Boundedness of fractional integral operators containing Mittag-Leffler functions via (<em>s,m</em>)-convexity

Ghulam Farid; Saira Bano Akbar; Shafiq Ur Rehman; Josip Pečarić; Ghulam Farid; Saira Bano Akbar; Shafiq Ur Rehman; Josip Pečarić

doi:10.3934/math.2020067

AIMS Mathematics

2020, Volume 5, Issue 2: 966-978. doi: 10.3934/math.2020067

Previous Article Next Article

Research article

Boundedness of fractional integral operators containing Mittag-Leffler functions via (s,m)-convexity

1.
Department of Mathematics, COMSATS University Islamabad, Attock Campus, Pakistan
2.
Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Pakistan
3.
Rudn University, Moscow, Russia

Received: 09 October 2019 Accepted: 14 December 2019 Published: 08 January 2020
MSC : 26A51, 26A33, 33E12

The objective of this paper is to derive the bounds of fractional integral operators which contain Mittag-Leffler functions in the kernels. By using (s, m)-convex functions bounds of these operators are evaluated which lead to obtain their boundedness and continuity. Moreover the presented results can be used to get various results for known fractional integrals and functions deducible from (s, m)-convexity. Also a version of Hadamard type inequality is established for (s, m)-convex functions via generalized fractional integrals.
- convex function,
- (s,m)-convex function,
- Mittag-Leffler function,
- generalized fractional integral operators
Citation: Ghulam Farid, Saira Bano Akbar, Shafiq Ur Rehman, Josip Pečarić. Boundedness of fractional integral operators containing Mittag-Leffler functions via (s,m)-convexity[J]. AIMS Mathematics, 2020, 5(2): 966-978. doi: 10.3934/math.2020067

Related Papers:

Abstract

The objective of this paper is to derive the bounds of fractional integral operators which contain Mittag-Leffler functions in the kernels. By using (s, m)-convex functions bounds of these operators are evaluated which lead to obtain their boundedness and continuity. Moreover the presented results can be used to get various results for known fractional integrals and functions deducible from (s, m)-convexity. Also a version of Hadamard type inequality is established for (s, m)-convex functions via generalized fractional integrals.

References

[1]	Y. Adjabi, F. Jarad, T. Abdeljawad, On generalized fractional operators and a Gronwall type inequality with applications, Filomat, 31 (2017), 5457-5473. doi: 10.2298/FIL1717457A
[2]	G. A. Anastassiou, Generalized fractional Hermit-Hadamard inequalities involving m-convexity and (s,m)-convexity, Ser. Math. Inform., 28 (2013), 107-126.
[3]	M. Andrić, G. Farid and J. Pečarić, A further extension of Mittag-Leffler function, Fract. Calc. Appl. Anal., 21 (2018), 1377-1395. doi: 10.1515/fca-2018-0072
[4]	M. Arshad, J. Choi, S. Mubeen, et al. A New Extension of MittagLeffler function, Commun. Korean Math. Soc., 33 (2018), 549-560.
[5]	I. A. Baloch, I. Iscan, Some Hermite-Hadamard type inequalities for harmonically (s,m)-convex functions in second sense, arXiv:1604.08445v1.
[6]	V. I. Burenkov, A. Gogatishvili, V. S. Guliyev, et al. Necessary and sufficient conditions for the boundedness of genuine singular integral operators in local Morrey-type space, Eurasian Math. J., 1 (2010), 32-53.
[7]	V. I. Burenkov, V. S. Guliyev, A. Serbetci, et al. Boundedness of the Riesz potential in local Morreytype spaces, Potential Anal., 35 (2011), 67-87. doi: 10.1007/s11118-010-9205-x
[8]	H. Chen, U. N. Katugampola, Hermite-Hadamard and Hermite-Hadamard-Fejér type inequalities for generalized fractional integrals, J. Math. Anal. Appl., 446 (2017), 1274-1291. doi: 10.1016/j.jmaa.2016.09.018
[9]	F. Deringoz, V. S. Guliyev, G. S. Samko, Boundedness of the maximal operator and its commutators on vanishing generalized Orlicz-Morrey spaces, Ann. Acad. Sci. Fenn., Math., 40 (2015), 535-549. doi: 10.5186/aasfm.2015.4029
[10]	N. Eftekhari, Some remarks on (s, m)-convexity in the second sense, J. Math. inequal., 8 (2014), 489-495.
[11]	G. Farid, Some Riemann-Liouville fractional integral inequalities for convex functions, The Journal of Analysis, (2018), 1-8.
[12]	G. Farid, U. N. Katugampola, M. Usman, Ostrowski type fractional integral inequalities for s-Godunova-Levin functions via Katugampola fractional integrals, Open J. Math. Sci., 1 (2017), 97-110. doi: 10.30538/oms2017.0010
[13]	G. Farid, K. A. Khan, N. Latif, et al. General fractional integral inequalities for convex and m-convex functions via an extended generalized Mittag-Leffler function, J. Inequal. Appl., 2018 (2018), 243.
[14]	V. S. Guliyev, N. N. Garakhanova, I. Ekincioglu, Pointwise and integral estimates for the fractional integrals on the Laguerre hypergroup, Math. Inequal. Appl., 15 (2012), 513-524.
[15]	H. J. Haubold, A. M. Mathai,, R. K. Saxena, Mittag-Leffler functions and their applications, J. Appl. Math., 2011 (2011), 298628.
[16]	H. Hudzik, L. Maligranda, Some remarks on s-convex functions, Aequ. Math., 48 (1994), 100-111. doi: 10.1007/BF01837981
[17]	S. M. Kang, G. Farid, W. Nazeer, et al. (h - m)-convex functions and associated fractional Hadamard and Fejér-Hadamard inequalities via an extended generalized Mittag-Leffler function, J. Inequal. Appl., 2019 (2019), 78.
[18]	V. Mihesan, A generalization of the convexity, Seminar on Functional Equations, Approx. Convex., Cluj-Napoca, Romania, 1993.
[19]	V. C Miguel, Fejér type inequalities for (s, m)-convex functions in second sense, Appl. Math. Inf. Sci., 10 (2016), 1689-1696.
[20]	S. M. Kang, G. Farid, W. Nazeer, et al. Hadamard and Fejér-Hadamard inequalities for extended generalized fractional integrals involving special functions J. Inequal. Appl., 2018 (2018), 119.
[21]	G. Mittag-Leffler, Sur la nouvelle fonction E_α(x), C. R. Acad. Sci. Paris., 137 (1903), 554-558.
[22]	C. P. Niculescu, L. E. Persson, Convex functions and their applications: A contemporary approach, Springer Science & Business Media, Inc., 2006.
[23]	J. Park, New Ostrowski-like type inequalities for differentable (s, m)-convex mappings, Int. J. Pure Appl. Math., 78 (2012), 1077-1089.
[24]	J. Pecarić, F. Proschan, Y. L. Tong, Convex functions, partial orderings, and statistical applications, Academics Press, New York, 1992.
[25]	T. R. Prabhakar, A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J., 19 (1971), 7-15.
[26]	G. Rahman, D. Baleanu, M. A. Qurashi, et al. The extended Mittag-Leffler function via fractional calculus, J. Nonlinear Sci. Appl., 10 (2017), 4244-4253. doi: 10.22436/jnsa.010.08.19
[27]	A. W. Roberts, D. E. Varberg, Convex functions, Academic Press, New York, 1973.
[28]	T. O. Salim and A. W. Faraj, A Generalization of Mittag-Leffler function and integral operator associated with integral calculus, J. Frac. Calc. Appl., 3 (2012), 1-13.
[29]	A. K. Shukla and J. C. Prajapati, On a generalization of Mittag-Leffler function and its properties, J. Math. Anal. Appl., 336 (2007), 797-811. doi: 10.1016/j.jmaa.2007.03.018
[30]	H. M. Srivastava and Z. Tomovski, Fractional calculus with an integral operator containing generalized Mittag- Leffler function in the kernel, Appl. Math. Comput., 211 (2009), 198-210.
[31]	G. Toader, Some generalizations of the convexity, Proc. Colloq. Approx. Optim., (1984), 329-338.
[32]	S. Ullah, G. Farid, K. A. Khan, et al. Generalized fractional inequalities for quasi-convex functions, Adv. Difference Equ., 2019 (2019), 15.

Reader Comments

Your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)