Some variants on Mercer's Hermite-Hadamard like inclusions of interval-valued functions for strong Kernel

Jamshed Nasir; Saber Mansour; Shahid Qaisar; Hassen Aydi; Jamshed Nasir; Saber Mansour; Shahid Qaisar; Hassen Aydi

doi:10.3934/math.2023506

AIMS Mathematics

2023, Volume 8, Issue 5: 10001-10020. doi: 10.3934/math.2023506

Previous Article Next Article

Research article

Some variants on Mercer's Hermite-Hadamard like inclusions of interval-valued functions for strong Kernel

1.
Virtual University of Pakistan, Department of Mathematics, Lahore Campus, Pakistan
2.
Department of Mathematics, Umm Al-Qura University, Faculty of Applied sciences, P.O. Box 14035, Holly Makkah 21955, Saudi Arabia
3.
COMSATS University of Pakistan, Department of Mathematics, Sahiwal Campus, Pakistan
4.
Institut Supérieur d'Informatique et des Techniques de Communication, Université de Sousse, 4000, Tunisia
5.
China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
6.
Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, South Africa

Received: 04 January 2023 Revised: 10 February 2023 Accepted: 14 February 2023 Published: 23 February 2023
MSC : 26A33, 26A51, 26D15

Using Atangana-Baleanu ($ AB $) fractional integral operators, we first establish some fractional Hermite-Hadamard-Mercer inclusions for interval-valued functions in this study. In this, some fresh developments of the Hermite-Hadamard inequality for fractional integral operators are presented. A few instances are also given to support our conclusions. The resulting results could provide new insight into a variety of integral inequalities for fuzzy interval-valued functions, fractional interval-valued functions, and the optimization issues they raise. Finally, matrices-related applications are also shown.
- convex function,
- Mercer's Hermite-Hadamard inequality,
- Atangana-Baleanu fractional integral operator,
- interval-valued function
Citation: Jamshed Nasir, Saber Mansour, Shahid Qaisar, Hassen Aydi. Some variants on Mercer's Hermite-Hadamard like inclusions of interval-valued functions for strong Kernel[J]. AIMS Mathematics, 2023, 8(5): 10001-10020. doi: 10.3934/math.2023506

Related Papers:

Abstract

Using Atangana-Baleanu ($ AB $) fractional integral operators, we first establish some fractional Hermite-Hadamard-Mercer inclusions for interval-valued functions in this study. In this, some fresh developments of the Hermite-Hadamard inequality for fractional integral operators are presented. A few instances are also given to support our conclusions. The resulting results could provide new insight into a variety of integral inequalities for fuzzy interval-valued functions, fractional interval-valued functions, and the optimization issues they raise. Finally, matrices-related applications are also shown.

References

[1]	K. Mehren, P. Agarwal, New Hermite-Hadamard type integral inequalities for the convex functions and theirs applications, J. Comput. Appl. Math., 350 (2019), 274–285. https://doi.org/10.1016/j.cam.2018.10.022 doi: 10.1016/j.cam.2018.10.022
[2]	D. S. Mitrinović, J. E. Pečarić, A. M. Fink, Classical and New Inequalities in Analysis, Dordrecht: Springer, 1993. https://doi.org/10.1007/978-94-017-1043-5
[3]	A. Mcd Mercer, A variant of Jensen's inequality, J. Ineq. Pure Appl. Math., 4 (2003), 73.
[4]	A. Matković, J. Pečarić, I. Perić, A variant of Jensen's inequality of Mercers type for operators with applications, Linear Algebra Appl., 418 (2006), 551–564. https://doi.org/10.1016/j.laa.2006.02.030 doi: 10.1016/j.laa.2006.02.030
[5]	M. Niezgoda, A generalization of Mercer's result on convex functions, Nonlinear Anal. Theory Methods Appl., 71 (2009), 2771–2779. https://doi.org/10.1016/j.na.2009.01.120 doi: 10.1016/j.na.2009.01.120
[6]	M. Kian, Operator Jensen inequality for superquadratic functions, Linear Algebra Appl., 456 (2014), 82–87. https://doi.org/10.1016/j.laa.2012.12.011 doi: 10.1016/j.laa.2012.12.011
[7]	M. Kian, M. Moslehian, Refinements of the operator Jensen-Mercer inequality, Electron. J. Linear Algebra, 26 (2013), 742–753. https://doi.org/10.13001/1081-3810.1684 doi: 10.13001/1081-3810.1684
[8]	J. Zhao, S. I. Butt, J. Nasir, Z. Wang, I. Tlili, Hermite-Jensen-Mercer type inequalities for Caputo fractional derivatives, J. Funct. Spaces, 2020 (2020), 7061549. https://doi.org/10.1155/2020/7061549 doi: 10.1155/2020/7061549
[9]	H. R. Moradi, S. Furuichi, Improvement and generalization of some Jensen-Mercer-type inequalities, J. Math. Ineq., 14 (2020), 377–383. http://doi.org/10.7153/jmi-2020-14-24 doi: 10.7153/jmi-2020-14-24
[10]	M. Niezgoda, A generalization of Mercer's result on convex functions, Nonlinear Anal. Theory Methods Appl., 71 (2009), 2771–2779. https://doi.org/10.1016/j.na.2009.01.120 doi: 10.1016/j.na.2009.01.120
[11]	M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 73–85. https://doi.org/10.12785/pfda/010201 doi: 10.12785/pfda/010201
[12]	T. Abdeljawad, D. Baleanu, On fractional derivatives with exponential kernel and their discrete versions, Rep. Math. Phys., 80 (2017), 11–27. https://doi.org/10.1016/S0034-4877(17)30059-9 doi: 10.1016/S0034-4877(17)30059-9
[13]	A. Atangana, D. Baleanu, New fractional derivatices with non-local and non-singular kernel: Theory and application to heat transfer model, Therm. Sci., 20 (2016), 763–769. http://doi.org/10.2298/TSCI160111018A doi: 10.2298/TSCI160111018A
[14]	Y. Chalco-Cano, A. Flores-Franulic, H. Roman-Flores, Ostrowski type inequalities for interval-valued functions using generalized Hukuhara derivative, Comput. Appl. Math., 31 (2012), 457–472. https://doi.org/10.1590/S1807-03022012000300002 doi: 10.1590/S1807-03022012000300002
[15]	Y. Chalco-Cano, W. A. Lodwick, W. Condori-Equice, Ostrowski type inequalities and applications in numerical integration for interval-valued functions, Soft Comput., 19 (2015), 3293–3300. https://doi.org/10.1007/s00500-014-1483-6 doi: 10.1007/s00500-014-1483-6
[16]	H. Budak, T. Tunc, M. Z. Sarikaya, Fractional Hermite-Hadamard-type inequalities for interval-valued functions, Proc. Am. Math. Soc., 148 (2020), 705–718.
[17]	T. M. Costa, Jensen's inequality type integral for fuzzy interval-valued functions, Fuzzy Sets Syst., 327 (2017), 31–47. https://doi.org/10.1016/j.fss.2017.02.001 doi: 10.1016/j.fss.2017.02.001
[18]	T. M. Costa, H. Roman-Flores, Some integral inequalities for fuzzy-interval-valued functions, Inf. Sci., 420 (2017), 110–125. https://doi.org/10.1016/j.ins.2017.08.055 doi: 10.1016/j.ins.2017.08.055
[19]	D. Zhao, T. An, G. Ye, W. Liu, New Jensen and Hermite-Hadamard type inequalities for h-convex interval-valued functions, J. Inequal. Appl., 2018 (2018), 302. https://doi.org/10.1186/s13660-018-1896-3 doi: 10.1186/s13660-018-1896-3
[20]	J. P. Aubin, A. Cellina, Differential Inclusions, Berlin, Heidelberg: Springer, 1984. https://doi.org/10.1007/978-3-642-69512-4
[21]	H. Kara, M. A. Ali, H. Budak, Hermite-Hadamard-Mercer type inclusions for interval-valued functions via Riemann-Liouville fractional integrals, Turk. J. Math., 46 (2022), 2193–2207. https://doi.org/10.55730/1300-0098.3263 doi: 10.55730/1300-0098.3263
[22]	S. Markov, On the algebraic properties of convex bodies and some applications, J. Convex Anal., 7 (2000), 129–166.
[23]	V. Lupulescu, Fractional calculus for interval-valued functions, Fuzzy Sets Syst., 265 (2015), 63–85. https://doi.org/10.1016/j.fss.2014.04.005 doi: 10.1016/j.fss.2014.04.005
[24]	R. E. Moore, Interval Analysis, Hoboken: Prentice-Hall, 1966.
[25]	T. Du, T. Zhou, On the fractional double integral inclusion relations having exponential kernels via interval-valued co-ordinated convex mappings, Chaos Solitons Fractals, 156 (2022), 111846. https://doi.org/10.1016/j.chaos.2022.111846 doi: 10.1016/j.chaos.2022.111846
[26]	R. E. Moore, R. B. Kearfott, M. J. Cloud, Introduction to Interval Analysis, SIAM, 2009. https://doi.org/10.1137/1.9780898717716
[27]	D. Zhao, G. Ye, W. Liu, D. F. Torres, Some inequalities for interval-valued functions on time scales, Soft Comput., 23 (2019), 6005–6015. https://doi.org/10.1007/s00500-018-3538-6 doi: 10.1007/s00500-018-3538-6
[28]	E. Sadowska, Hadamard inequality and a refinement of Jensen inequality for set-valued functions, Results. Math., 32 (1997), 332–337. https://doi.org/10.1007/BF03322144 doi: 10.1007/BF03322144
[29]	W. W. Breckner, Continuity of generalized convex and generalized concave set-valued functions, Revue d'analyse numérique et de théorie de l'approximation, 22 (1993), 39–51.
[30]	R. Osuna-Gómez, M. D. Jime´nez-Gamero, Y. Chalco-Cano, M. A. Rojas-Medar, Hadamard and Jensen inequalities for $s$-convex fuzzy processes, In: Soft Methodology and Random Information Systems, Berlin, Heidelberg: Springer, 2004. https://doi.org/10.1007/978-3-540-44465-7_80
[31]	I. B. Sial, S. Mei, M. A. Ali, H. Budak, A new variant of Jensen inclusion and Hermite-Hadamard type inclusions for interval-valued functions, In press.
[32]	M. Sababheh, Convex functions and means of matrices, J. Math. Ineq., 20 (2017), 29–47. https://doi.org/10.7153/mia-20-03 doi: 10.7153/mia-20-03

Reader Comments

Your name:*

Email:*
© 2023 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)