Hadamard and Fejér-Hadamard inequalities for generalized $ k $-fractional integrals involving further extension of Mittag-Leffler function

Ye Yue; Ghulam Farid; Ayșe Kübra Demirel; Waqas Nazeer; Yinghui Zhao; Ye Yue; Ghulam Farid; Ayșe Kübra Demirel; Waqas Nazeer; Yinghui Zhao

doi:10.3934/math.2022043

AIMS Mathematics

2022, Volume 7, Issue 1: 681-703. doi: 10.3934/math.2022043

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Research article

Hadamard and Fejér-Hadamard inequalities for generalized $ k $-fractional integrals involving further extension of Mittag-Leffler function

1.
School of Science, Shijiazhuang University, Shijiazhuang 050035, China
2.
Department of Mathematics, COMSATS University Islamabad, Attock Campus, Pakistan
3.
Department of Mathematics, Ordu University, Ordu, Turkey
4.
Department of Mathematics, Government College University, Lahore, Pakistan

Received: 06 July 2021 Accepted: 30 August 2021 Published: 15 October 2021
MSC : 26A51, 26A33, 33E12, 26D15, 26A51, 26B25

In this paper, $ k $-fractional integral operators containing further extension of Mittag-Leffler function are defined firstly. Then, the first and second version of Hadamard and Fejér-Hadamard inequalities for generalized $ k $-fractional integrals are obtained. Finally, by using these generalized $ k $-fractional integrals containing Mittag-Leffler functions, results for $ p $-convex functions are obtained. The results for convex functions can be deduced by taking $ p = 1 $.
- Hadamard inequality,
- Fejér-Hadamard inequality,
- convex function,
- $ p $-convex function,
- $ k $-fractional integrals,
- Mittag-Leffler function
Citation: Ye Yue, Ghulam Farid, Ayșe Kübra Demirel, Waqas Nazeer, Yinghui Zhao. Hadamard and Fejér-Hadamard inequalities for generalized $ k $-fractional integrals involving further extension of Mittag-Leffler function[J]. AIMS Mathematics, 2022, 7(1): 681-703. doi: 10.3934/math.2022043

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Abstract

In this paper, $ k $-fractional integral operators containing further extension of Mittag-Leffler function are defined firstly. Then, the first and second version of Hadamard and Fejér-Hadamard inequalities for generalized $ k $-fractional integrals are obtained. Finally, by using these generalized $ k $-fractional integrals containing Mittag-Leffler functions, results for $ p $-convex functions are obtained. The results for convex functions can be deduced by taking $ p = 1 $.

References

[1]	M. Andric, G. Farid, J. Pecaric, A further extension of Mittag-Leffler function, Fract. Calculus Appl. Anal., 21 (2018), 1377–1395. doi: 10.1515/fca-2018-0072. doi: 10.1515/fca-2018-0072
[2]	G. Farid, A unified integral operator and further its consequences, Open J. Math. Anal., 4 (2020), 1–7. doi: 10.30538/psrp-oma2020.0047. doi: 10.30538/psrp-oma2020.0047
[3]	M. Yussouf, G. Farid, K. A. Khan, C. Y. Jung, Hadamard and Fejér-Hadamard inequalities for further generalized fractional integrals involving Mittag-Leffler functions, J. Math., 2021 (2021), 5589405. doi: 10.1155/2021/5589405. doi: 10.1155/2021/5589405
[4]	C. P. Niculescu, L. E. Persson, Convex functions and their applications, A contemporary approach, New York, Springer, 2006.
[5]	I. Iscan, Ostrowski type inequalities for $p$-convex functions, NTMSCI, 4 (2016), 140–150. doi: 10.20852/NTMSCI.2016318838. doi: 10.20852/NTMSCI.2016318838
[6]	M. Kunt, I. Iscan, Hermite-Hadamard-Fejér type inequalities for $p$-convex functions via fractional integrals, Iran J. Sci. Technol. Trans. Sci., 42 (2018), 2079–2089. doi: 10.1016/j.ajmsc.2016.11.001. doi: 10.1016/j.ajmsc.2016.11.001
[7]	B. G. Pachpatte, Mathematical inequalities, Elsevier, volume 67, 2005.
[8]	L. Fejér, Uberdie Fourierreihen II, Math Naturwiss Anz Ungar. Akad. Wiss, 24 (1906), 369–390.
[9]	T. Tunç, H. Budak, F. Usta, M. Z. Sarikaya, On new generalized fractional integral operators and related fractional inequalities, Konuralp J. Math., 8 (2020), 268–278.
[10]	S. Mubeen, G. M. Habibullah, $k$-fractional integrals and applications, Int. J. Contemp. Math. Sci., 7 (2012), 89–94.
[11]	R. K. Raina, On generalized wright's hypergeometric functions and fractional calculus operators, East Asian Math. J., 21 (2005), 191–203.
[12]	A. Akkurt, M. E. Yildirim, H. Yildirim, On some integral inequalities for $(k, h)$-Riemann-Liouville fractional integral, NTMSCI, 4 (2016), 138–146. doi: 10.20852/ntmsci.2016217824. doi: 10.20852/ntmsci.2016217824
[13]	W. Nazeer, G. Farid, Z. Salleh, H. Yasmeen, Generalized Riemann-Liouville fractional integral inequalities of Hadamard-type for $(\alpha, h-m)$-$p$-convex functions (submitted).
[14]	G. Abbas, G. Farid, Hadamard and Fejér-Hadamard type inequalities for harmonically convex functions via generalized fractional integrals, J. Anal., 25 (2017), 107–119. doi: 10.1007/s41478-017-0032-y. doi: 10.1007/s41478-017-0032-y
[15]	G. Farid, A. U. Rehman, S. Mehmood, Hadamard and Fejér-Hadamard type integral inequalities for harmonically convex functions via an extended generalized Mittag-Leffler function, J. Math. Comput. Sci., 8 (2018), 630–643.
[16]	I. Iscan, S. Wu, Hemite-Hadamard type inequalities for harmonically convex functions via fractional integrals, Appl. Math. Comput., 238 (2014), 237–244. doi: 10.1016/j.amc.2014.04.020. doi: 10.1016/j.amc.2014.04.020
[17]	X. Qiang, G. Farid, M. Yussouf, K. A. Khan, A. U. Rehman, New generalized fractional versions of Hadamard and Fejér inequalities for harmonically convex functions, J. Inequal. Appl., 2020 (2020), 1–13. doi: 10.1186/s13660-020-02457-y. doi: 10.1186/s13660-020-02457-y
[18]	G. Farid, G. Abbas, Generalizations of some Hermite-Hadamard-Fejér type inequalities for $p$-convex functions via generalized fractional integrals, J. Fract. Calculus Appl., 9 (2018), 56–76.

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