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  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 & Continuous Dynamical Systems - S, 2021, 14 (7) : 2119-2135. doi: 10.3934/dcdss.2021063
References:
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[25]

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.   Google Scholar

[10]

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

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[46]

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