In this paper, we propose a new framework of weighted generalized proportional fractional integral operator with respect to a monotone function $ \Psi, $ we develop novel modifications of the aforesaid operator. Moreover, contemplating the so-called operator, we determine several notable weighted Chebyshev and Grüss type inequalities with respect to increasing, positive and monotone functions $ \Psi $ by employing traditional and forthright inequalities. Furthermore, we demonstrate the applications of the new operator with numerous integral inequalities by inducing assumptions on $ \omega $ and $ \Psi $ verified the superiority of the suggested scheme in terms of efficiency. Additionally, our consequences have a potential association with the previous results. The computations of the proposed scheme show that the approach is straightforward to apply and computationally very user-friendly and accurate.
Citation: Shuang-Shuang Zhou, Saima Rashid, Asia Rauf, Fahd Jarad, Y. S. Hamed, Khadijah M. Abualnaja. Efficient computations for weighted generalized proportional fractional operators with respect to a monotone function[J]. AIMS Mathematics, 2021, 6(8): 8001-8029. doi: 10.3934/math.2021465
In this paper, we propose a new framework of weighted generalized proportional fractional integral operator with respect to a monotone function $ \Psi, $ we develop novel modifications of the aforesaid operator. Moreover, contemplating the so-called operator, we determine several notable weighted Chebyshev and Grüss type inequalities with respect to increasing, positive and monotone functions $ \Psi $ by employing traditional and forthright inequalities. Furthermore, we demonstrate the applications of the new operator with numerous integral inequalities by inducing assumptions on $ \omega $ and $ \Psi $ verified the superiority of the suggested scheme in terms of efficiency. Additionally, our consequences have a potential association with the previous results. The computations of the proposed scheme show that the approach is straightforward to apply and computationally very user-friendly and accurate.
| [1] | R. Gorenflo, F. Mainardi, I. Podlubny, Fractional differential equations, Academic Press, 1999,683–699. |
| [2] | R. Hilfer, Applications of fractional calculus in physics, Word Scientific, 2000. |
| [3] | A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and application of fractional differential equations, elsevier, 2006. |
| [4] | R. L. Magin, Fractional calculus in bioengineering, Begell House, 2006. |
| [5] | S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives: Theory and applications, Gordon and Breach, Yverdon, 1993. |
| [6] |
F. Jarad, T. Abdeljawad, D. Baleanu, Caputo-type modification of the Hadamard fractional derivative, Adv. Differ. Equ., 2012 (2012), 1–8. doi: 10.1186/1687-1847-2012-1
|
| [7] | U. N. Katugampola, New approach to generalized fractional integral, Appl. Math. Comput., 218 (2010), 860–865. |
| [8] | U. N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl., 6 (2014), 1–15. |
| [9] |
F. Jarad, T. Abdeljawad, 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
|
| [10] |
S. S. Zhou, S. Rashid, S. Parveen, A. O. Akdemir, Z. Hammouch, New computations for extended weighted functionals within the Hilfer generalized proportional fractional integral operators, AIMS Mathematics, 6 (2021), 4507–4525. doi: 10.3934/math.2021267
|
| [11] | M. Al-Qurashi, S. Rashid, S. Sultana, H. Ahmad, K. A. Gepreel, New formulation for discrete dynamical type inequalities via $\hbar$-discrete fractional operator pertaining to nonsingular kernel, Math. Biosci. Eng., 18 (2021), 1794–1812. DOI: 10.3934/mbe.2021093. |
| [12] | Y. M. Chu, S. Rashid, J. Singh, A novel comprehensive analysis on generalized harmonically $\Psi$-convex with respect to Raina's function on fractal set with applications, Math. Method. Appl. Sci., 2021, DOI: 10.1002/mma.7346. |
| [13] |
S. Rashid, Y. M. Chu, J. Singh, D. Kumar, A unifying computational framework for novel estimates involving discrete fractional calculus approaches, Alex. Eng. J., 60 (2021), 2677–2685. doi: 10.1016/j.aej.2021.01.003
|
| [14] | S. Rashid, Z. Hammouch, R. Ashraf, Y. M. Chu, New computation of unified bounds via a more general fractional operator using generalized Mittag-Leffler function in the kernel, Comp. Model. Eng., 126 (2021), 359–378. |
| [15] |
O. P. Agrawal, Generalized Multiparameters fractional variational calculus, Int. J. Differ. Equ., 2012 (2012), 1–38. doi: 10.1186/1687-1847-2012-1
|
| [16] | O. P. Agrawal, Some generalized fractional calculus operators and their applications in integral equations, Fract. Calc. Appl. Anal., 15 (2012), 700–711. |
| [17] |
M. Al-Refai, A. M. Jarrah, Fundamental results on weigted Caputo-Fabrizio fractional derivative, Chaos Soliton. Fract., 126 (2019), 7–11. doi: 10.1016/j.chaos.2019.05.035
|
| [18] |
M. Al-Refai, On weighted Atangana-Baleanu fractional operators, Adv. Differ. Equ., 2020 (2020), 1–11. doi: 10.1186/s13662-019-2438-0
|
| [19] |
F. Jarard, T. Abdeljawad, K. Shah, On the weighted fractional operators of a function with respect to another function, Fractals, 28 (2020), 2040011. doi: 10.1142/S0218348X20400113
|
| [20] |
Y. Zhang, X. Xing Liu, M. R. Belic, W. Zhong, Y. P. Zhang, M. Xiao, Propagation Dynamics of a Light Beam in a Fractional Schrödinger Equation, Phys. Rev. Lett., 115 (2015), 180403. doi: 10.1103/PhysRevLett.115.180403
|
| [21] |
Y. Zhang, H. Zhong, M. R. Belic, Y. Zhu, W. P. Zhong, Y. Zhang, et al. PT symmetry in a fractional Schrödinger equation, Laser Photonics Rev., 10 (2016), 526–531. doi: 10.1002/lpor.201600037
|
| [22] | S. Belarbi, Z. Dahmani, On some new fractional integral inequalities, J. Inequal. Pure Appl. Math., 10 (2009), 1–12. |
| [23] |
S. I. Butt, A. O. Akdemir, M. Y. Bhatti, M. Nadeem, New refinements of Chebyshev-Polya-Szego-type inequalities via generalized fractional integral operators, J. Inequal. Appl., 2020 (2020), 1–13. doi: 10.1186/s13660-019-2265-6
|
| [24] |
S. Rashid, F. Jarad, H. Kalsoom, Y. M. Chu, On Polya-Szego and Cebysev type inequalities via generalized k-fractional integrals, Adv. Differ. Equ., 2020 (2020), 1–18. doi: 10.1186/s13662-019-2438-0
|
| [25] | E. Set, Z. Dahmani, İ. Mumcu, New extensions of Chebyshev type inequalities using generalized Katugampola integrals via Polya-Szego inequality, IJOCTA, 8 (2018), 137–144. |
| [26] | Z. Dahmani, New inequalities in fractional integrals, Int. J. Nonlinear Sci., 9 (2010), 493–497. |
| [27] | V. Chinchane, D. Pachpatte, On some integral inequalities using Hadamard fractional integral, J. Mat., 1 (2012), 62–66. |
| [28] | K. Brahim, S. Taf, On some fractional $q$-integral inequalities, J. Mat., 3 (2013), 21–26. |
| [29] |
S. B. Chen, S. Rashid, M. A. Noor, R. Ashraf, Y. M. Chu, A new approach on fractional calculus and probability density function, AIMS Mathematics, 5 (2020), 7041–7054. doi: 10.3934/math.2020451
|
| [30] | P. L. Chebyshev, Sur les expressions approximatives des integrales definies par les autres prises entre les mmes limites, Proc. Math. Soc. Charkov, 2 (1882), 93–98. |
| [31] |
G. Grüss, Uber das Maximum des absoluten Betrages von $\frac{1}{b_{1}-a_{1}}\int\limits_{a_{1}}^{b_{1}}f_{1}(\varkappa)g_{1}(\varkappa)d\varkappa\leq\Big(\frac{1}{b_{1}-a_{1}}\Big)^{2}\int\limits_{a_{1}}^{b_{1}}f_{1}(\varkappa)d\varkappa\int\limits_{a_{1}}^{b_{1}}g_{1}(\varkappa)d\varkappa$, Math. Z., 39 (1935), 215–226. doi: 10.1007/BF01201355
|
| [32] | D. S. Mitrinovic, J. E. Pecaric, A. M. Fink, Classical and new inequalities in analysis, Springer, Dordrecht, 1993. |
| [33] |
S. Rashid, T. Abdeljawad, F. Jarad, 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] |
T. H. Zhao, M. K. Wang, Y. M. Chu, A sharp double inequality involving generalized complete elliptic integral of the first kind, AIMS Mathematics, 5 (2020), 4512–4528. doi: 10.3934/math.2020290
|
| [35] |
M. Adil Khan, J. E. Pecaric, Y. M. Chu, Refinements of Jensen's and McShane's inequalities with applications, AIMS Mathematics, 5 (2020), 4931–4945. doi: 10.3934/math.2020315
|
| [36] |
S. S. Dragomir, Quasi Grüss type inequalities for continuous functions of selfadjoint operators in Hilbert spaces, Filomat, 27 (2013), 277–289. doi: 10.2298/FIL1302277D
|
| [37] | S. S. Dragomir, Some integral inequalities of Grüss type, Indian J. Pure Appl. Math., 4 (1998), 397–415. |
| [38] | Z. Dahmani, L. Tabharit, S. Taf, New generalisations of Grüss inequality using Riemann-Liouville fractional integrals, Bull. Math. Anal. Appl., 2 (2010), 93–99. |
| [39] | Z. Dahmani, A. Benzidane, New weighted Grüss type inequalities via $(\alpha, \beta)$ fractional q-integral inequalities, IJIAS, 1 (2012), 76–83. |
| [40] | Z. Dahmani, Some results associate with fractional integrals involving the extended Chebyshev functional, Acta Univ. Apulens, 27 (2011), 217–224 |
| [41] |
Z. Dahmani, L. Tabharit, S. Taf, New results using fractional integrals, Journal of Interdisciplinary Mathematics, 13 (2010), 601–606. doi: 10.1080/09720502.2010.10700721
|
| [42] | E. Set, M. Tomar, M. Z. Sarikaya, On generalized Grüss type inequalities for k-fractional integrals, Appl. Math. Comput., 269 (2015), 29–34. |
| [43] |
S. B. Chen, S. Rashid, M. A. Noor, Z. Hammouch, Y. M. Chu, New fractional approaches for n-polynomial $p$-convexity with applications in special function theory, Adv. Differ. Equ., 2020 (2020), 1–31. doi: 10.1186/s13662-019-2438-0
|
| [44] |
T. Abdeljawad, S. Rashid, Z. Hammouch, İ. İşcan, Y. M. Chu, Some new Simpson-type inequalities for generalized $p$-convex function on fractal sets with applications, Adv. Differ. Equ., 2020 (2020), 1–26. doi: 10.1186/s13662-019-2438-0
|
| [45] |
F. Jarad, T. Abdeljawad, S. Rashid, Z. Hammouch, More properties of the proportional fractional integrals and derivatives of a function with respect to another function, Adv. Differ. Equ., 2020 (2020), 1–16. doi: 10.1186/s13662-019-2438-0
|
| [46] |
S. Rashid, F. Jarad, M. A. Noor, H. Kalsoom, Y. M. Chu, Inequalities by means of generalized proportional fractional integral operators with respect to another function, Mathematics, 7 (2019), 1225. doi: 10.3390/math7121225
|
| [47] |
F. Jarad, M. A. Alqudah, T. Abdeljawad, On more generalized form of proportional fractional operators, Open Math., 18 (2020), 167–176. doi: 10.1515/math-2020-0014
|
| [48] |
F. Jarad, T. Abdeljawad, 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
|
| [49] |
G. Rahman, T. Abdeljawad, F. Jarad, A. Khan, K. S. Nisar, Certain inequalities via generalized proportional Hadamard fractional integral operators, Adv. Differ. Equ., 2019 (2019), 1–10. doi: 10.1186/s13662-018-1939-6
|
| [50] |
T. U. Khan, M. Adil Khan, Generalized conformable fractional operators, J. Comput. Appl. Math., 346 (2019), 378–389. doi: 10.1016/j.cam.2018.07.018
|
| [51] |
F. Jarad, E. Ugurlu, T. Abdeljawad, D. Baleanu, On a new class of fractional operators, Adv. Differ. Equ., 2017 (2017), 1–16. doi: 10.1186/s13662-016-1057-2
|
| [52] |
G. J. O. Jameson, The incomplete gamma functions, The Mathematical Gazette, 100 (2016), 298–306. doi: 10.1017/mag.2016.67
|
| [53] | N. N. Lebedev, Special functions and their applications Prentice-Hall, INC. Englewood Cliffs, 1965. |
| [54] | D. R. Anderson, D. J. Ulness, Newly defined conformable derivatives, Adv. Dyn. Syst. Appl., 10 (2015), 109–137. |
Shuang-Shuang Zhou, Saima Rashid, Asia Rauf, Fahd Jarad, Y. S. Hamed, Khadijah M. Abualnaja. Efficient computations for weighted generalized proportional fractional operators with respect to a monotone function[J]. AIMS Mathematics, 2021, 6(8): 8001-8029. doi: 10.3934/math.2021465
DownLoad: