Some fractional integral inequalities involving extended Mittag-Leffler function with applications

Sabir Hussain; Rida Khaliq; Sobia Rafeeq; Azhar Ali; Jongsuk Ro; Sabir Hussain; Rida Khaliq; Sobia Rafeeq; Azhar Ali; Jongsuk Ro

doi:10.3934/math.20241689

AIMS Mathematics

2024, Volume 9, Issue 12: 35599-35625. doi: 10.3934/math.20241689

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Some fractional integral inequalities involving extended Mittag-Leffler function with applications

1.
Department of Mathematics, University of Engineering and Technology, 54890, Lahore, Pakistan
2.
Department of Basic Sciences and Humanities (Mathematics), MNS University of Engineering and Technology, 60600, Multan, Pakistan
3.
Department of Mathematics and Computer Science, Freie University Berlin, 14163, Berlin, Germany
4.
School of Electrical and Electronics Engineering, Chung-Ang University, Dongjak-gu, Seoul, 06974, Republic of Korea
5.
Department of Intelligent Energy and Industry, Chung-Ang University, Dongjak-gu, Seoul, 06974, Republic of Korea

Received: 12 September 2024 Revised: 08 December 2024 Accepted: 11 December 2024 Published: 23 December 2024
MSC : 33E12, 26D15, 26A33, 15A45, 33C10

Integral inequalities and the Mittag-Leffler function play a crucial role in many branches of mathematics and applications, including fractional calculus, mathematical physics, and engineering. In this paper, we introduced an extended generalized Mittag-Leffler function that involved several well-known Mittag-Leffler functions as a special case. We also introduced an associated generalized fractional integral to obtain some estimates for fractional integral inequalities of the Hermite-Hadamard and Hermite-Hadamard-Fejér types. This article offered several analytical tools that will be useful to anyone working in this field. To demonstrate the veracity of our findings, we offered a few numerical and graphical examples. A few applications of modified Bessel functions and unitarily invariant norm of matrices were also given.
Citation: Sabir Hussain, Rida Khaliq, Sobia Rafeeq, Azhar Ali, Jongsuk Ro. Some fractional integral inequalities involving extended Mittag-Leffler function with applications[J]. AIMS Mathematics, 2024, 9(12): 35599-35625. doi: 10.3934/math.20241689

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Abstract

Integral inequalities and the Mittag-Leffler function play a crucial role in many branches of mathematics and applications, including fractional calculus, mathematical physics, and engineering. In this paper, we introduced an extended generalized Mittag-Leffler function that involved several well-known Mittag-Leffler functions as a special case. We also introduced an associated generalized fractional integral to obtain some estimates for fractional integral inequalities of the Hermite-Hadamard and Hermite-Hadamard-Fejér types. This article offered several analytical tools that will be useful to anyone working in this field. To demonstrate the veracity of our findings, we offered a few numerical and graphical examples. A few applications of modified Bessel functions and unitarily invariant norm of matrices were also given.

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