July  2021, 14(7): 2119-2135. doi: 10.3934/dcdss.2021063

More new results on integral inequalities for generalized $ \mathcal{K} $-fractional conformable Integral operators

1. 

Department of Mathematics, Huzhou University, Huzhou 313000, China

2. 

Government College University, Faisalabad, Pakistan

3. 

Department of Mathematics, Faculty of Arts and Sciences, Cankaya University Ankara, Turkey

4. 

Department of Mathematics, COMSATS University Islamabad, Islamabad, Pakistan

5. 

Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China

* Corresponding author: Saima Rashid

Received  December 2019 Revised  June 2020 Published  July 2021 Early access  May 2021

This paper aims to investigate the several generalizations by newly proposed generalized $ \mathcal{K} $-fractional conformable integral operator. Based on these novel ideas, we derived a novel framework to study for $ \breve{C} $eby$ \breve{s} $ev and P$ \acute{o} $lya-Szeg$ \ddot{o} $ type inequalities by generalized $ \mathcal{K} $-fractional conformable integral operator. Several special cases are apprehended in the light of generalized fractional conformable integral. This novel strategy captures several existing results in the relative literature. We also aim at showing important connections of the results here with those including Riemann-Liouville fractional integral operator.

Citation: Yu-Ming Chu, Saima Rashid, Fahd Jarad, Muhammad Aslam Noor, Humaira Kalsoom. More new results on integral inequalities for generalized $ \mathcal{K} $-fractional conformable Integral operators. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2119-2135. doi: 10.3934/dcdss.2021063
References:
[1]

T. Abdeljawad, J. Alzabut and F. Jarad, A generalized Lyapunov-type inequality in the frame of conformable derivatives, Adv. Differ. Eqs., 2017 (2017), Paper No. 321, 10 pp. doi: 10.1186/s13662-017-1383-z.

[2]

T. AbdeljawadF. Jarad and J. Alzabut, Fractional proportional differences with memory, Eur. Phys. J. Spec. Top., 226 (2017), 3333-3354.  doi: 10.1140/epjst/e2018-00053-5.

[3]

P. Agarwal, M. Jleli and M. Tomar, Certain Hermite-Hadamard type inequalities via generalized $\mathcal{K}$-fractional integrals, J. Inequal. Appl., 2017 (2017), Paper No. 55, 10 pp. doi: 10.1186/s13660-017-1318-y.

[4]

M. Altaf Khan, Z. Hammouch and D. Baleanu, Modeling the dynamics of hepatitis E via the Caputo-Fabrizio derivative, Math. Model. Nat. Phenomena, 14 (2019), Paper No. 311, 19 pp. doi: 10.1051/mmnp/2018074.

[5]

D. R. Anderson and D. J. Ulness, Newly defined conformable derivatives, Adv. Dyn. Syst. Appl., 10 (2015), 109-137. 

[6]

N. A. Asif, Z. Hammouch, M. B. Riaz and H. Bulut, Analytical solution of a Maxwell fluid with slip effects in view of the Caputo-Fabrizio derivative, The European Phys. J. Plus, 133 (2018), 272. doi: 10.1140/epjp/i2018-12098-6.

[7]

D. Baleanu, K. Diethelm and E. Scalas, Fractional Calculus: Models and Numerical Methods. World Scientific, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012. doi: 10.1142/9789814355216.

[8]

S. Belarbi and Z. Dahmani, On some new fractional integral inequalities, J. Ineq. Pure Appl. Math., 10 (2009), Article 86, 5 pages.

[9]

P. L. $\breve{C}$eby$\breve{s}$ev, Sur les expressions approximatives des integrales definies par les autres prises entre les memes limites, Proc. Math. Soc. Charkov, 2 (1982), 93-98. 

[10]

Z. Dahmani, New inequalities in fractional integrals, Int. J. of Nonlinear Sci., 9 (2010), 493-497. 

[11]

Z. DahmaniO. Mechouar and S. Brahami, Certain inequalities related to the $\breve{C}$eby$\breve{s}$ev functional involving a type Riemann-Liouville operator, Bull. of Math. Anal. and Appl., 3 (2011), 38-44. 

[12]

S. S. Dragomir and N. T. Diamond, Integral inequalities of Gr$\ddot{u}$ss tspe via Polya-Szeg$\ddot{O}$ and Shisha-Mond results., East Asian Math. J., 19 (2003), 27-39. 

[13]

N. Doming, S. Rashid, A. O. Akdemir, D. Baleanue 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.

[14]

F. JaradT. Abdeljawad and Z. Hammouch, On a class of ordinary differential equations in the frame of Atangana–Baleanu fractional derivative, Chaos, Solitons & Fractals, 117 (2018), 16-20.  doi: 10.1016/j.chaos.2018.10.006.

[15]

H. Kalsoom, S. Rashid, M. Idrees, F. Safdar, S. Akram, D. Baleanu and Y. -M. Chu, Post quantum integral inequalities of Hermite-Hadamard-type associated with Co-ordinated higher-order generalized strongly pre-Invex and quasi-pre-invex mappings, Symmetry, 12 (2020), 443. doi: 10.3390/sym12030443.

[16]

H. Kalsoom, S. Rashid, M. Idrees, D. Baleanu and Y. -M. Chu, 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.

[17]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Studies, Vol. 204, Elsevier (North-Holland) Science Publishers, Amsterdam, London and New York, 2006.

[18]

V. S. Kiryakova, Generalized Fractional Calculus and Applications, Pitman Research Notes in Mathematics Series, 301. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1994.

[19]

H. Khan, T. Abdeljawad, C. Tunç, A. Alkhazzan and A. Khan, Minkowski's inequality for the $AB$-fractional integral operator, J. Inequal. Appl., 2019 (2019), Paper No. 96, 12 pp. doi: 10.1186/s13660-019-2045-3.

[20]

U. N. Katugampola, New approach to generalized fractional integral, Appl. Math. Comput., 218 (2011), 860-865.  doi: 10.1016/j.amc.2011.03.062.

[21]

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.

[22]

J.-F. Li, S. Rashid, J.-B. Liu, A. O. Akdemir and F. Safdar, Inequalities involving conformable approach for exponentially convex functions and their applications, J. Fun. Spaces, 2020 (2020), Art ID 6517068, 17 pages. doi: 10.1155/2020/6517068.

[23]

W. J. LiuQ. A. Ngo and V. N. Huy., Several interesting integral inequalities, J. Math. Inequal., 3 (2009), 201-212.  doi: 10.7153/jmi-03-20.

[24]

T. U. Khan and M. A. Khan, Generalized conformable fractional integral operators, J. Comput. Appl. Math., 346 (2019), 378-389.  doi: 10.1016/j.cam.2018.07.018.

[25]

M. A. Khan, Y. Khurshid, T. S. Du and Y. -M. Chu, Generalization of Hermite-Hadamard type inequalities via conformable fractional integrals, J. Funct. Spaces, 2018 (2018), Art ID: 5357463, 12 pages. doi: 10.1155/2018/5357463.

[26]

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.

[27]

M. E. OzdemirE. SetA. O. Akdemir and M. Z. Sarikaya, Some new Chebyshev tspe inequalities for functions whose derivatives belongs to Lp spaces, Afr. Mat, 26 (2015), 1609-1619.  doi: 10.1007/s13370-014-0312-5.

[28] I. Podlubny, Fractional Differential Equations,, Academic Press, London, 1999. 
[29]

G. Pólya and G. Szeg$\ddot{o}$, Aufgaben und Lehrsatze aus der Analssis, Band I: Reihen. Integralrechnung. Funktionentheorie. (German) Dritte berichtigte Auflage. Die Grundlehren der Mathematischen Wissenschaften, Band 19 Springer-Verlag, Berlin-New York, 1964.

[30]

S. K. Nisar, G. Rahman and K. Mehrez, Chebyshev type inequalities via generalized fractional conformable integrals, J. Inequal. Appl., 2019 (2019), Paper No. 245, 9 pp. doi: 10.1186/s13660-019-2197-1.

[31]

S. K. NtouyasP. Agarwal and J. Tariboon, On Polya-Szeg$\ddot{o}$ and $\breve{C}$eby$\breve{s}$ev type inequalities involving the Riemann-Liouville fractional integral operators, J. Math. Inequal, 10 (2016), 491-504.  doi: 10.7153/jmi-10-38.

[32]

D. Oregan and B. Samet, Lyapunov-type inequalities for a class of fractional differential equations, J. Inequal. Appl., 2015 (2015), 1-10.  doi: 10.1186/s13660-015-0769-2.

[33]

S. Rashid, T. Abdeljawed, 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.

[34]

S. RashidR. AshrafM. A. NoorK. I. Noor and Y. -M. Chu, New weighted generalizations for differentiable exponentially convex mapping with application, AIMS Math., 5 (2020), 3525-3546.  doi: 10.3934/math.2020229.

[35]

S. RashidA. O. AkdemirF. JaradM. 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.

[36]

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 2018, Hitec Taxila, Pakistan, 2018.

[37]

S. Rashid, Z. Hammouch, K. Kalsoom, R. Ashraf and Y.-M. Chu, New investigation on the generalized $K$-fractional integral operators, Front. Phys., 25 (2020). doi: 10.3389/fphy.2020.00025.

[38]

S. Rashid, H. Kalsoom, Z. Hammouch, R. Ashraf, D. Baleanu and Y.-M. Chu, New multi-parametrized estimates having pth-order differentiability in fractional calculus for predominating h-convex functions in Hilbert space, Symmetry, 222 (2020), 222. doi: 10.3390/sym12020222.

[39]

S. Rashid, F. Jarad and Y. -M. Chu, A note on reverse Minkowski inequality via generalized proportional fractional integral operator with respect to another function, Math. Prob. Engineering, 2020 (2020), Article ID 7630260, 12 pages. doi: 10.1155/2020/7630260.

[40]

S. Rashid, F. Jarad, H. Kalsoom and Y.-M. Chu, On Polya-Szego and Cebysev type inequalities via generalized $K$-fractional integrals, Adv. Differ. Equ., 2020 (2020), Article ID 125, 18 pages. doi: 10.1186/s13662-020-02583-3.

[41]

S. Rashid, F. Jarad, M. A. Noor, H. Kalsoom and Y. -M. Chu, Inequalities by means of generalized proportional fractional integral operators with respect to another function, Mathematics, 7 (2019), 1225.

[42]

S. Rashid, F. Jarad, M. A. Noor, K. I. Noor, D. Baleanu and J. -B. Liu, On Gruss inequalities within generalized $K$-fractional integrals, Adv. Differ. Eq., 2020 (2020), Paper No. 203, 18 pp. doi: 10.1186/s13662-020-02644-7.

[43]

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, 1448 (2019), 1448. doi: 10.3390/sym11121448.

[44]

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), Paper No. 299, 17 pp. doi: 10.1186/s13660-019-2248-7.

[45]

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, 7 (2019), 956. doi: 10.3390/math7100956.

[46]

E. SetZ. Dahmani and I. Mumcu, New extensions of Chebyshev type inequalities using generalized Katugampola integrals via Polya-Szego inequality, Inter. J. Opt. Cont. Theor. Appl., 8 (2018), 132-144.  doi: 10.11121/ijocta.01.2018.00541.

[47]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Reading, 1993.

[48]

M. -K. Wang and Y.-M. Chu, Refinements of transformation inequal- ities for zero-balanced hypergeometric functions, Acta Math. Sci. B, 37 (2017), 607-622.  doi: 10.1016/S0252-9602(17)30026-7.

[49]

T. -H. Zhao, L. Shi and Y. -M. Chu, Convexity and concavity of the modified Bessel functions of the first kind with respect to Hölder means, RACSAM, 114 (2020), Article ID 96, 14 pages. doi: 10.1007/s13398-020-00825-3.

[50]

S. -S. Zhou, S. Rashid, S. S. Dragomir, M. A. Latif, A. O. Akdemir and J. -B. Liu, Some new inequalities involving $K$-fractional integral for certain classes of functions and their applications, J. Fun. Spaces, 2020 (2020), Article ID 5285147, 14 pages. doi: 10.1155/2020/5285147.

show all references

References:
[1]

T. Abdeljawad, J. Alzabut and F. Jarad, A generalized Lyapunov-type inequality in the frame of conformable derivatives, Adv. Differ. Eqs., 2017 (2017), Paper No. 321, 10 pp. doi: 10.1186/s13662-017-1383-z.

[2]

T. AbdeljawadF. Jarad and J. Alzabut, Fractional proportional differences with memory, Eur. Phys. J. Spec. Top., 226 (2017), 3333-3354.  doi: 10.1140/epjst/e2018-00053-5.

[3]

P. Agarwal, M. Jleli and M. Tomar, Certain Hermite-Hadamard type inequalities via generalized $\mathcal{K}$-fractional integrals, J. Inequal. Appl., 2017 (2017), Paper No. 55, 10 pp. doi: 10.1186/s13660-017-1318-y.

[4]

M. Altaf Khan, Z. Hammouch and D. Baleanu, Modeling the dynamics of hepatitis E via the Caputo-Fabrizio derivative, Math. Model. Nat. Phenomena, 14 (2019), Paper No. 311, 19 pp. doi: 10.1051/mmnp/2018074.

[5]

D. R. Anderson and D. J. Ulness, Newly defined conformable derivatives, Adv. Dyn. Syst. Appl., 10 (2015), 109-137. 

[6]

N. A. Asif, Z. Hammouch, M. B. Riaz and H. Bulut, Analytical solution of a Maxwell fluid with slip effects in view of the Caputo-Fabrizio derivative, The European Phys. J. Plus, 133 (2018), 272. doi: 10.1140/epjp/i2018-12098-6.

[7]

D. Baleanu, K. Diethelm and E. Scalas, Fractional Calculus: Models and Numerical Methods. World Scientific, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012. doi: 10.1142/9789814355216.

[8]

S. Belarbi and Z. Dahmani, On some new fractional integral inequalities, J. Ineq. Pure Appl. Math., 10 (2009), Article 86, 5 pages.

[9]

P. L. $\breve{C}$eby$\breve{s}$ev, Sur les expressions approximatives des integrales definies par les autres prises entre les memes limites, Proc. Math. Soc. Charkov, 2 (1982), 93-98. 

[10]

Z. Dahmani, New inequalities in fractional integrals, Int. J. of Nonlinear Sci., 9 (2010), 493-497. 

[11]

Z. DahmaniO. Mechouar and S. Brahami, Certain inequalities related to the $\breve{C}$eby$\breve{s}$ev functional involving a type Riemann-Liouville operator, Bull. of Math. Anal. and Appl., 3 (2011), 38-44. 

[12]

S. S. Dragomir and N. T. Diamond, Integral inequalities of Gr$\ddot{u}$ss tspe via Polya-Szeg$\ddot{O}$ and Shisha-Mond results., East Asian Math. J., 19 (2003), 27-39. 

[13]

N. Doming, S. Rashid, A. O. Akdemir, D. Baleanue 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.

[14]

F. JaradT. Abdeljawad and Z. Hammouch, On a class of ordinary differential equations in the frame of Atangana–Baleanu fractional derivative, Chaos, Solitons & Fractals, 117 (2018), 16-20.  doi: 10.1016/j.chaos.2018.10.006.

[15]

H. Kalsoom, S. Rashid, M. Idrees, F. Safdar, S. Akram, D. Baleanu and Y. -M. Chu, Post quantum integral inequalities of Hermite-Hadamard-type associated with Co-ordinated higher-order generalized strongly pre-Invex and quasi-pre-invex mappings, Symmetry, 12 (2020), 443. doi: 10.3390/sym12030443.

[16]

H. Kalsoom, S. Rashid, M. Idrees, D. Baleanu and Y. -M. Chu, 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.

[17]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Studies, Vol. 204, Elsevier (North-Holland) Science Publishers, Amsterdam, London and New York, 2006.

[18]

V. S. Kiryakova, Generalized Fractional Calculus and Applications, Pitman Research Notes in Mathematics Series, 301. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1994.

[19]

H. Khan, T. Abdeljawad, C. Tunç, A. Alkhazzan and A. Khan, Minkowski's inequality for the $AB$-fractional integral operator, J. Inequal. Appl., 2019 (2019), Paper No. 96, 12 pp. doi: 10.1186/s13660-019-2045-3.

[20]

U. N. Katugampola, New approach to generalized fractional integral, Appl. Math. Comput., 218 (2011), 860-865.  doi: 10.1016/j.amc.2011.03.062.

[21]

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.

[22]

J.-F. Li, S. Rashid, J.-B. Liu, A. O. Akdemir and F. Safdar, Inequalities involving conformable approach for exponentially convex functions and their applications, J. Fun. Spaces, 2020 (2020), Art ID 6517068, 17 pages. doi: 10.1155/2020/6517068.

[23]

W. J. LiuQ. A. Ngo and V. N. Huy., Several interesting integral inequalities, J. Math. Inequal., 3 (2009), 201-212.  doi: 10.7153/jmi-03-20.

[24]

T. U. Khan and M. A. Khan, Generalized conformable fractional integral operators, J. Comput. Appl. Math., 346 (2019), 378-389.  doi: 10.1016/j.cam.2018.07.018.

[25]

M. A. Khan, Y. Khurshid, T. S. Du and Y. -M. Chu, Generalization of Hermite-Hadamard type inequalities via conformable fractional integrals, J. Funct. Spaces, 2018 (2018), Art ID: 5357463, 12 pages. doi: 10.1155/2018/5357463.

[26]

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.

[27]

M. E. OzdemirE. SetA. O. Akdemir and M. Z. Sarikaya, Some new Chebyshev tspe inequalities for functions whose derivatives belongs to Lp spaces, Afr. Mat, 26 (2015), 1609-1619.  doi: 10.1007/s13370-014-0312-5.

[28] I. Podlubny, Fractional Differential Equations,, Academic Press, London, 1999. 
[29]

G. Pólya and G. Szeg$\ddot{o}$, Aufgaben und Lehrsatze aus der Analssis, Band I: Reihen. Integralrechnung. Funktionentheorie. (German) Dritte berichtigte Auflage. Die Grundlehren der Mathematischen Wissenschaften, Band 19 Springer-Verlag, Berlin-New York, 1964.

[30]

S. K. Nisar, G. Rahman and K. Mehrez, Chebyshev type inequalities via generalized fractional conformable integrals, J. Inequal. Appl., 2019 (2019), Paper No. 245, 9 pp. doi: 10.1186/s13660-019-2197-1.

[31]

S. K. NtouyasP. Agarwal and J. Tariboon, On Polya-Szeg$\ddot{o}$ and $\breve{C}$eby$\breve{s}$ev type inequalities involving the Riemann-Liouville fractional integral operators, J. Math. Inequal, 10 (2016), 491-504.  doi: 10.7153/jmi-10-38.

[32]

D. Oregan and B. Samet, Lyapunov-type inequalities for a class of fractional differential equations, J. Inequal. Appl., 2015 (2015), 1-10.  doi: 10.1186/s13660-015-0769-2.

[33]

S. Rashid, T. Abdeljawed, 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.

[34]

S. RashidR. AshrafM. A. NoorK. I. Noor and Y. -M. Chu, New weighted generalizations for differentiable exponentially convex mapping with application, AIMS Math., 5 (2020), 3525-3546.  doi: 10.3934/math.2020229.

[35]

S. RashidA. O. AkdemirF. JaradM. 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.

[36]

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 2018, Hitec Taxila, Pakistan, 2018.

[37]

S. Rashid, Z. Hammouch, K. Kalsoom, R. Ashraf and Y.-M. Chu, New investigation on the generalized $K$-fractional integral operators, Front. Phys., 25 (2020). doi: 10.3389/fphy.2020.00025.

[38]

S. Rashid, H. Kalsoom, Z. Hammouch, R. Ashraf, D. Baleanu and Y.-M. Chu, New multi-parametrized estimates having pth-order differentiability in fractional calculus for predominating h-convex functions in Hilbert space, Symmetry, 222 (2020), 222. doi: 10.3390/sym12020222.

[39]

S. Rashid, F. Jarad and Y. -M. Chu, A note on reverse Minkowski inequality via generalized proportional fractional integral operator with respect to another function, Math. Prob. Engineering, 2020 (2020), Article ID 7630260, 12 pages. doi: 10.1155/2020/7630260.

[40]

S. Rashid, F. Jarad, H. Kalsoom and Y.-M. Chu, On Polya-Szego and Cebysev type inequalities via generalized $K$-fractional integrals, Adv. Differ. Equ., 2020 (2020), Article ID 125, 18 pages. doi: 10.1186/s13662-020-02583-3.

[41]

S. Rashid, F. Jarad, M. A. Noor, H. Kalsoom and Y. -M. Chu, Inequalities by means of generalized proportional fractional integral operators with respect to another function, Mathematics, 7 (2019), 1225.

[42]

S. Rashid, F. Jarad, M. A. Noor, K. I. Noor, D. Baleanu and J. -B. Liu, On Gruss inequalities within generalized $K$-fractional integrals, Adv. Differ. Eq., 2020 (2020), Paper No. 203, 18 pp. doi: 10.1186/s13662-020-02644-7.

[43]

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, 1448 (2019), 1448. doi: 10.3390/sym11121448.

[44]

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), Paper No. 299, 17 pp. doi: 10.1186/s13660-019-2248-7.

[45]

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, 7 (2019), 956. doi: 10.3390/math7100956.

[46]

E. SetZ. Dahmani and I. Mumcu, New extensions of Chebyshev type inequalities using generalized Katugampola integrals via Polya-Szego inequality, Inter. J. Opt. Cont. Theor. Appl., 8 (2018), 132-144.  doi: 10.11121/ijocta.01.2018.00541.

[47]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Reading, 1993.

[48]

M. -K. Wang and Y.-M. Chu, Refinements of transformation inequal- ities for zero-balanced hypergeometric functions, Acta Math. Sci. B, 37 (2017), 607-622.  doi: 10.1016/S0252-9602(17)30026-7.

[49]

T. -H. Zhao, L. Shi and Y. -M. Chu, Convexity and concavity of the modified Bessel functions of the first kind with respect to Hölder means, RACSAM, 114 (2020), Article ID 96, 14 pages. doi: 10.1007/s13398-020-00825-3.

[50]

S. -S. Zhou, S. Rashid, S. S. Dragomir, M. A. Latif, A. O. Akdemir and J. -B. Liu, Some new inequalities involving $K$-fractional integral for certain classes of functions and their applications, J. Fun. Spaces, 2020 (2020), Article ID 5285147, 14 pages. doi: 10.1155/2020/5285147.

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