Some novel inequalities involving Atangana-Baleanu fractional integral operators and applications

Miguel Vivas-Cortez; Muhammad Uzair Awan; Sehrish Rafique; Muhammad Zakria Javed; Artion Kashuri; Miguel Vivas-Cortez; Muhammad Uzair Awan; Sehrish Rafique; Muhammad Zakria Javed; Artion Kashuri

doi:10.3934/math.2022678

AIMS Mathematics

2022, Volume 7, Issue 7: 12203-12226. doi: 10.3934/math.2022678

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Research article

Some novel inequalities involving Atangana-Baleanu fractional integral operators and applications

1.
Escuela de Ciencias Fisicas y Matemáticas, Facultad de Ciencias Exactas y Naturales, Pontificia Universidad Católica del Ecuador, Av. 12 de Octubre 1076, Apartado, Quito 17-01-2184, Ecuador
2.
Department of Mathematics, Government College University, Faisalabad, Pakistan
3.
Department of Mathematics, Faculty of Technical Science, University "Ismail Qemali", 9400 Vlora, Albania

Received: 24 January 2022 Revised: 26 February 2022 Accepted: 08 March 2022 Published: 22 April 2022
MSC : 05A30, 26A33, 26A51, 34A08, 26D07, 26D10, 26D15

As we know, Atangana and Baleanu developed great fractional integral operators which used the generalized Mittag-Leffler function as non-local and non-singular kernel. Inspired by these integral operators, we derive in this paper two new fractional integral identities involving Atangana-Baleanu fractional integrals. Using these identities as auxiliary results, we establish new fractional counterparts of classical inequalities essentially using first and second order differentiable higher order strongly $ n $-polynomial convex functions. We also discuss several important special cases of the main results. In order to show the efficiency of our main results, we offer applications for special means and for differentiable functions of first and second order that are in absolute value bounded.
- Atangana-Baleanu fractional integrals,
- higher order strongly $ n $-polynomial convex,
- Hölder's inequality,
- power mean inequality,
- special means,
- bounded functions
Citation: Miguel Vivas-Cortez, Muhammad Uzair Awan, Sehrish Rafique, Muhammad Zakria Javed, Artion Kashuri. Some novel inequalities involving Atangana-Baleanu fractional integral operators and applications[J]. AIMS Mathematics, 2022, 7(7): 12203-12226. doi: 10.3934/math.2022678

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Abstract

As we know, Atangana and Baleanu developed great fractional integral operators which used the generalized Mittag-Leffler function as non-local and non-singular kernel. Inspired by these integral operators, we derive in this paper two new fractional integral identities involving Atangana-Baleanu fractional integrals. Using these identities as auxiliary results, we establish new fractional counterparts of classical inequalities essentially using first and second order differentiable higher order strongly $ n $-polynomial convex functions. We also discuss several important special cases of the main results. In order to show the efficiency of our main results, we offer applications for special means and for differentiable functions of first and second order that are in absolute value bounded.

References

[1]	P. O. Mohammed, T. Abdeljawad, M. A. Alqudah, F. Jarad, New discrete inequalities of Hermite-Hadamard type for convex functions, Adv. Differ. Equ., 122 (2021). https://doi.org/10.1186/s13662-021-03290-3 doi: 10.1186/s13662-021-03290-3
[2]	P. O. Mohammed, M. Z. Sarikaya, D. Baleanu, On the generalized Hermite-Hadamard inequalities via the tempered fractional integrals, Symmetry, 12 (2020), 595. https://doi.org/10.3390/sym12040595 doi: 10.3390/sym12040595
[3]	P. O. Mohammed, T. Abdeljawad, D. Baleanu, A. Kashuri, F. Hamasalh, P. Agarwal, New fractional inequalities of Hermite-Hadamard type involving the incomplete gamma functions, J. Inequal. Appl., 263 (2020). https://doi.org/10.1186/s13660-020-02538-y doi: 10.1186/s13660-020-02538-y
[4]	G. Rahman, K. S. Nisar, T. Abdeljawad, S. Ullah, Certain fractional proportional integral inequalities via convex functions, Mathematics, 8 (2020), 1–11. https://doi.org/10.3390/math8020222 doi: 10.3390/math8020222
[5]	G. Rahman, T. Abdeljawad, F. Jarad, K. S. Nisar, Bounds of generalized proportional fractional integrals in general form via convex functions and their applications, Mathematics, 8 (2020). https://doi.org/10.3390/math8010113 doi: 10.3390/math8010113
[6]	M. Z. Sarikaya, E. Set, H. Yaldiz, N. Basak, Hermite-Hadamard's inequalities for fractional integrals and related fractional inequalities, Math. Comput. Model., 57 (2013), 2403–2407. https://doi.org/10.1016/j.mcm.2011.12.048 doi: 10.1016/j.mcm.2011.12.048
[7]	H. Budak, P. Agarwal, New generalized midpoint type inequalities for fractional integral, Miskolc Math. Notes, 20 (2019), 781–793. https://doi.org/10.18514/MMN.2019.2525 doi: 10.18514/MMN.2019.2525
[8]	Y. M. Chu, M. U. Awan, M. Z. Javed, A. G. Khan, Bounds for the remainder in Simpson's inequality via $n$-polynomial convex functions of higher order using Katugampola fractional integrals, J. Math., 2020 (2020).
[9]	Y. M. Chu, M. U. Awan, S. Talib, M. A. Noor, K. I. Noor, Generalizations of Hermite-Hadamard like inequalities involving $\chi_{k}$-Hilfer fractional integrals, Adv. Differ. Equ., 594 (2020).
[10]	A. Kashuri, M. U. Awan, M. A. Noor, Fractional integral identity, estimation of its bounds and some applications to trapezoidal quadrature rule, Filomat, 34 (2020), 2629–2641.
[11]	K. Liu, J. R. Wang, D. O'Regan, On the Hermite-Hadamard type inequality for $\psi$-Riemann-Liouville fractional integrals via convex functions, J. Inequal. Appl., 27 (2019).
[12]	H. K. Onalan, A. O. Akdemir, M. A. Ardic, D. Baleanu, On new general versions of Hermite-Hadamard type integral inequalities via fractional integral operators with Mittag-Leffler kernel, J. Inequal. Appl., 186 (2021).
[13]	S. Talib, M. U. Awan, Estimations of upper bounds for $n$-th order differentiable functions involving $\chi$-Riemann-Liouville integrals via $\gamma$-preinvex functions, Math. Prob. Eng., 2021 (2021), 6882882.
[14]	S. Wu, M. U. Awan, M. V. Mihai, M. A. Noor, S. Talib, Estimates of upper bound for a $k$-th order differentiable functions involving Riemann-Liouville integrals via higher order strongly $h$-preinvex functions, J. Inequal. Appl., 227 (2019).
[15]	S. Wu, M. U. Awan, M. U. Ullah, S. Talib, A. Kashuri, Some integral inequalities for $n$-polynomial $\zeta$-preinvex functions, J. Funct. Spaces, 2021 (2021), 6697729.
[16]	Y. Zhang, T. S. Du, H. Wang, Some new $k$-fractional integral inequalities containing multiple parameters via generalized $(s, m)$-preinvexity, Ital. J. Pure Appl. Mat., 40 (2018), 510–527.
[17]	C. J. Huang, G. Rahman, K. S. Nisar, A. Ghaffar, F. Qi, Some inequalities of the Hermite-Hadamard type for $k$-fractional conformable integrals, Aust. J. Math. Anal. Appl., 16 (2019).
[18]	G. Rahman, T. Abdeljawad, F. Jarad, A. Khan, K. S. Nisar, Certain inequalities via generalized proportional Hadamard fractional integral operators, Adv. Differ. Equ., 454 (2019). https://doi.org/10.1186/s13662-019-2381-0 doi: 10.1186/s13662-019-2381-0
[19]	G. Rahman, A. Khan, T. Abdeljawad, K. S. Nisar, The Minkowski inequalities via generalized proportional fractional integral operators, Adv. Differ. Equ., 287 (2019). https://doi.org/10.1186/s13662-019-2229-7 doi: 10.1186/s13662-019-2229-7
[20]	G. Rahman, K. S. Nisar, F. Qi, Some new inequalities of the Gruss type for conformable fractional integrals, AIMS Math., 3 (2018), 575–583. https://doi.org/10.3934/Math.2018.4.575 doi: 10.3934/Math.2018.4.575
[21]	K. S. Nisar, G. Rahman, K. Mehrez, Chebyshev type inequalities via generalized fractional conformable integrals, J. Inequal. Appl., 245 (2019). https://doi.org/10.1186/s13660-019-2197-1 doi: 10.1186/s13660-019-2197-1
[22]	K. S. Nisar, A. Tassaddiq, G. Rahman, A. Khan, Some inequalities via fractional conformable integral operators, J. Inequal. Appl., 217 (2019). https://doi.org/10.1186/s13660-019-2170-z doi: 10.1186/s13660-019-2170-z
[23]	K. A. Abro, I. Khan, K. S. Nisar, Novel technique of Atangana and Baleanu for heat dissipation in transmission line of electrical circuit, Chaos Soliton. Fract., 129 (2019), 40–45. https://doi.org/10.1016/j.chaos.2019.08.001 doi: 10.1016/j.chaos.2019.08.001
[24]	K. S. Nisar, F. Qi, G. Rahman, S. Mubeen, M. Arshad, Some inequalities involving the extended gamma function and the Kummer confluent hypergeometric $k$-function, J. Inequal. Appl., 135 (2018). https://doi.org/10.1186/s13660-018-1717-8 doi: 10.1186/s13660-018-1717-8
[25]	G. Rahman, Z. Ullah, A. Khan, E. Set, K. S. Nisar, Certain Chebyshev-type inequalities involving fractional conformable integral operators, Mathematics, 7 (2019). https://doi.org/10.3390/math7040364 doi: 10.3390/math7040364
[26]	T. Toplu, M. Kadakal, İ. İșcan, On $n$-polynomial convexity and some related inequalities, AIMS Math., 5 (2020), 1304–1318. https://doi.org/10.3934/math.2020089 doi: 10.3934/math.2020089
[27]	B. T. Polyak, Existence theorems and convergence of minimizing sequences for extremal problems with constraints, Dokl. Akad. Nauk SSSR, 166 (1966), 287–290.
[28]	D. Baleanu, H. Khan, H. Jafari, R. A. Khan, M. Alipour, On existence results for solutions of a coupled system of hybrid boundary value problems with hybrid conditions, Adv. Differ. Equ., 318 (2015). https://doi.org/10.1186/s13662-015-0651-z doi: 10.1186/s13662-015-0651-z
[29]	H. Khan, W. Chen, H. Sun, Analysis of positive solution and Hyers-Ulam stability for a class of singular fractional differential equations with $p$-Laplacian in Banach space, Math. Method. Appl. Sci., 2018. https://doi.org/10.1002/mma.4835. doi: 10.1002/mma.4835
[30]	A. Khan, H. Khan, J. F. G. Aguilar, T. Abdeljawad, Existence and Hyers-Ulam stability for a nonlinear singular fractional differential equations with Mittag-Leffler kernel, Chaos Soliton. Fract., 127 (2019), 422–427. https://doi.org/10.1016/j.chaos.2019.07.026 doi: 10.1016/j.chaos.2019.07.026
[31]	A. Ekinci, M. E. Özdemir, E. Set, New integral inequalities of Ostrowski type for quasi-convex functions with applications, Turk. J. Sci., 5 (2020), 290–304.
[32]	A. O. Akdemir, S. I. Butt, M. Nadeem, M. A. Ragusa, New general variants of Chebyshev type inequalities via generalized fractional integral operators, Mathematics, 9 (2021). https://doi.org/10.3390/math9020122 doi: 10.3390/math9020122
[33]	S. Kızıl, M. A. Ardiç, Inequalities for strongly convex functions via Atangana-Baleanu integral operators, Turk. J. Sci., 6 (2021), 96–109.
[34]	A. Ekinci, M. E. Özdemir, Some new integral inequalities via Riemann-Liouville integral operators, Appl. Comput. Math., 3 (2019), 288–295.
[35]	S. I. Butt, M. Nadeem, G. Farid, On Caputo fractional derivatives via exponential $s$-convex functions, Turk. J. Sci., 5 (2020), 140–146.
[36]	K. S. Miller, B. Ross, An introduction to the fractional Calculus and fractional differential equations, Wiley, New York, NY, USA, 1993.
[37]	A. Atangana, I. Koca, Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos Soliton. Fract., 89 (2016), 447–454. https://doi.org/10.1016/j.chaos.2016.02.012 doi: 10.1016/j.chaos.2016.02.012
[38]	A. Atangana, D. Baleanu, New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Therm. Sci., 20 (2016), 763–769.
[39]	T. Abdeljawad, D. Baleanu, Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel, J. Nonlinear Sci. Appl., 10 (2017), 1098–1107. https://doi.org/10.22436/jnsa.010.03.20 doi: 10.22436/jnsa.010.03.20
[40]	H. Ahmad, M. Tariq, S. K. Sahoo, S. Askar, A. E. Abouelregal, K. M. Khedher, Refinements of Ostrowski type integral inequalities involving Atangana-Baleanu fractional integral operator, Symmetry, 13 (2021). https://doi.org/10.3390/sym13112059 doi: 10.3390/sym13112059
[41]	A. A. Lupaș, A. Cǎtaș, Fuzzy differential subordination of the Atangana-Baleanu fractional integral, Symmetry, 13 (2021).
[42]	A. A. Lupaș, A. Cǎtaș, An application of the principle of differential subordination to analytic functions involving Atangana-Baleanu fractional integral of Bessel functions, Symmetry, 13 (2021).
[43]	A. Khan, D. Khan, I. Khan, M. Taj, I. Ullah, A. M. Aldawsari, et al., MHD flow and heat transfer in sodium alginate fluid with thermal radiation and porosity effects: Fractional model of Atangana-Baleanu derivative of non-local and non-singular kernel, Symmetry, 11 (2019). https://doi.org/10.3390/sym11101295 doi: 10.3390/sym11101295
[44]	E. Uçar, S. Uçar, F. Evirgen, N. Özdemir, A fractional SAIDR model in the frame of Atangana-Baleanu derivative, Fractal Fract., 5 (2021).
[45]	C. N. Angstmann, B. A. Jacobs, B. I. Henry, Z. Xu, Intrinsic discontinuities in solutions of evolution equations involving fractional Caputo-Fabrizio and Atangana-Baleanu operators, Mathematics, 8 (2020). https://doi.org/10.3390/math8112023 doi: 10.3390/math8112023
[46]	D. Baleanu, R. Darzi, B. Agheli, Existence results for Langevin equation involving Atangana-Baleanu fractional operators, Mathematics, 8 (2020). https://doi.org/10.3390/math8030408 doi: 10.3390/math8030408
[47]	J. B. Liu, S. I. Butt, J. Nasir, A. Aslam, A. Fahad, J. Soontharanon, Jensen-Mercer variant of Hermite-Hadamard type inequalities via Atangana-Baleanu fractional operator, AIMS Math., 7 (2022), 2123–2141. https://doi.org/10.3934/math.2022121 doi: 10.3934/math.2022121
[48]	A. I. K. Butt, W. Ahmad, M. Rafiq, D. Baleanu, Numerical analysis of Atangana-Baleanu fractional model to understand the propagation of a novel corona virus pandemic, Alex. Eng. J., 61 (2022), 7007–7027. https://doi.org/10.1016/j.aej.2021.12.042 doi: 10.1016/j.aej.2021.12.042
[49]	X. S. Zhou, C. X. Huang, H. J. Hu, L. Liu, Inequality estimates for the boundedness of multilinear singular and fractional integral operators, J. Inequal. Appl., 303 (2013). https://doi.org/10.1186/1029-242X-2013-303 doi: 10.1186/1029-242X-2013-303
[50]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Math. Stud., 204 (2006).
[51]	D. Baleanu, A. Fernandez, On fractional operators and their classifications, Mathematics, 7 (2019). https://doi.org/10.3390/math7090830 doi: 10.3390/math7090830
[52]	H. M. Srivastava, P. W. Karlsson, Multiple gaussian hypergeometric series, Halsted Press (Ellis Horwood Limited, Chichester): Chichester, UK, 1985.
[53]	N. S. Barnett, P. Cerone, S. S. Dragomir, J. Roumeliotis, Some inequalities for the dispersion of a random variable whose pdf is defined on a finite interval, J. Inequal. Pure Appl. Math., 2 (2001), 1–18.
[54]	N. S. Barnett, S. S. Dragomir, Some elementary inequalities for the expectation and variance of a random variable whose pdf is defined on a finite interval, RGMIA Res. Rep. Colloq., 2 (1999), 1–7.
[55]	P. Cerone, S. S. Dragomir, On some inequalities for the expectation and variance, Korean J. Comput. Appl. Math., 2 (2000), 357–380. https://doi.org/10.1007/BF02941972 doi: 10.1007/BF02941972
[56]	J. E. Pečarič, F. Proschan, Y. L. Tong, Convex functions, partial ordering and statistical applications, Academic Press: New York, NY, USA, 1991.

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