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.

Keywords:

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

References

[1]	A. Wiman, Uber den fundamentalsatz in der teorie der funktionen $E_{\alpha}(x)$ , Acta Math., 29 (1905), 191–201. https://doi.org/10.1007/BF02403202 doi: 10.1007/BF02403202
[2]	E. M. Wright, On the coefficients of power series having exponential singularities, J. Lond. Math. Soc., 8 (1933), 71–79. https://doi.org/10.1112/jlms/s1-8.1.71 doi: 10.1112/jlms/s1-8.1.71
[3]	T. R. Prabhakar, A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J., 19 (1971), 7–15.
[4]	A. K. Shukla, J. C. Prajapati, On a generalization of Mittag-Leffler function and its properties, J. Math. Anal. Appl., 336 (2007), 797–811. https://doi.org/10.1016/j.jmaa.2007.03.018 doi: 10.1016/j.jmaa.2007.03.018
[5]	H. M. Srivastava, Z. Tomovski, Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel, Appl. Math. Comput., 211 (2009), 198–210. https://doi.org/10.1016/j.amc.2009.01.055 doi: 10.1016/j.amc.2009.01.055
[6]	T. O. Salim, A. W. Faraj, A generalization of Mittag-Leffler function and integral operator associated with fractional calculus, J. Fract. Calc. Appl., 3 (2012), 1–13.
[7]	M. Andric, G. Farid, J. Pecaric, A further extension of Mittag-Leffler function, Fract. Calc. Appl. Anal., 21 (2018), 1377–1395. https://doi.org/10.1515/fca-2018-0072 doi: 10.1515/fca-2018-0072
[8]	D. Bansal, K. Mehrez, On a new class of functions related with Mittag-Leffler and Wright functions and their properties, Commun. Korean Math. S., 35 (2020), 1123–1132. https://doi.org/10.4134/CKMS.c200022 doi: 10.4134/CKMS.c200022
[9]	R. K. Raina, On generalized Wright's hypergeometric functions and fractional calculus operators, East Asian Math. J., 21 (2005), 191–203.
[10]	M. E. Shahed, A. Salem, An extension of Wright function and its properties, J. Math., 2015 (2015), 950728. https://doi.org/10.1155/2015/950728 doi: 10.1155/2015/950728
[11]	M. A. Pathan, M. G. B. Saad, Mittag-Leffler-type function of arbitrary order and their application in the fractional kinetic equation, Partial Differ. Eq. Appl., 4 (2023), 15. https://doi.org/10.1007/s42985-023-00234-2 doi: 10.1007/s42985-023-00234-2
[12]	B. Shiri, D. Baleanu, All linear fractional derivatives with power functions' convolution kernel and interpolation properties, Chaos Soliton. Fract., 170 (2023), 113399. https://doi.org/10.1016/j.chaos.2023.113399 doi: 10.1016/j.chaos.2023.113399
[13]	H. Askari, A. Ansari, Asymptotic analysis of three-parameter Mittag-Leffler function with large parameters, and application to sub-diffusion equation involving Bessel operator, Fract. Calc. Appl. Anal., 27 (2024), 1162–1185. https://doi.org/10.1007/s13540-024-00263-7 doi: 10.1007/s13540-024-00263-7
[14]	R. Gorenflo, A. A. Kilbas, F. Mainardi, S. V. Rogosin, Mittag-Leffler functions, related topics and applications, Berlin: Springer, 2014. https://doi.org/10.1007/978-3-662-43930-2-6
[15]	G. Rajchakit, P. Chanthorn, M. Niezabitowski, R. Raja, D. Baleanu, A. Pratap, Impulsive effects on stability and passivity analysis of memristor-based fractional-order competitive neural networks, Neurocomputing, 417 (2020), 290–301. https://doi.org/10.1016/j.neucom.2020.07.036 doi: 10.1016/j.neucom.2020.07.036
[16]	G. Abbas, G. Farid, Some integral inequalities for m-convex functions via generalized fractional integral operator containing generalized Mittag-Leffler function, Cogent Math., 3 (2016), 1269589. https://doi.org/10.1080/23311835.2016.1269589 doi: 10.1080/23311835.2016.1269589
[17]	M. Andric, Fractional integral inequalities of Hermite-Hadamard type for (h, g; m)-convex functions with extended Mittag-Leffler function, Fractal Fract., 6 (2022), 1–15. https://doi.org/10.3390/fractalfract6060301 doi: 10.3390/fractalfract6060301
[18]	M. V. Cortez, A. Latif, R. Hussain, Some fractional integral inequalities by way of Raina fractional integrals, Symmetry, 15 (2023), 1935. https://doi.org/10.3390/sym15101935 doi: 10.3390/sym15101935
[19]	A. Khan, H. M. Akhtar, K. S. Nisar, D. L. Suthar, Pathway fractional integral formula involving an extended Mittag-Leffler function, Analysis, 42 (2022), 141–147. https://doi.org/10.1515/anly-2021-0039 doi: 10.1515/anly-2021-0039
[20]	T. Du, Y. Long, The multi-parameterized integral inequalities for multiplicative Riemann-Liouville fractional integrals, J. Math. Anal. Appl, 541 (2025), 128692. https://doi.org/10.1016/j.jmaa.2024.128692 doi: 10.1016/j.jmaa.2024.128692
[21]	H. M. Srivastava, M. K. Bansal, P. Harjule, A class of fractional integral operators involving a certain general multiindex Mittag-Leffler function, Ukr. Math. J., 75 (2024), 1255–1271. https://doi.org/10.1007/s11253-023-02259-7 doi: 10.1007/s11253-023-02259-7
[22]	H. Chen, U. N. Katugampola, Hermite-Hdamard and Hermite-Hadamard-Fejér type inequalities for generalized fractional integrals, J. Math. Anal. Appl., 446 (2017), 1274–1291. https://doi.org/10.1016/j.jmaa.2016.09.018 doi: 10.1016/j.jmaa.2016.09.018
[23]	S. Hussain, S. Rafeeq, Some new Hermite-Hadamard type integral inequalities for functions whose $n$ -th derivatives are logarithmically relative $h$ -preinvex, Miskolc Math. Notes, 18 (2017), 837–849. https://doi.org/10.18514/MMN.2017.1831 doi: 10.18514/MMN.2017.1831
[24]	J. E. Pecaric, F. Proschan, Y. L. Tong, Convex functions, partial orderings and statistical applications, San Diego: Academic Press Limited, 1 (1992).
[25]	S. Hussain, S. Rafeeq, Y. M. Chu, S. Khalid, S. Saleem, On some new generalized fractional Bullen-type inequalities with applications, J. Inequal. Appl., 2022 (2022), 138. https://doi.org/10.1186/s13660-022-02878-x doi: 10.1186/s13660-022-02878-x
[26]	A. A. Al-Gonah, W. K. Mohammed, A new forms of extended hypergeometric functions and their properties, Eng. Appl. Sci. Lett., 4 (2021), 30–41.
[27]	E. D. Rainville, Special functions, New York: The Macmillan Company, 1 (1960).
[28]	M. Sababheh, Convex functions and means of matrices, Math. Inequal. Appl., 20 (2017), 29–47. https://doi.org/10.7153/mia-20-03
[29]	G. N. Watson, A treatise on the theory of Bessel functions, London: Cambridge University Press, 1 (1922).

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