An innovative perspective on fractional inequalities through fractional operators and extended convexity

Muhammad Imran; Ahsan Mehmood; Shahid Mubeen; Muhammad Samraiz; Ishtiaq Ali; Muhammad Imran; Ahsan Mehmood; Shahid Mubeen; Muhammad Samraiz; Ishtiaq Ali

doi:10.3934/math.2026161

AIMS Mathematics

2026, Volume 11, Issue 2: 4008-4042. doi: 10.3934/math.2026161

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An innovative perspective on fractional inequalities through fractional operators and extended convexity

1.
Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan; imranmath84@gmail.com, muhammad.samraiz@uos.edu.pk
2.
School of Mathematical Sciences, Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, East China Normal University, 500 Dongchuan Road, Shanghai 200241, China; mehmoodahsan154@gmail.com
3.
Department of Mathematics, Baba Guru Nanak Univeristy, Nankana Sahib, Pakistan; smjhanda@gmail.com
4.
Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al-Ahsa 31982, Saudi Arabia; iamirzada@kfu.edu.sa

Received: 02 November 2025 Revised: 27 December 2025 Accepted: 28 January 2026 Published: 09 February 2026
MSC : 26A33, 26D10, 35J05

The theory of integral inequalities is significantly advanced by the relationship between fractional calculus and convexity. The current study used extended convex functions and Katugampola fractional operators to derive Hermite-Hadamard, Fejér-Hermite-Hadamard-type inequalities as well as a few other fractional integral inequalities. We used two extended-type convex functions: log-convex and exponentially trigonometric convex functions. Our conclusions were supported by tabular data and graphical representations, which offer numerical and visual validation of the explored results. This study improves mathematical analysis and expands the scope of these inequalities by emphasizing their applicability across different forms of convexity. The knowledge acquired is highly valuable for theoretical investigation as well as real-world applications in a variety of scientific fields.
- log-convex function,
- exponentially trigonometric convex function,
- Katugampola fractional integrals,
- Hölder's inequality,
- Hermite-Hadamard inequalities
Citation: Muhammad Imran, Ahsan Mehmood, Shahid Mubeen, Muhammad Samraiz, Ishtiaq Ali. An innovative perspective on fractional inequalities through fractional operators and extended convexity[J]. AIMS Mathematics, 2026, 11(2): 4008-4042. doi: 10.3934/math.2026161

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Abstract

The theory of integral inequalities is significantly advanced by the relationship between fractional calculus and convexity. The current study used extended convex functions and Katugampola fractional operators to derive Hermite-Hadamard, Fejér-Hermite-Hadamard-type inequalities as well as a few other fractional integral inequalities. We used two extended-type convex functions: log-convex and exponentially trigonometric convex functions. Our conclusions were supported by tabular data and graphical representations, which offer numerical and visual validation of the explored results. This study improves mathematical analysis and expands the scope of these inequalities by emphasizing their applicability across different forms of convexity. The knowledge acquired is highly valuable for theoretical investigation as well as real-world applications in a variety of scientific fields.

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