Milne-Type inequalities via expanded fractional operators: A comparative study with different types of functions

Abd-Allah Hyder; Hüseyin Budak; Mohamed A. Barakat; Abd-Allah Hyder; Hüseyin Budak; Mohamed A. Barakat

doi:10.3934/math.2024551

AIMS Mathematics

2024, Volume 9, Issue 5: 11228-11246. doi: 10.3934/math.2024551

Previous Article Next Article

Research article

Milne-Type inequalities via expanded fractional operators: A comparative study with different types of functions

1.
Department of Mathematics, College of Science, King Khalid University, P.O. Box 9004, Abha 61413, Saudi Arabia
2.
Department of Mathematics, Faculty of Science and Arts, Düzce University, Düzce, Türkiye
3.
Department of Computer Science, College of Al Wajh, University of Tabuk, Saudi Arabia
4.
Department of Mathematics, Faculty of Sciences, Al-Azhar University, Assiut 71524, Egypt

Received: 25 December 2023 Revised: 10 February 2024 Accepted: 19 February 2024 Published: 21 March 2024
MSC : 26D15, 26D10, 26D07

This study focused on deriving Milne-type inequalities using expanded fractional integral operators. We began by establishing a key equality associated with these operators. Using this equality, we explored Milne-type inequalities for functions with convex derivatives, supported by an illustrative example for clarity. Additionally, we investigated Milne-type inequalities for bounded and Lipschitzian functions utilizing fractional expanded integrals. Finally, we extended our exploration to Milne-type inequalities involving functions of bounded variation.
- fractional integrals,
- Milne-type inequalities,
- functions of bounded variation,
- convex functions,
- bounded functions
Citation: Abd-Allah Hyder, Hüseyin Budak, Mohamed A. Barakat. Milne-Type inequalities via expanded fractional operators: A comparative study with different types of functions[J]. AIMS Mathematics, 2024, 9(5): 11228-11246. doi: 10.3934/math.2024551

Related Papers:

Abstract

This study focused on deriving Milne-type inequalities using expanded fractional integral operators. We began by establishing a key equality associated with these operators. Using this equality, we explored Milne-type inequalities for functions with convex derivatives, supported by an illustrative example for clarity. Additionally, we investigated Milne-type inequalities for bounded and Lipschitzian functions utilizing fractional expanded integrals. Finally, we extended our exploration to Milne-type inequalities involving functions of bounded variation.

References

[1]	B. Çelik, M. Ç. Gürbüz, M. E. Özdemir, E. Set, On integral inequalities related to the weighted and the extended Chebyshev functionals involving different fractional operators, J. Inequal. Appl., 2020 (2020), 1–10. https://doi.org/10.1186/s13660-020-02512-8 doi: 10.1186/s13660-020-02512-8
[2]	M. A. Barakat, A. H. Soliman, A. Hyder, Langevin equations with generalized proportional Hadamard-Caputo fractional derivative, Comput. Intel. Neurosc., 2021 (2021). https://doi.org/10.1155/2021/6316477 doi: 10.1155/2021/6316477
[3]	T. S. Du, Y. Peng, Hermite-Hadamard type inequalities for multiplicative Riemann- Liouville fractional integrals, J. Comput. Appl. Math., 440 (2024), 115582. https://doi.org/10.1016/j.cam.2023.115582 doi: 10.1016/j.cam.2023.115582
[4]	F. Ertuğral, M. Z. Sarikaya, H. Budak, On Hermite-Hadamard type inequalities associated with the generalized fractional integrals, Filomat, 36 (2022), 3981–3993. https://doi.org/10.2298/FIL2212981E doi: 10.2298/FIL2212981E
[5]	Y. Peng, H. Fu, T. S. Du, Estimations of bounds on the multiplicative fractional integral inequalities having exponential kernels, Commun. Math. Stat., 2022. https://doi.org/10.1007/s40304-022-00285-8 doi: 10.1007/s40304-022-00285-8
[6]	A. A. Almoneef, M. A. Barakat, A. Hyder, Analysis of the fractional HIV model under proportional Hadamard-Caputo operators, Fractal Fract., 7 (2023), 220. https://doi.org/10.3390/fractalfract7030220 doi: 10.3390/fractalfract7030220
[7]	M. A. Khan, M. Hanif, Z. A. H. Khan, K. Ahmad, Y. M. Chu, Association of Jensen's inequality for $s$-convex function with Csiszár divergence, J. Inequal. Appl., 2019 (2019), 1–14. https://doi.org/10.1186/s13660-019-2112-9 doi: 10.1186/s13660-019-2112-9
[8]	M. A. Khan, J. Pecaric, Y. M. Chu, Refinements of Jensen's and McShane's inequalities with applications, AIMS Math., 5 (2020), 4931–4945. https://doi.org/10.3934/math.2020315 doi: 10.3934/math.2020315
[9]	M. Iqbal, M. I. Bhatti, K. Nazeer, Generalization of inequalities analogous to Hermite–Hadamard inequality via fractional integrals, B. Korean Math. Soc., 52 (2015), 707–716. https://doi.org/10.4134/BKMS.2015.52.3.707 doi: 10.4134/BKMS.2015.52.3.707
[10]	M. Z. Sarikaya, H. Budak, Some Hermite-Hadamard type integral inequalities for twice differentiable mappings via fractional integrals, Facta Univ. Ser. Math., 29 (2014), 371–384.
[11]	M. Z. Sarikaya, H. Yildirim, On Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals, Misk. Math. Notes, 17 (2016), 1049–1059.
[12]	M. U. Awan, M. Z. Javed, M. T. Rassias, M. A. Noor, K. I. Noor, Simpson type inequalities and applications, J. Anal., 29 (2021), 1403–1419. https://doi.org/10.1007/s41478-021-00319-4 doi: 10.1007/s41478-021-00319-4
[13]	M. Z. Sarikaya, E. Set, M. E. Özdemir, On new inequalities of Simpson's type for $s$-convex functions, Comput. Math. Appl., 60 (2010), 2191–2199. https://doi.org/10.1016/j.camwa.2010.07.033 doi: 10.1016/j.camwa.2010.07.033
[14]	T. S. Du, Y. J. Li, Z. Q. Yang, A generalization of Simpson's inequality via differentiable mapping using expanded $(s, m)$-convex functions, Appl. Math. Comput., 293 (2017), 358–369. https://doi.org/10.1016/j.amc.2016.08.045 doi: 10.1016/j.amc.2016.08.045
[15]	S. S. Dragomir, On Simpson's quadrature formula for mappings of bounded variation and applications, Tamkang J. Math., 30 (1999), 53–58. https://doi.org/10.5556/j.tkjm.30.1999.4207 doi: 10.5556/j.tkjm.30.1999.4207
[16]	M. Iqbal, S. Qaisar, S. Hussain, On Simpson's type inequalities utilizing fractional integrals, J. Comput. Anal. Appl., 23 (2017), 1137–1145.
[17]	S. Hussain, S. Qaisar, More results on Simpson's type inequality through convexity for twice differentiable continuous mappings, SpringerPlus, 5 (2016), 1–9. https://doi.org/10.1186/s40064-016-1683-x doi: 10.1186/s40064-016-1683-x
[18]	J. Nasir, S. Qaisar, S. I. Butt, A. Qayyum, Some Ostrowski type inequalities for mappings whose second derivatives are preinvex function via fractional integral operator, AIMS Math., 7 (2022), 3303–3320. https://doi.org/10.3934/math.2022184 doi: 10.3934/math.2022184
[19]	M. Z. Sarikaya, E. Set, M. E. Özdemir, On new inequalities of Simpson's type for functions whose second derivatives absolute values are convex, J. Appl. Math. Stat. Inf., 9 (2013), 37–45. https://doi.org/10.2478/jamsi-2013-0004 doi: 10.2478/jamsi-2013-0004
[20]	R. Hilfer, Applications of fractional calculus in physics, Singapore: World Scientific, 2000.
[21]	F. Jarad, E. Uğurlu, T. Abdeljawad, D. Baleanu, On a new class of fractional operators, Adv. Differ. Equ., 2017 (2017), 1–16. https://doi.org/10.1186/s13662-017-1306-z doi: 10.1186/s13662-017-1306-z
[22]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, Amsterdam: Elsevier, 2006.
[23]	A. Hyder, M. A. Barakat, A. Fathallah, Enlarged integral inequalities through recent fractional generalized operators, J. Inequal. Appl., 2022 (2022), 95. https://doi.org/10.1186/s13660-022-02831-y doi: 10.1186/s13660-022-02831-y
[24]	A. Hyder, M. A. Barakat, Novel improved fractional operators and their scientific applications, Adv. Differ. Equ., 2021 (2021), 1–24. https://doi.org/10.1186/s13662-021-03547-x doi: 10.1186/s13662-021-03547-x
[25]	R. Gorenflo, F. Mainardi, Fractional calculus: Integral and differential equations of fractional order, Wien: Springer-Verlag, 1997.
[26]	S. K. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, NewYork: Wiley, 1993.
[27]	H. M. Srivastava, J. Choi, Zeta and $q$-Zeta functions and associated series and integrals, Amsterdam: Elsevier Science Publishers, 2011.
[28]	M. V. Mihai, M. U. Awan, M. A. Noor, J. K. Kim, K. I. Noor, Hermite-Hadamard inequalities and their applications, J. Inequal. Appl., 2018 (2018), 309. https://doi.org/10.1186/s13660-018-1895-4 doi: 10.1186/s13660-018-1895-4

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)