New inequalities via Caputo-Fabrizio integral operator with applications

Hong Yang; Shahid Qaisar; Arslan Munir; Muhammad Naeem; Hong Yang; Shahid Qaisar; Arslan Munir; Muhammad Naeem

doi:10.3934/math.2023989

AIMS Mathematics

2023, Volume 8, Issue 8: 19391-19412. doi: 10.3934/math.2023989

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New inequalities via Caputo-Fabrizio integral operator with applications

1.
School of Computer Science, Chengdu University, Chengdu, 610106, China
2.
Key Laboratory of Pattern Recognition and Intelligent Information Processing of Sichuan, Chengdu University, Chengdu, 610106, China
3.
Department of Mathematics, COMSATS University Islamabad, Sahiwal Campus, Pakistan

Received: 03 March 2023 Revised: 16 May 2023 Accepted: 21 May 2023 Published: 08 June 2023
MSC : 26A33, 26D10, 26D15, 34K38

Fractional integral inequalities have become one of the most useful and expansive tools for the development of many fields of pure and applied mathematics over the past few years. Many authors have just recently introduced various generalized inequalities that involved the fractional integral operators. The main goal of the present study is to incorporate the concept of strongly $ \left(s, m\right) $-convex functions and Hermite-Hadamard inequality with Caputo-Fabrizio integral operator. Also, we consider a new identity for twice differentiable mapping in the context of Caputo-Fabrizio fractional integral operator. Then, considering this identity as an auxiliary result, new mid-point version using well known inequalities like Hölder, power-mean, Young are presented. Moreover, some graphs of obtained inequalities are given for better understanding by the reader. Finally, we discussed some applications to matrix inequalities and spacial means.
- Hermite-Hadamard inequality,
- convex function,
- Caputo-Fabrizio fractional integral operator,
- Jensen inequality,
- Hölder inequality
Citation: Hong Yang, Shahid Qaisar, Arslan Munir, Muhammad Naeem. New inequalities via Caputo-Fabrizio integral operator with applications[J]. AIMS Mathematics, 2023, 8(8): 19391-19412. doi: 10.3934/math.2023989

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Abstract

Fractional integral inequalities have become one of the most useful and expansive tools for the development of many fields of pure and applied mathematics over the past few years. Many authors have just recently introduced various generalized inequalities that involved the fractional integral operators. The main goal of the present study is to incorporate the concept of strongly $ \left(s, m\right) $-convex functions and Hermite-Hadamard inequality with Caputo-Fabrizio integral operator. Also, we consider a new identity for twice differentiable mapping in the context of Caputo-Fabrizio fractional integral operator. Then, considering this identity as an auxiliary result, new mid-point version using well known inequalities like Hölder, power-mean, Young are presented. Moreover, some graphs of obtained inequalities are given for better understanding by the reader. Finally, we discussed some applications to matrix inequalities and spacial means.

References

[1]	Y. C. Kwun, M. S. Saleem, M. Ghafoor, Hermite-Hadamard type inequalities for functions whose derivatives are $\eta $-convex via fractional integrals, J. Inequal. Appl., 2019 (2019), 44. https://doi.org/10.1186/s13660-019-1993-y doi: 10.1186/s13660-019-1993-y
[2]	M. Tariq, H. Ahmad, S. K. Sahoo, L. S. Aljoufi, S. Khan, A novel comprehensive analysis of the refinements of Hermite-Hadamard type integral inequalities involving special functions, J. Math. Comput. Sci., 26 (2022), 330–348. https://doi.org/10.22436/jmcs.026.04.02 doi: 10.22436/jmcs.026.04.02
[3]	M. Raees, M. Anwar, G. Farid, Error bounds associated with different versions of Hadamard inequalities of mid-point type, J. Math. Comput. Sci., 23 (2021), 213–229. https://doi.org/.10.22436/jmcs.023.03.05 doi: 10.22436/jmcs.023.03.05
[4]	O. Almutairi, A. Kiliçman, New generalized Hermite-Hadamard inequality and related integral inequalities involving Katugampola type fractional integrals, Symmetry, 12 (2020), 568. https://doi.org/10.3390/sym12040568 doi: 10.3390/sym12040568
[5]	M. B. Khan, H. M. Srivastava, P. O. Mohammed, K. Nonlaopon, Y. S. Hamed, Some new Jensen schur and Hermite-Hadamard inequalities for log convex fuzzy interval-valued functions, AIMS Math., 7 (2022), 4338–4358. https://doi.org/10.3934/math.2022241 doi: 10.3934/math.2022241
[6]	H. M. Srivastava, S. K. Sahoo, P. O. Mohammed, D. Baleanu, B. Kodamasingh, Hermite-Hadamard type inequalities for interval-valued preinvex functions via fractional integral operators, Int. J. Comput. Intell. Syst., 15 (2022), 8. https://doi.org/10.1007/s44196-021-00061-6 doi: 10.1007/s44196-021-00061-6
[7]	H. Budak, H. Kara, M. Z. Sarikaya, M. E. Kirimcs, New extensions of the Hermite-Hadamard inequalities involving Riemann-Liouville fractional integrals, Miskolc Math. Notes, 21 (2020), 665–678. https://doi.org/10.18514/mmn.2020.3073 doi: 10.18514/mmn.2020.3073
[8]	P. Bedi, A. Kumar, T. Abdeljawad, A. Khan, J. F. Gómez-Aguilar, Mild solutions of coupled hybrid fractional order system with Caputo-Hadamard derivatives, Fractals, 29 (2021), 2150158. https://doi.org/10.1142/S0218348X21501589 doi: 10.1142/S0218348X21501589
[9]	H. Khan, T. Abdeljawad, J. F. Gómez-Aguilar, H. Tajadodi, A. Khan, Fractional order volterra integro-differential equation with Mittag-Leffler kernel, Fractals, 29 (2021), 2150154. https://doi.org/10.1142/S0218348X21501541 doi: 10.1142/S0218348X21501541
[10]	O. Martínez-Fuentes, F. Meléndez-Vázquez, G. Fernández-Anaya, J. F. Gómez-Aguilar, Analysis of fractional-order nonlinear dynamic systems with general analytic kernels: lyapunov stability and inequalities, Mathematics, 9 (2021), 2084. https://doi.org/10.3390/math9172084
[11]	A. Asma, J. F. Gómez-Aguilar, G. Rahman, M. Javed, Stability analysis for fractional order implicit $\psi $-Hilfer differential equations, Math. Method. Appl. Sci., 45 (2022), 2701–2712. https://doi.org/10.1002/mma.7948 doi: 10.1002/mma.7948
[12]	R. Dhayal, J. F. Gómez-Aguilar, J. Torres-Jiménez, Stability analysis of Atangana-Baleanu fractional stochastic differential systems with impulses, Int. J. Syst. Sci., 53 (2022), 3481–3495. htps://doi.org/10.1080/00207721.2022.2090638 doi: 10.1080/00207721.2022.2090638
[13]	A. González-Calderón, L. Vivas-Cruz, M. A. Taneco-Hernández, J. F. Gómez-Aguilar, Assessment of the performance of the hyperbolic-nilt method to solve fractional differential equations, Math. Comput. Simulat., 206 (2023), 375–390. https://doi.org/10.1016/j.matcom.2022.11.022 doi: 10.1016/j.matcom.2022.11.022
[14]	S. S. Dragomir, C. Pearce, Selected topics on Hermite-Hadamard inequalities and applications, 2003.
[15]	V. G. Mihesan, A generalization of the convexity, seminar on functional equations, Approx and convex, 1993.
[16]	H. Hudzik, L. Maligranda, Some remarks on $s$-convex functions, Aeq. Math., 48 (1994), 100–111. https://doi.org/10.1007/BF01837981 doi: 10.1007/BF01837981
[17]	N. Eftekhari, Some remarks on $(s, m)$-convexity in the second sense, J. Math. Inequal., 8 (2014), 489–495. https://doi.org/10.7153/jmi-08-36 doi: 10.7153/jmi-08-36
[18]	X. Wang, M. S. Saleem, K. N. Aslam, X. Wu, T. Zhou, On Caputo-Fabrizio fractional integral inequalities of Hermite-Hadamard type for modified $h$-convex functions, J. Math., 2020 (2020), 8829140. https://doi.org/10.1155/2020/8829140 doi: 10.1155/2020/8829140
[19]	S. I. Butt, M. Nadeem, G. Farid, On Caputo fractional derivatives via exponential $(s, m)$-convex functions, Eng. Appl. Sci. Lett., 3 (2020), 32–39. https://doi.org/10.30538/psrp-easl2020.0039 doi: 10.30538/psrp-easl2020.0039
[20]	S. Kemali, G. Tinaztepe, I. Y. Isik, S. S. Evcan, New integral inequalities for s-convex functions in the second sense via Caputo fractional derivative and Caputo-Fabrizio integral operator, Rocky Mt. J. Math., preprint.
[21]	A. M. K. Abbasi, M. Anwar, Hermite-Hadamard inequality involving Caputo-Fabrizio fractional integrals and related inequalities via $ s$-convex functions in the second sense, AIMS Math., 7 (2022), 18565–18575. https://doi.org/10.3934/math.20221020 doi: 10.3934/math.20221020
[22]	Q. Li, M. S. Saleem, P. Yan, M. S. Zahoor, M. Imran, On strongly convex functions via Caputo-Fabrizio-type fractional integral and some applications, J. Math., 2021 (2021), 6625597. https://doi.org/10.1155/2021/6625597 doi: 10.1155/2021/6625597
[23]	B. T. Poljak, Existence theorems and convergence of minimizing sequences in extremum problems with restrictions, Soviet Math. Doklady, 7 (1966), 72–75. https://doi.org/10.1137/0502051 doi: 10.1137/0502051
[24]	J. Hua, B. Y. Xi, F. Qi, Some new inequalities of Simpson type for strongly $s$-convex functions, Afr. Mat, 26 (2015), 741–752. https://doi.org/10.1007/s13370-014-0242-2 doi: 10.1007/s13370-014-0242-2
[25]	M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73–85. https://doi.org/10.12785/pfda/010201 doi: 10.12785/pfda/010201
[26]	T. Abdeljawad, D. Baleanu, On fractional derivatives with exponential kernel and their discrete versions, Rep. Math. Phys., 80 (2017), 11–27. https://doi.org/10.1016/S0034-4877(17)30059-9 doi: 10.1016/S0034-4877(17)30059-9
[27]	S. S. Dragomir, R. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett., 11 (1998), 91–95. https://doi.org/10.1016/S0893-9659(98)00086-X doi: 10.1016/S0893-9659(98)00086-X
[28]	M. Z. Sarikaya, H. Yildirim, On Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals, Miskolc Math. Notes, 17 (2016), 1049–1059. https://doi.org/10.18514/mmn.2017.1197 doi: 10.18514/mmn.2017.1197
[29]	S. S. Dragomir, Some inequalities of Hermite-Hadamard type for convex functions and Riemann-Liouville fractional integrals, RGMIA Res. Rep. Coll, preprint.
[30]	S. K. Sahoo, P. O. Mohammed, B. Kodamasingh, M. Tariq, Y. S. Hamed, New fractional integral inequalities for convex functions pertaining to Caputo-Fabrizio Operator, Fractal Fract., 6 (2022), 171. https://doi.org/10.3390/fractalfract6030171 doi: 10.3390/fractalfract6030171
[31]	M. Sababheh, Convexity and matrix means, Linear Algebra Appl., 506 (2016), 588–602. https://doi.org/10.1016/j.laa.2016.06.027
[32]	M. Z. Sarikaya, N. Aktan, On the generalization of some integral inequalities and their applications, Math. Comput. Model., 54 (2011), 2175–2182. https://doi.org/10.1016/j.mcm.2011.05.026 doi: 10.1016/j.mcm.2011.05.026

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