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October  2021, 14(10): 3703-3718. doi: 10.3934/dcdss.2021020

## Some new bounds analogous to generalized proportional fractional integral operator with respect to another function

 1 Government College University, Faisalabad, Pakistan 2 Department of Mathematics, Faculty of Arts and Sciences, Cankaya University Ankara, Turkey 3 Department of Medical Research, China Medical University Hospital, Taichung 40402, Taiwan, École Normale Supérieure, Moulay Ismail University, Meknes, Morocco

* Corresponding author: Zakia Hammouch.

Received  November 2019 Revised  January 2020 Published  October 2021 Early access  March 2021

The present article deals with the new estimates in the view of generalized proportional fractional integral with respect to another function. In the present investigation, we focus on driving certain new classes of integral inequalities utilizing a family of positive functions $n(n\in\mathbb{N})$ for this newly defined operator. From the computed outcomes, we concluded some new variants for classical generalized proportional fractional and other integrals as remarks. These variants are connected with some existing results in the literature. Certain interesting consequent results of the main theorems are also pointed out.

Citation: Saima Rashid, Fahd Jarad, Zakia Hammouch. Some new bounds analogous to generalized proportional fractional integral operator with respect to another function. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3703-3718. doi: 10.3934/dcdss.2021020
##### References:
 [1] T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57-66.  doi: 10.1016/j.cam.2014.10.016. [2] T. Abdeljawad and D. Baleanu, Monotonicity results for fractional difference operators with discrete exponential kernels, Adv. Differ. Equ., 2017 (2017), 78-87.  doi: 10.1186/s13662-017-1126-1. [3] T. Abdeljawad and D. Baleanu, On fractional derivatives with exponential kernel and their discrete versions, Rep. Math. Phys., 80 (2017), 11-27.  doi: 10.1016/S0034-4877(17)30059-9. [4] M. Adil Khan, Y.-M. Chu, T. U. Khan and J. Khan, Some new inequalities of Hermite-Hadamard type for $s$-convex functions with applications, Open Math., 15 (2017), 1414-1430. [5] M. Adil Khan, Y.-M. Chu, A. Kashuri, R. Liko and G. Ali, Conformable fractional integrals versions of Hermite-Hadamard inequalities and their generalizations, J. Funct. Spaces, 2018 (2018), 6928130, 9 pp. doi: 10.1155/2018/6928130. [6] M. Adil Khan, A. Iqbal, M. Suleman and Y.-M. Chu, Hermite-Hadamard type inequalities for fractionalintegrals via Green's function, J. Inequal. Appl., 2018 (2018), 161-176.  doi: 10.1186/s13660-018-1751-6. [7] R. Almeida, A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci. Numer. Simulat, 44 (2017), 460-481.  doi: 10.1016/j.cnsns.2016.09.006. [8] J. Alzabut, T. Abdeljawad, F. Jarad and W. Sudsutad, A Gronwall inequality via the generalized proportional fractional derivative with applications, J. Inequal. Appl., 2019 (2019), 101-113.  doi: 10.1186/s13660-019-2052-4. [9] D. R. Anderson and D. J. Ulness, Newly defined conformable derivatives, Adv. Dyn. Syst. Appl., 10 (2015), 109-137. [10] A. Atangana and K. M. Owolabi, New numerical approach for fractional differential equations, Math. Model. Nat. Phenom., 13 (2018), 3-24.  doi: 10.1051/mmnp/2018010. [11] D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus, Models and Numerical Methods, World Scientific: Singapore, 2012. doi: 10.1142/9789814355216. [12] S. Bhatter, A. Mathur, D. Kumar and J. Singh, A new analysis of fractional Drinfeld-Sokolov-Wilson model with exponential memory, Physica A, 537 (2020), 122578, 13 pp. doi: 10.1016/j.physa.2019.122578. [13] Y.-M. Chu, M. Adil Khan, T. Ali and S. S. Dragomir, Inequalities for $GA$-fractional differentiable functions, J. Inequal. Appl., 2017 (2017), 93-105.  doi: 10.1186/s13660-017-1371-6. [14] Z. Dahmani, New classes of integral inequalities of fractional order, Matematiche, 69 (2011), 237-247.  doi: 10.4418/2014.69.1.18. [15] F. Jarad, T. Abdeljawad and J. Alzabut, Generalized fractional derivatives generated by a class of local proportional derivatives, Eur. Phys. J. Spec. Top., 226 (2017), 3457-3471.  doi: 10.1140/epjst/e2018-00021-7. [16] F. Jarad, M. A. Alqudah and T. Abdeljawad, On more general forms of proportional fractional operators, Open Math., 18 (2020), 167-176. [17] F. Jarad, U. Ugurlu, T. Abdeljawad and D. Baleanu, On a new class of fractional operators, Adv. Differ. Equ, 2017 (2017), 247-263.  doi: 10.1186/s13662-017-1306-z. [18] H. Kalsoom, S. Rashid, M. Idrees, Y.-M. Chu and D. Baleanu, Two variable quantum integral inequalities of Simpson-type based on higher order generalized strongly preinvex and quasi preinvex functions, Symmetry, 12 (2020), 51. doi: 10.3390/sym12010051. [19] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Studies, 2006. [20] R. Khalil, M. Al Horani, A. Yousef and M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70.  doi: 10.1016/j.cam.2014.01.002. [21] D. Kumar, J. Singh and D. Baleanu, On the analysis of vibration equation involving a fractional derivative with Mittag-Leffler law, Math. Methods. Appl. Scis., 43 (2020), 443-457. [22] D. Kumar, J. Singh, K. Tanwar and D. Baleanu, A new fractional exothermic reactions model having constant heat source in porous media with power, exponential and Mittag-Leffler Laws, Inter. J. Heat. Mass. Transfer, 138 (2019), 1222-1227.  doi: 10.1016/j.ijheatmasstransfer.2019.04.094. [23] M. A. Latif, S. Rashid, S. S. Dragomir and Y.-M. Chu, Hermite-Hadamard type inequalities for co-ordinated convex and qausi-convex functions and their applications, J. Inequal. Appl, 2019 (2019), 317. doi: 10.1186/s13660-019-2272-7. [24] W. J. Liu, Q. A. Ngo and V. N. Huy, Several interesting integral inequalities, J. Math. Inequal., 3 (2009), 201-212.  doi: 10.7153/jmi-03-20. [25] J. Losada and J. J. Nieto, Properties of a new fractional derivative without singular kernel, Prog. Fract. Differ. Appl., 1 (2015), 87-92. [26] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, New York, 1993. [27] D. Nie, S. Rashid, A. O. Akdemir, D. Baleanu and J. -B. Liu, On some new weighted inequalities for differentiable exponentially convex and exponentially quasi-convex functions with applications, Mathematics, 7 (2019), 727. doi: 10.3390/math7080727. [28] M. A. Noor, K. I. Noor and S. Rashid, Some new classes of preinvex functions and inequalities, Mathematics, 7 (2019), 29. doi: 10.3390/math7010029. [29] D. Oregan and B. Samet, Lyapunov-type inequalities for a class of fractional differential equations, J. Inequal. Appl., 2015 (2015), 247-257.  doi: 10.1186/s13660-015-0769-2. [30] K. M. Owolabi, Numerical approach to fractional blow-up equations with Atangana-Baleanu derivative in Riemann-Liouville sense, Math. Model. Nat. Phenom., 13 (2018), 7-34.  doi: 10.1051/mmnp/2018006. [31] K. M. Owolabi, Analysis and numerical simulation of multicomponent system with Atangana-Baleanu fractional derivative, Chaos, Solitons & Fractals, 115 (2018), 127-134.  doi: 10.1016/j.chaos.2018.08.022. [32] K. M. Owolabi, Numerical patterns in system of integer and non-integer order derivatives, Chaos, Solitons & Fractals, 115 (2018), 143-153.  doi: 10.1016/j.chaos.2018.08.010. [33] K. M. Owolabi, Mathematical modelling and analysis of love dynamics: A fractional approach, Physica A: Stat. Mech. Appl., 525 (2019), 849-865.  doi: 10.1016/j.physa.2019.04.024. [34] K. M. Owolabi and A. Atangana, Computational study of multi-species fractional reaction-diffusion system with ABC operator, Chaos. Solitons & Fractals., 128 (2019), 280-289.  doi: 10.1016/j.chaos.2019.07.050. [35] K. M. Owolabi and A. Atangana, Numerical Methods for Fractional Differentiation, Springer Series in Computational Mathematics book series (SSCM), 2019. doi: 10.1007/978-981-15-0098-5. [36] K. M. Owolabi and A. Atangana, On the formulation of Adams-Bashforth scheme with Atangana-Baleanu-Caputo fractional derivative to model chaotic problems, Chaos, 29 (2019), 023111, 12 pp. doi: 10.1063/1.5085490. [37] K. M. Owolabi and Z. Hammouch, Mathematical modeling and analysis of two-variable system with noninteger-order derivative, Chaos, 29 (2019), 013145, 15 pp. doi: 10.1063/1.5086909. [38] K. M. Owolabi and Z. Hammouch, Spatiotemporal patterns in the Belousov-Zhabotinskii reaction systems with Atangana-Baleanu fractional order derivative, Physica A: Stat. Mech. Appl., 523 (2019), 1072-1090.  doi: 10.1016/j.physa.2019.04.017. [39] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego CA, 1999. [40] G. Rahman, T. Abdeljawad, A. Khan and K. S. Nisar, Some fractional proportional integral inequalities, J. Inequal. Appl., 2019 (2019), 244-257.  doi: 10.1186/s13660-019-2199-z. [41] G. Rahman, T. Abdeljawad, F. Jarad, A. Khan and K. S. Nisar, Certain inequalities via generalized proportional Hadamard fractional integral operators, Adv. Diff. Eqs, 2019 (2019), 454-464.  doi: 10.1186/s13662-019-2381-0. [42] S. Rashid, M. A. Noor, K. I. Noor and F. Safdar, Integral inequalities for generalized preinvex functions, Punjab. Univ. J. Math., 51 (2019), 77-91. [43] S. Rashid, M. A. Noor, K. I. Noor, F. Safdar and Y.-M. Chu, Hermite-Hadamard inequalities for the class of convex functions on time scale, Mathematics., 956 (2019). [44] S. Rashid, T. Abdeljawad, F. Jarad and M. A. Noor, Some estimates for generalized Riemann-Liouville fractional integrals of exponentially convex functions and their applications, Mathematics, 7 (2019), 807. doi: 10.3390/math7090807. [45] S. Rashid, A. O. Akdemir, F. Jarad, M. A. Noor and K. I. Noor, Simpson's type integral inequalities for $k$-fractional integrals and their applications, AIMS. Math., 4 (2019), 1087-1100.  doi: 10.3934/math.2019.4.1087. [46] S. Rashid, A. O. Akdemir, M. A. Noor and K. I. Noor, Generalization of inequalities analogous to preinvex functions via extended generalized Mittag-Leffler functions, in Proceedings of the International Conference on Applied and Engineering Mathematics?Second International Conference, ICAEM 2019, Hitec Taxila, Pakistan, (2019), 256–263. doi: 10.1109/ICAEM.2019.8853807. [47] S. Rashid, F. Jarad, M. A. Noor and H. Kalsoom, Inequalities by means of generalized proportional fractional integral operators with respect to another function, Mathematics, 7 (2020), 1225. doi: 10.3390/math7121225. [48] S. Rashid, M. A. Latif, Z. Hammouch and Y.-M. Chu, Fractional integral inequalities for strongly $h$-preinvex functions for a kth order differentiable functions, Symmetry, 11 (2019), 1448. doi: 10.3390/sym11121448. [49] S. Rashid, M. A. Noor and K. I. Noor, New Estimates for Exponentially Convex functions via conformable fractional operator, Fractal Fract., 3 (2019), 19. doi: 10.3390/fractalfract3020019. [50] S. Rashid, M. A. Noor and K. I. Noor, Some generalize Riemann-Liouville fractional estimates involving function having exponentially convexity property, Punjab. Univ. J. Math., 51 (2019), 1-15. [51] S. Rashid, M. A. Noor and K. I. Noor, Fractional exponentially $m$-convex functions and inequalities, Int. J. Anal.Appl., 17 (2019), 464-478. [52] S. Rashid, M. A. Noor and K. I. Noor, Inequalities pertaining fractional approach through exponentially convex functions, Fractal Fract., 37 (2019). doi: 10.3390/fractalfract3030037. [53] S. Rashid, M. A. Noor, K. I. Noor and A. O. Akdemir, Some new generalizations for exponentially $s$-convexfunctions and inequalities via fractional operators, Fractal Fract., 24 (2019). doi: 10.3390/fractalfract3020024. [54] S. Rashid, F. Safdar, A. O. Akdemir, M. A. Noor and K. I. Noor, Some new fractional integral inequalities for exponentially $m$-convex functions via extended generalized Mittag-Leffler function, J. Inequal. Appl., 2019 (2019), 299-316.  doi: 10.1186/s13660-019-2248-7. [55] G. Samko, A. A. Kilbas and I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon, (1993) [56] J. Singh, D. Kumar and D. Baleanu, New aspects of fractional Biswas-Milovic model with Mittag-Leffler law, Math. Modelling. Natural. Phenomena, 14 (2019), 303-326.  doi: 10.1051/mmnp/2018068. [57] Y. -Q. Song, M. Adil Khan, S. Zaheer Ullah and Y.-M. Chu, Integral inequalities involving strongly convex functions, J. Funct. Spaces, 2018 (2018), 6595921, 8 pp. doi: 10.1155/2018/6595921.

show all references

##### References:
 [1] T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57-66.  doi: 10.1016/j.cam.2014.10.016. [2] T. Abdeljawad and D. Baleanu, Monotonicity results for fractional difference operators with discrete exponential kernels, Adv. Differ. Equ., 2017 (2017), 78-87.  doi: 10.1186/s13662-017-1126-1. [3] T. Abdeljawad and D. Baleanu, On fractional derivatives with exponential kernel and their discrete versions, Rep. Math. Phys., 80 (2017), 11-27.  doi: 10.1016/S0034-4877(17)30059-9. [4] M. Adil Khan, Y.-M. Chu, T. U. Khan and J. Khan, Some new inequalities of Hermite-Hadamard type for $s$-convex functions with applications, Open Math., 15 (2017), 1414-1430. [5] M. Adil Khan, Y.-M. Chu, A. Kashuri, R. Liko and G. Ali, Conformable fractional integrals versions of Hermite-Hadamard inequalities and their generalizations, J. Funct. Spaces, 2018 (2018), 6928130, 9 pp. doi: 10.1155/2018/6928130. [6] M. Adil Khan, A. Iqbal, M. Suleman and Y.-M. Chu, Hermite-Hadamard type inequalities for fractionalintegrals via Green's function, J. Inequal. Appl., 2018 (2018), 161-176.  doi: 10.1186/s13660-018-1751-6. [7] R. Almeida, A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci. Numer. Simulat, 44 (2017), 460-481.  doi: 10.1016/j.cnsns.2016.09.006. [8] J. Alzabut, T. Abdeljawad, F. Jarad and W. Sudsutad, A Gronwall inequality via the generalized proportional fractional derivative with applications, J. Inequal. Appl., 2019 (2019), 101-113.  doi: 10.1186/s13660-019-2052-4. [9] D. R. Anderson and D. J. Ulness, Newly defined conformable derivatives, Adv. Dyn. Syst. Appl., 10 (2015), 109-137. [10] A. Atangana and K. M. Owolabi, New numerical approach for fractional differential equations, Math. Model. Nat. Phenom., 13 (2018), 3-24.  doi: 10.1051/mmnp/2018010. [11] D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus, Models and Numerical Methods, World Scientific: Singapore, 2012. doi: 10.1142/9789814355216. [12] S. Bhatter, A. Mathur, D. Kumar and J. Singh, A new analysis of fractional Drinfeld-Sokolov-Wilson model with exponential memory, Physica A, 537 (2020), 122578, 13 pp. doi: 10.1016/j.physa.2019.122578. [13] Y.-M. Chu, M. Adil Khan, T. Ali and S. S. Dragomir, Inequalities for $GA$-fractional differentiable functions, J. Inequal. Appl., 2017 (2017), 93-105.  doi: 10.1186/s13660-017-1371-6. [14] Z. Dahmani, New classes of integral inequalities of fractional order, Matematiche, 69 (2011), 237-247.  doi: 10.4418/2014.69.1.18. [15] F. Jarad, T. Abdeljawad and J. Alzabut, Generalized fractional derivatives generated by a class of local proportional derivatives, Eur. Phys. J. Spec. Top., 226 (2017), 3457-3471.  doi: 10.1140/epjst/e2018-00021-7. [16] F. Jarad, M. A. Alqudah and T. Abdeljawad, On more general forms of proportional fractional operators, Open Math., 18 (2020), 167-176. [17] F. Jarad, U. Ugurlu, T. Abdeljawad and D. Baleanu, On a new class of fractional operators, Adv. Differ. Equ, 2017 (2017), 247-263.  doi: 10.1186/s13662-017-1306-z. [18] H. Kalsoom, S. Rashid, M. Idrees, Y.-M. Chu and D. Baleanu, Two variable quantum integral inequalities of Simpson-type based on higher order generalized strongly preinvex and quasi preinvex functions, Symmetry, 12 (2020), 51. doi: 10.3390/sym12010051. [19] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Studies, 2006. [20] R. Khalil, M. Al Horani, A. Yousef and M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70.  doi: 10.1016/j.cam.2014.01.002. [21] D. Kumar, J. Singh and D. Baleanu, On the analysis of vibration equation involving a fractional derivative with Mittag-Leffler law, Math. Methods. Appl. Scis., 43 (2020), 443-457. [22] D. Kumar, J. Singh, K. Tanwar and D. Baleanu, A new fractional exothermic reactions model having constant heat source in porous media with power, exponential and Mittag-Leffler Laws, Inter. J. Heat. Mass. Transfer, 138 (2019), 1222-1227.  doi: 10.1016/j.ijheatmasstransfer.2019.04.094. [23] M. A. Latif, S. Rashid, S. S. Dragomir and Y.-M. Chu, Hermite-Hadamard type inequalities for co-ordinated convex and qausi-convex functions and their applications, J. Inequal. Appl, 2019 (2019), 317. doi: 10.1186/s13660-019-2272-7. [24] W. J. Liu, Q. A. Ngo and V. N. Huy, Several interesting integral inequalities, J. Math. Inequal., 3 (2009), 201-212.  doi: 10.7153/jmi-03-20. [25] J. Losada and J. J. Nieto, Properties of a new fractional derivative without singular kernel, Prog. Fract. Differ. Appl., 1 (2015), 87-92. [26] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, New York, 1993. [27] D. Nie, S. Rashid, A. O. Akdemir, D. Baleanu and J. -B. Liu, On some new weighted inequalities for differentiable exponentially convex and exponentially quasi-convex functions with applications, Mathematics, 7 (2019), 727. doi: 10.3390/math7080727. [28] M. A. Noor, K. I. Noor and S. Rashid, Some new classes of preinvex functions and inequalities, Mathematics, 7 (2019), 29. doi: 10.3390/math7010029. [29] D. Oregan and B. Samet, Lyapunov-type inequalities for a class of fractional differential equations, J. Inequal. Appl., 2015 (2015), 247-257.  doi: 10.1186/s13660-015-0769-2. [30] K. M. Owolabi, Numerical approach to fractional blow-up equations with Atangana-Baleanu derivative in Riemann-Liouville sense, Math. Model. Nat. Phenom., 13 (2018), 7-34.  doi: 10.1051/mmnp/2018006. [31] K. M. Owolabi, Analysis and numerical simulation of multicomponent system with Atangana-Baleanu fractional derivative, Chaos, Solitons & Fractals, 115 (2018), 127-134.  doi: 10.1016/j.chaos.2018.08.022. [32] K. M. Owolabi, Numerical patterns in system of integer and non-integer order derivatives, Chaos, Solitons & Fractals, 115 (2018), 143-153.  doi: 10.1016/j.chaos.2018.08.010. [33] K. M. Owolabi, Mathematical modelling and analysis of love dynamics: A fractional approach, Physica A: Stat. Mech. Appl., 525 (2019), 849-865.  doi: 10.1016/j.physa.2019.04.024. [34] K. M. Owolabi and A. Atangana, Computational study of multi-species fractional reaction-diffusion system with ABC operator, Chaos. Solitons & Fractals., 128 (2019), 280-289.  doi: 10.1016/j.chaos.2019.07.050. [35] K. M. Owolabi and A. Atangana, Numerical Methods for Fractional Differentiation, Springer Series in Computational Mathematics book series (SSCM), 2019. doi: 10.1007/978-981-15-0098-5. [36] K. M. Owolabi and A. Atangana, On the formulation of Adams-Bashforth scheme with Atangana-Baleanu-Caputo fractional derivative to model chaotic problems, Chaos, 29 (2019), 023111, 12 pp. doi: 10.1063/1.5085490. [37] K. M. Owolabi and Z. Hammouch, Mathematical modeling and analysis of two-variable system with noninteger-order derivative, Chaos, 29 (2019), 013145, 15 pp. doi: 10.1063/1.5086909. [38] K. M. Owolabi and Z. Hammouch, Spatiotemporal patterns in the Belousov-Zhabotinskii reaction systems with Atangana-Baleanu fractional order derivative, Physica A: Stat. Mech. Appl., 523 (2019), 1072-1090.  doi: 10.1016/j.physa.2019.04.017. [39] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego CA, 1999. [40] G. Rahman, T. Abdeljawad, A. Khan and K. S. Nisar, Some fractional proportional integral inequalities, J. Inequal. Appl., 2019 (2019), 244-257.  doi: 10.1186/s13660-019-2199-z. [41] G. Rahman, T. Abdeljawad, F. Jarad, A. Khan and K. S. Nisar, Certain inequalities via generalized proportional Hadamard fractional integral operators, Adv. Diff. Eqs, 2019 (2019), 454-464.  doi: 10.1186/s13662-019-2381-0. [42] S. Rashid, M. A. Noor, K. I. Noor and F. Safdar, Integral inequalities for generalized preinvex functions, Punjab. Univ. J. Math., 51 (2019), 77-91. [43] S. Rashid, M. A. Noor, K. I. Noor, F. Safdar and Y.-M. Chu, Hermite-Hadamard inequalities for the class of convex functions on time scale, Mathematics., 956 (2019). [44] S. Rashid, T. Abdeljawad, F. Jarad and M. A. Noor, Some estimates for generalized Riemann-Liouville fractional integrals of exponentially convex functions and their applications, Mathematics, 7 (2019), 807. doi: 10.3390/math7090807. [45] S. Rashid, A. O. Akdemir, F. Jarad, M. A. Noor and K. I. Noor, Simpson's type integral inequalities for $k$-fractional integrals and their applications, AIMS. Math., 4 (2019), 1087-1100.  doi: 10.3934/math.2019.4.1087. [46] S. Rashid, A. O. Akdemir, M. A. Noor and K. I. Noor, Generalization of inequalities analogous to preinvex functions via extended generalized Mittag-Leffler functions, in Proceedings of the International Conference on Applied and Engineering Mathematics?Second International Conference, ICAEM 2019, Hitec Taxila, Pakistan, (2019), 256–263. doi: 10.1109/ICAEM.2019.8853807. [47] S. Rashid, F. Jarad, M. A. Noor and H. Kalsoom, Inequalities by means of generalized proportional fractional integral operators with respect to another function, Mathematics, 7 (2020), 1225. doi: 10.3390/math7121225. [48] S. Rashid, M. A. Latif, Z. Hammouch and Y.-M. Chu, Fractional integral inequalities for strongly $h$-preinvex functions for a kth order differentiable functions, Symmetry, 11 (2019), 1448. doi: 10.3390/sym11121448. [49] S. Rashid, M. A. Noor and K. I. Noor, New Estimates for Exponentially Convex functions via conformable fractional operator, Fractal Fract., 3 (2019), 19. doi: 10.3390/fractalfract3020019. [50] S. Rashid, M. A. Noor and K. I. Noor, Some generalize Riemann-Liouville fractional estimates involving function having exponentially convexity property, Punjab. Univ. J. Math., 51 (2019), 1-15. [51] S. Rashid, M. A. Noor and K. I. Noor, Fractional exponentially $m$-convex functions and inequalities, Int. J. Anal.Appl., 17 (2019), 464-478. [52] S. Rashid, M. A. Noor and K. I. Noor, Inequalities pertaining fractional approach through exponentially convex functions, Fractal Fract., 37 (2019). doi: 10.3390/fractalfract3030037. [53] S. Rashid, M. A. Noor, K. I. Noor and A. O. Akdemir, Some new generalizations for exponentially $s$-convexfunctions and inequalities via fractional operators, Fractal Fract., 24 (2019). doi: 10.3390/fractalfract3020024. [54] S. Rashid, F. Safdar, A. O. Akdemir, M. A. Noor and K. I. Noor, Some new fractional integral inequalities for exponentially $m$-convex functions via extended generalized Mittag-Leffler function, J. Inequal. Appl., 2019 (2019), 299-316.  doi: 10.1186/s13660-019-2248-7. [55] G. Samko, A. A. Kilbas and I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon, (1993) [56] J. Singh, D. Kumar and D. Baleanu, New aspects of fractional Biswas-Milovic model with Mittag-Leffler law, Math. Modelling. Natural. Phenomena, 14 (2019), 303-326.  doi: 10.1051/mmnp/2018068. [57] Y. -Q. Song, M. Adil Khan, S. Zaheer Ullah and Y.-M. Chu, Integral inequalities involving strongly convex functions, J. Funct. Spaces, 2018 (2018), 6595921, 8 pp. doi: 10.1155/2018/6595921.
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