Abstract

This research paper investigated weighted Milne-type inequalities utilizing Riemann-Liouville fractional integrals across diverse function classes. A key contribution lies in the establishment of a fundamental integral equality, facilitated by the use of a nonnegative weighted function, which is pivotal for deriving the main results. The paper systematically proved weighted Milne-type inequalities for various function classes, including differentiable convex functions, bounded functions, Lipschitzian functions, and functions of bounded variation. The obtained results not only contribute to the understanding of Milne-type inequalities but also offer insights that pave the way for potential future research in the considered topics. Furthermore, it is evident that the results obtained encompass numerous findings that were previously presented in various studies as special cases.

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