Applications of Mittag-Leffler functions in computing fractional integral inequalities

Hala H. Taha; Josip Pečarić; Jongsuk Ro; Ghulam Farid; Hala H. Taha; Josip Pečarić; Jongsuk Ro; Ghulam Farid

doi:10.3934/math.20261203

AIMS Mathematics

2026, Volume 11, Issue 9: 30355-30373. doi: 10.3934/math.20261203

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Applications of Mittag-Leffler functions in computing fractional integral inequalities

1.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
2.
Croatian Academy of Sciences and Arts, Zagreb, Croatia
3.
School of Electrical and Electronics Engineering, Chung-Ang University, Dongjak-gu, Seoul 06974, Korea
4.
Department of Intelligent Energy and Industry, Chung-Ang University, Dongjak-gu, Seoul 06974, Korea
5.
Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Pakistan

Received: 27 May 2026 Revised: 17 July 2026 Accepted: 28 August 2026 Published: 17 September 2026
MSC : 26A24, 26A33, 26A51, 26B15, 33E12

This paper develops fractional integral inequalities generated by integral operators whose kernels contain an extended generalized Mittag-Leffler function. Under two-sided derivative bounds, we establish Ostrowski-type, Ostrowski-Grüss-type, and Hermite-Hadamard-type estimates, together with midpoint consequences and error bounds. The contribution is not limited to a formal replacement of the classical power kernel: The kernel parameters enter the normalization, the fractional means, and the derivative-controlled remainder terms simultaneously, thereby producing a unified family of estimates. A concrete application to a fractional relaxation equation illustrates how the results provide computable bounds for the discrepancy between a pointwise state and its Mittag-Leffler-weighted fractional mean. Classical Riemann-Liouville inequalities are recovered through suitable parameter choices.
- Ostrowski inequality,
- Hadamard inequality,
- Grüss inequality,
- bounds
Citation: Hala H. Taha, Josip Pečarić, Jongsuk Ro, Ghulam Farid. Applications of Mittag-Leffler functions in computing fractional integral inequalities[J]. AIMS Mathematics, 2026, 11(9): 30355-30373. doi: 10.3934/math.20261203

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Abstract

This paper develops fractional integral inequalities generated by integral operators whose kernels contain an extended generalized Mittag-Leffler function. Under two-sided derivative bounds, we establish Ostrowski-type, Ostrowski-Grüss-type, and Hermite-Hadamard-type estimates, together with midpoint consequences and error bounds. The contribution is not limited to a formal replacement of the classical power kernel: The kernel parameters enter the normalization, the fractional means, and the derivative-controlled remainder terms simultaneously, thereby producing a unified family of estimates. A concrete application to a fractional relaxation equation illustrates how the results provide computable bounds for the discrepancy between a pointwise state and its Mittag-Leffler-weighted fractional mean. Classical Riemann-Liouville inequalities are recovered through suitable parameter choices.

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