Some new Jensen, Schur and Hermite-Hadamard inequalities for log convex fuzzy interval-valued functions

Muhammad Bilal Khan; Hari Mohan Srivastava; Pshtiwan Othman Mohammed; Kamsing Nonlaopon; Y. S. Hamed; Muhammad Bilal Khan; Hari Mohan Srivastava; Pshtiwan Othman Mohammed; Kamsing Nonlaopon; Y. S. Hamed

doi:10.3934/math.2022241

AIMS Mathematics

2022, Volume 7, Issue 3: 4338-4358. doi: 10.3934/math.2022241

Previous Article Next Article

Research article Special Issues

Some new Jensen, Schur and Hermite-Hadamard inequalities for log convex fuzzy interval-valued functions

1.
Department of Mathematics, COMSATS University Islamabad, Islamabad 44000, Pakistan
2.
Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada
3.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
4.
Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, AZ1007 Baku, Azerbaijan
5.
Section of Mathematics, International Telematic University Uninettuno, I-00186 Rome, Italy
6.
Department of Mathematics, College of Education, University of Sulaimani, Sulaimani 46001, Kurdistan Region, Iraq
7.
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand
8.
Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia

Received: 01 October 2021 Revised: 05 December 2021 Accepted: 13 December 2021 Published: 20 December 2021
MSC : 26A33, 26A51, 26D10

The inclusion relation and the order relation are two distinct ideas in interval analysis. Convexity and nonconvexity create a significant link with different sorts of inequalities under the inclusion relation. For many classes of convex and nonconvex functions, many works have been devoted to constructing and refining classical inequalities. However, it is generally known that log-convex functions play a significant role in convex theory since they allow us to deduce more precise inequalities than convex functions. Because the idea of log convexity is so important, we used fuzzy order relation $\left(\preceq \right)$ to establish various discrete Jensen and Schur, and Hermite-Hadamard (H-H) integral inequality for log convex fuzzy interval-valued functions (L-convex F-I-V-Fs). Some nontrivial instances are also offered to bolster our findings. Furthermore, we show that our conclusions include as special instances some of the well-known inequalities for L-convex F-I-V-Fs and their variant forms. Furthermore, we show that our conclusions include as special instances some of the well-known inequalities for L-convex F-I-V-Fs and their variant forms. These results and different approaches may open new directions for fuzzy optimization problems, modeling, and interval-valued functions.
Citation: Muhammad Bilal Khan, Hari Mohan Srivastava, Pshtiwan Othman Mohammed, Kamsing Nonlaopon, Y. S. Hamed. Some new Jensen, Schur and Hermite-Hadamard inequalities for log convex fuzzy interval-valued functions[J]. AIMS Mathematics, 2022, 7(3): 4338-4358. doi: 10.3934/math.2022241

Related Papers:

Abstract

The inclusion relation and the order relation are two distinct ideas in interval analysis. Convexity and nonconvexity create a significant link with different sorts of inequalities under the inclusion relation. For many classes of convex and nonconvex functions, many works have been devoted to constructing and refining classical inequalities. However, it is generally known that log-convex functions play a significant role in convex theory since they allow us to deduce more precise inequalities than convex functions. Because the idea of log convexity is so important, we used fuzzy order relation $\left(\preceq \right)$ to establish various discrete Jensen and Schur, and Hermite-Hadamard (H-H) integral inequality for log convex fuzzy interval-valued functions (L-convex F-I-V-Fs). Some nontrivial instances are also offered to bolster our findings. Furthermore, we show that our conclusions include as special instances some of the well-known inequalities for L-convex F-I-V-Fs and their variant forms. Furthermore, we show that our conclusions include as special instances some of the well-known inequalities for L-convex F-I-V-Fs and their variant forms. These results and different approaches may open new directions for fuzzy optimization problems, modeling, and interval-valued functions.

References

[1]	G. D. Anderson, M. K. Vamanamurthy, M. Vuorinen, Generalized convexity and inequalities, J. Math. Anal. Appl., 335 (2007), 1294-1308. https://doi.org/10.1016/j.jmaa.2007.02.016 doi: 10.1016/j.jmaa.2007.02.016
[2]	M. Avci, H. Kavurmaci, M. E. Özdemir, New inequalities of Hermite-Hadamard type via s-convex functions in the second sense with applications, Appl. Math. Comput., 217 (2011), 5171-5176. https://doi.org/10.1016/j.amc.2010.11.047 doi: 10.1016/j.amc.2010.11.047
[3]	F. Chen, S. Wu, Integral inequalities of Hermite-Hadamard type for products of two h-convex functions, Abstr. Appl. Anal., 6 (2014), 1-6.
[4]	W. Liu, New integral inequalities involving beta function via P-convexity, Miskolc Math. Notes, 15 (2014), 585-591. https://doi.org/10.18514/MMN.2014.660 doi: 10.18514/MMN.2014.660
[5]	J. Hadamard, É tude sur les propriétés des fonctions entières et en particulier d'une fonction considérée par Riemann, J. Math. Pure. Appl., 5 (1893), 171-216.
[6]	C. Hermite, Sur deux limites d'une intégrale définie, Mathesis, 3 (1883), 1-82.
[7]	M. Alomari, M. Darus, S. S. Dragomir, P. Cerone, Ostrowski type inequalities for functions whose derivatives are s-convex in the second sense, Appl. Math. Lett., 23 (2010), 1071-1076. https://doi.org/10.1016/j.aml.2010.04.038 doi: 10.1016/j.aml.2010.04.038
[8]	P. Cerone, S. S. Dragomir, Ostrowski type inequalities for functions whose derivatives satisfy certain convexity assumptions, Demonstr. Math., 37 (2004), 299-308. https://doi.org/10.1515/dema-2004-0208 doi: 10.1515/dema-2004-0208
[9]	M. A. Latif, S. Rashid, S. S. Dragomir, Y. M. Chu, Hermite-Hadamard type inequalities for co-ordinated convex and quasi-convex functions and their applications, J. Inequal. Appl., 2019 (2019), 317. https://doi.org/10.1186/s13660-019-2272-7 doi: 10.1186/s13660-019-2272-7
[10]	L. Fejér, Uber die fourierreihen, Ⅱ, Math. Naturwise. Anz, Ungar. Akad. Wiss., 24 (1906), 369-390.
[11]	S. Varošanec, On h-convexity, J. Math. Anal. Appl., 326 (2007), 303-311. https://doi.org/10.1016/j.jmaa.2006.02.086 doi: 10.1016/j.jmaa.2006.02.086
[12]	B. Y. Xi, F. Qi, Some integral inequalities of Hermite-Hadamard type for s-log convex functions, Acta Math. Sci. Ser. A (Chin. Ed.), 35 (2015), 515-524. https://doi.org/10.13140/RG.2.1.4385.9044 doi: 10.13140/RG.2.1.4385.9044
[13]	M. A. Noor, F. Qi, M. U. Awan, Some Hermite-Hadamard type inequalities for log-h-convex functions, Analysis, 33 (2013), 367-375. https://doi.org/10.1524/anly.2013.1223 doi: 10.1524/anly.2013.1223
[14]	M. A. Noor, Hermite-Hadamard integral inequalities for log-preinvex functions, J. Math. Anal. Approx. Theory, 2 (2007), 126-131.
[15]	J. E. Peajcariaac, F. Proschan, Y. L. Tong, Convex functions, partial orderings and statistical applications, Academic Press, 1992.
[16]	S. S. Dragomir C. E. M. Pearce, Selected topics on Hermite-Hadamard inequalities and applications, 2003.
[17]	S. S. Dragomir, Refinements of the Hermite-Hadamard integral inequality for log-convex functions, 2000.
[18]	S. S. Dragomir, B. Mond, Integral inequalities of Hadamard type for log convex functions, Demonstr. Math., 31 (1998), 354-364. https://doi.org/10.1515/dema-1998-0214 doi: 10.1515/dema-1998-0214
[19]	S. S. Dragomir, New inequalities of Hermite-Hadamard type for log convex functions, Khayyam J. Math., 3 (2017), 98-115. https://doi.org/10.22034/kjm.2017.47458 doi: 10.22034/kjm.2017.47458
[20]	S. S. Dragomir, J. Pecaric, L. E. Persson, Some inequalities of Hadamard type, Soochow J. Math., 21 (1995), 335-341.
[21]	S. S. Dragomir, A survey of Jensen type inequalities for log-convex functions of self adjoint operators in Hilbert spaces, Commun. Math. Anal., 10 (2011), 82-104.
[22]	Jr R. Goetschel, W. Voxman, Elementary fuzzy calculus, Fuzzy set. syst., 18 (1986), 31-43. https://doi.org/10.1016/0165-0114(86)90026-6 doi: 10.1016/0165-0114(86)90026-6
[23]	R. E. Moore, Interval analysis, Englewood Cliffs: Prentice Hall, 1966.
[24]	U. W. Kulish, W. Miranker, Computer arithmetic in theory and practice, New York: Academic Press, 1981.
[25]	S. Nanda, K. Kar, Convex fuzzy mappings, Fuzzy Set. Syst., 48 (1992), 129-132. https://doi.org/10.1016/0165-0114(92)90256-4 doi: 10.1016/0165-0114(92)90256-4
[26]	S. S. Chang, Variational inequality and complementarity problems theory and applications, Shanghai: Shanghai Scientific and Technological Literature Publishing House, 1991.
[27]	M. A. Noor, Fuzzy preinvex functions, Fuzzy Set. Syst., 64 (1994), 95-104. https://doi.org/10.1016/0165-0114(94)90011-6 doi: 10.1016/0165-0114(94)90011-6
[28]	B. Bede, Mathematics of fuzzy sets and fuzzy logic studies in fuzziness and soft computing, In: Studies in fuzziness and soft computing springer, 2013. https://doi.org/10.1007/978-3-642-35221-8
[29]	J. Cervelati, M. D. Jiménez-Gamero, F. Vilca-Labra, M. A. Rojas-Medar, Continuity for s-convex fuzzy processes, In: Soft methodology and random information systems, 2004,653-660. https://doi.org/10.1007/978-3-540-44465-7_81
[30]	Y. Chalco-Cano, M. A. Rojas-Medar, H. Román-Flores, M-convex fuzzy mappings and fuzzy integral mean, Comput. Math. Appl., 40 (2000), 1117-1126. https://doi.org/10.1016/S0898-1221(00)00226-1 doi: 10.1016/S0898-1221(00)00226-1
[31]	M. B. Khan, M. A. Noor, K. I. Noor, A. T. Ab Ghani, L. Abdullah, Extended perturbed mixed variational-like inequalities for fuzzy mappings, J. Math., 2021 (2021), 6652930. https://doi.org/10.1155/2021/6652930 doi: 10.1155/2021/6652930
[32]	M. L. Puri, D. A. Ralescu, Fuzzy random variables, J. Math. Anal. Appl., 114 (1986), 409-422. https://doi.org/10.1016/0022-247X(86)90093-4 doi: 10.1016/0022-247X(86)90093-4
[33]	H. Román-Flores, Y. Chalco-Cano, W. A. Lodwick, Some integral inequalities for interval-valued functions, Comput. Appl. Math., 37 (2018), 1306-1318. https://doi.org/10.1007/s40314-016-0396-7 doi: 10.1007/s40314-016-0396-7
[34]	H. Roman-Flores, Y. Chalco-Cano, G. N. Silva, A note on Gronwall type inequality for interval-valued functions, In: 2013 IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS), 2013, 1455-1458. https://doi.org/10.1109/IFSA-NAFIPS.2013.6608616
[35]	Y. Chalco-Cano, A. Flores-Franulic, H. Román-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
[36]	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
[37]	D. Zhang, C. Guo, D. Chen, G. Wang, Jensen's inequalities for set-valued and fuzzy set-valued functions, Fuzzy Set. Syst., 404 (2021), 178-204. https://doi.org/10.1016/j.fss.2020.06.003 doi: 10.1016/j.fss.2020.06.003
[38]	T. M. Costa, Jensen's inequality type integral for fuzzy-interval-valued functions, Fuzzy Set. Syst., 327 (2017), 31-47. https://doi.org/10.1016/j.fss.2017.02.001 doi: 10.1016/j.fss.2017.02.001
[39]	T. M. Costa, H. Roman-Flores, Some integral inequalities for fuzzy-interval-valued functions, Inform. Sci., 420 (2017), 110-125. https://doi.org/10.1016/j.ins.2017.08.055 doi: 10.1016/j.ins.2017.08.055
[40]	O. Kaleva, Fuzzy differential equations, Fuzzy set. Sys., 24 (1987), 301-317. https://doi.org/10.1016/0165-0114(87)90029-7 doi: 10.1016/0165-0114(87)90029-7
[41]	J. Matkowski, K. Nikodem, An integral Jensen inequality for convex multifunctions, Results Math., 26 (1994), 348-353. https://doi.org/10.1007/BF03323058 doi: 10.1007/BF03323058
[42]	P. Diamond, P. E. Kloeden, Metric spaces of fuzzy sets: Theory and applications, World Scient, 1994.
[43]	M. B. Khan, M. A. Noor, K. I. Noor, Y. M. Chu, Higher-order strongly preinvex fuzzy mappings and fuzzy mixed variational-like inequalities, Int. J. Comput. Intell. Syst., 14 (2021), 186-1870. https://doi.org/10.2991/ijcis.d.210616.001 doi: 10.2991/ijcis.d.210616.001
[44]	P. Liu, M. B. Khan, M. A. Noor, K. I. Noor, On strongly generalized preinvex fuzzy mappings, J. Math., 2021 (2021), 6657602. https://doi.org/10.1155/2021/6657602 doi: 10.1155/2021/6657602
[45]	P. Liu, M. B. Khan, M. A. Noor, K. I. Noor, New Hermite-Hadamard and Jensen inequalities for log-s-convex fuzzy-interval-valued functions in the second sense, Complex Intell. Syst., 2021 (2021), 1-15. https://doi.org/10.1007/s40747-021-00379-w doi: 10.1007/s40747-021-00379-w
[46]	M. B. Khan, M. A. Noor, L. Abdullah, K. I. Noor, New Hermite-Hadamard and Jensen inequalities for log-h-convex fuzzy-interval-valued functions, Int. J. Comput. Intell. Syst., 14 (2021), 1-16. https://doi.org/10.1007/s44196-021-00004-1 doi: 10.1007/s44196-021-00004-1
[47]	M. B. Khan, M. A. Noor, K. I. Noor, Y. M. Chu, New Hermite-Hadamard type inequalities for (h1, h2)-convex fuzzy-interval-valued functions, Adv. Differ. Equ., 2021 (2021), 6-20. https://doi.org/10.1186/s13662-021-03245-8 doi: 10.1186/s13662-021-03245-8
[48]	M. B. Khan, M. A. Noor, L. Abdullah, Y. M. Chu, Some new classes of preinvex fuzzy-interval-valued functions and inequalities, Int. J. Comput. Intell. Syst., 14 (2021), 1403-1418. https://doi.org/10.2991/ijcis.d.210409.001 doi: 10.2991/ijcis.d.210409.001
[49]	H. M. Srivastava, S. M. El-Deeb, Fuzzy differential subordinations based upon the Mittag-Leffler type Borel distribution, Symmetry, 13 (2021), 1-15. https://doi.org/10.3390/sym13061023 doi: 10.3390/sym13061023
[50]	M. B. Khan, H. M. Srivastava, P. O. Mohammad, J. L. G. Guirao, Fuzzy mixed variational-like and integral inequalities for strongly preinvex fuzzy mappings, Symmetry, 13 (2021), 1816. https://doi.org/10.3390/sym13101816 doi: 10.3390/sym13101816
[51]	M. B. Khan, P. O. Mohammed, M. A. Noor, Y. S. Hamed, New Hermite-Hadamard inequalities in fuzzy-interval fractional calculus and related inequalities, Symmetry, 13 (2021), 673. https://doi.org/10.3390/sym13040673 doi: 10.3390/sym13040673
[52]	M. B. Khan, M. A. Noor, H. M. Al-Bayatti, K. I. Noor, Some new inequalities for LR-Log-h-convex interval-valued functions by means of pseudo order relation, Appl. Math. Inform. Sci., 15 (2021), 459-470. https://doi.org/10.18576/amis/150408 doi: 10.18576/amis/150408
[53]	G. Sana, M. B. Khan, M. A. Noor, P. O. Mohammed, Y. M. Chu, Harmonically convex fuzzy-interval-valued functions and fuzzy-interval Riemann-Liouville fractional integral inequalities, Int. J. Comput. Intell. Syst., 14 (2021), 1809-1822. https://dx.doi.org/10.2991/ijcis.d.210620.001 doi: 10.2991/ijcis.d.210620.001
[54]	H. M. Srivastava, Z. H. Zhang, Y. D. Wu, Some further refinements and extensions of the Hermite-Hadamard and Jensen inequalities in several variables, Math. Comput. Model., 54 (2021), 2709-2717. https://dx.doi.org/10.1016/j.mcm.2011.06.057 doi: 10.1016/j.mcm.2011.06.057
[55]	M. B. Khan, M. A. Noor, K. I. Noor, H. Almusawa, K. S. Nisar, Exponentially preinvex fuzzy mappings and fuzzy exponentially mixed variational-like inequalities, Int. J. Nonlinear Anal. Appl., 19 (2021), 518-541. https://dx.doi.org/10.28924/2291-8639-19-2021-518 doi: 10.28924/2291-8639-19-2021-518
[56]	M. B. Khan, P. O. Mohammed, M. A. Noor, K. Abuahalnaja, Fuzzy integral inequalities on coordinates of convex fuzzy interval-valued functions, Math. Biosci. Eng., 18 (2021), 6552-6580. https://dx.doi.org/10.3934/mbe.2021325 doi: 10.3934/mbe.2021325
[57]	M. B. Khan, P. O. Mohammed, M. A. Noor, Y. Hameed. K. I. Noor, New fuzzy-interval inequalities in fuzzy-interval fractional calculus by means of fuzzy order relation, AIMS Mathematics, 6 (2021), 10964-10988. https://dx.doi.org/10.3934/math.2021637 doi: 10.3934/math.2021637
[58]	M. B. Khan, P. O. Mohammed, M. A. Noor, D. Baleanu, J. Guirao, Some new fractional estimates of inequalities for LR-p-convex interval-valued functions by means of pseudo order relation, Axioms, 10 (2021), 175. https://doi.org/10.3390/axioms10030175 doi: 10.3390/axioms10030175

Reader Comments

Your name:*

Email:*
© 2022 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)