Some fuzzy-interval integral inequalities for harmonically convex fuzzy-interval-valued functions

Muhammad Bilal Khan; Muhammad Aslam Noor; Thabet Abdeljawad; Bahaaeldin Abdalla; Ali Althobaiti; Muhammad Bilal Khan; Muhammad Aslam Noor; Thabet Abdeljawad; Bahaaeldin Abdalla; Ali Althobaiti

doi:10.3934/math.2022024

AIMS Mathematics

2022, Volume 7, Issue 1: 349-370. doi: 10.3934/math.2022024

Previous Article Next Article

Research article

Some fuzzy-interval integral inequalities for harmonically convex fuzzy-interval-valued functions

1.
Department of Mathematics, COMSATS University Islamabad, Islamabad 44000, Pakistan
2.
Department of Mathematics and Natural Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia
3.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan
4.
Department of Mathematics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia

Received: 01 September 2021 Accepted: 29 September 2021 Published: 12 October 2021
MSC : 26A33, 26A51, 26D10

It is well-known fact that fuzzy interval-valued functions (F-I-V-Fs) are generalizations of interval-valued functions (I-V-Fs), and inclusion relation and fuzzy order relation on interval space and fuzzy space are two different concepts. Therefore, by using fuzzy order relation (FOR), we derive inequalities of Hermite-Hadamard (H·H) and Hermite-Hadamard Fejér (H·H Fejér) like for harmonically convex fuzzy interval-valued functions by applying fuzzy Riemann integrals. Moreover, we establish the relation between fuzzy integral inequalities and fuzzy products of harmonically convex fuzzy interval-valued functions. The outcomes of this study are generalizations of many known results which can be viewed as an application of a defined new version of inequalities.
Citation: Muhammad Bilal Khan, Muhammad Aslam Noor, Thabet Abdeljawad, Bahaaeldin Abdalla, Ali Althobaiti. Some fuzzy-interval integral inequalities for harmonically convex fuzzy-interval-valued functions[J]. AIMS Mathematics, 2022, 7(1): 349-370. doi: 10.3934/math.2022024

Related Papers:

Abstract

It is well-known fact that fuzzy interval-valued functions (F-I-V-Fs) are generalizations of interval-valued functions (I-V-Fs), and inclusion relation and fuzzy order relation on interval space and fuzzy space are two different concepts. Therefore, by using fuzzy order relation (FOR), we derive inequalities of Hermite-Hadamard (H·H) and Hermite-Hadamard Fejér (H·H Fejér) like for harmonically convex fuzzy interval-valued functions by applying fuzzy Riemann integrals. Moreover, we establish the relation between fuzzy integral inequalities and fuzzy products of harmonically convex fuzzy interval-valued functions. The outcomes of this study are generalizations of many known results which can be viewed as an application of a defined new version of inequalities.

References

[1]	S. S. Dragomir, C. E. M. Pearce, Selected topics on Hermite-Hadamard inequalities and applications, RGMIA monographs, Victoria University, 2004.
[2]	J. E. Pečarić, F. Proschan, Y. L. Tong, Convex functions, partial orderings, and statistical applications, Elsevier, 1992.
[3]	F. X. Chen, A note on Hermite-Hadamard inequalities for products of convex functions, J. Appl. Math., 2013 (2013), 935020. doi: 10.1155/2013/935020. doi: 10.1155/2013/935020
[4]	S. S. Dragomir, Inequalities of Hermite-Hadamard type for h-convex functions on linear spaces, Proyecciones, 34 (2015), 323-341. doi: 10.4067/S0716-09172015000400002. doi: 10.4067/S0716-09172015000400002
[5]	S. S. Dragomir, Two mappings in connection to Hadamard's inequalities, J. Math. Anal. Appl., 167 (1992), 49-56. doi: 10.1016/0022-247X(92)90233-4. doi: 10.1016/0022-247X(92)90233-4
[6]	S. Dragomir, J. Pecaric, L. E. Persson, Some inequalities of Hadamard type, Soochow J. Math., 21 (1995), 335-341.
[7]	B. Pachpatte, On some inequalities for convex functions, RGMIA Res. Rep. Collect., 6 (2003), 1-9.
[8]	J. R. Wang, X. Z. Li, C. Zhu, Refinements of Hermite-Hadamard type inequalities involving fractional integrals, Bull. Belg. Math. Soc. Simon Stevin, 20 (2013), 655-666. doi: 10.36045/bbms/1382448186. doi: 10.36045/bbms/1382448186
[9]	M. Z. Sarikaya, F. Ertugral, On the generalized Hermite-Hadamard inequalities, Ann. Univ. Craioval Math. Comput. Sci. Ser., 47 (2020), 193-213.
[10]	M. Z. Sarikaya, H. Yildirim, On generalization of the Riesz potential, Indian J. Math. Math. Sci., 3 (2007), 231-235.
[11]	F. Ertugral, M. Z. Sarikaya, Simpson type integral inequalities for generalized fractional integral, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., 113 (2019), 3115-3124. doi: 10.1007/s13398-019-00680-x. doi: 10.1007/s13398-019-00680-x
[12]	K. L. Tseng, S. R. Hwang, New Hermite-Hadamard-type inequalities and their applications, Filomat, 30 (2016), 3667-3680. doi: 10.2298/FIL1614667T. doi: 10.2298/FIL1614667T
[13]	R. E. Moore, Interval analysis, Prentice Hall, Englewood Cliffs, 1966.
[14]	Y. Chalco-Cano, A. Flores-Franulič, H. Román-Flores, Ostrowski type inequalities for interval-valued functions using generalized Hukuhara derivative, Comput. Appl. Math., 31 (2012), 457-472. 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. doi: 10.1007/s00500-014-1483-6.
[16]	H. Román-Flores, Y. Chalco-Cano, W. A. Lodwick, Some integral inequalities for interval-valued functions, Comput. Appl. Math., 37 (2018), 1306-1318. doi: 10.1007/s40314-016-0396-7. doi: 10.1007/s40314-016-0396-7
[17]	T. M. Costa, Jensen's inequality type integral for fuzzy-interval-valued functions, Fuzzy Sets Syst., 327 (2017), 31-47. doi: 10.1016/j.fss.2017.02.001. doi: 10.1016/j.fss.2017.02.001
[18]	T. M. Costa, H. Román-Flores, Some integral inequalities for fuzzy-interval-valued functions, Inf. Sci., 420 (2017), 110-125. doi: 10.1016/j.ins.2017.08.055.
[19]	A. Flores-Franulič, Y. Chalco-Cano, H. Román-Flores, An Ostrowski type inequality for interval-valued functions, In: 2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS), (2013), 1459-1462. doi: 10.1109/IFSA-NAFIPS.2013.6608617.
[20]	H. Román-Flores, Y. Chalco-Cano, G. N. Silva, A note on Gronwall type inequality for interval-valued functions, In: 2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS), (2013), 1455-1458. doi: 10.1109/IFSA-NAFIPS.2013.6608616.
[21]	E. Sadowska, Hadamard inequality and a refinement of Jensen inequality for set valued functions, Results Math., 32 (1997), 332-337. doi: 10.1007/BF03322144. doi: 10.1007/BF03322144
[22]	F. C. Mitroi, K. Nikodem, S. Wasowicz, Hermite-Hadamard inequalities for convex set-valued functions, Demonstr. Math., 46 (2013), 655-662. doi: 10.1515/dema-2013-0483. doi: 10.1515/dema-2013-0483
[23]	K. Nikodem, J. L. Sánchez, L. Sánchez, Jensen and Hermite-Hadamard inequalities for strongly convex set-valued maps, Math. Aeterna, 4 (2014), 979-987.
[24]	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. Equations, 2021 (2021), 6-20. doi: 10.1186/s13662-021-03245-8. doi: 10.1186/s13662-021-03245-8
[25]	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. doi: 10.3390/sym13040673. doi: 10.3390/sym13040673
[26]	M. B. Khan, P. O. Mohammed, M. A. Noor, A. M. Alsharif, K. I. Noor, New fuzzy-interval inequalities in fuzzy-interval fractional calculus by means of fuzzy order relation, AIMS Math., 6 (2021), 10964-10988. doi: 10.3934/math.2021637. doi: 10.3934/math.2021637
[27]	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. doi: 10.2991/ijcis.d.210409.001. doi: 10.2991/ijcis.d.210409.001
[28]	M. B. Khan, L. Abdullah, M. A. Noor, K. I. Noor, New Hermite-Hadamard and Jensen inequalities for log-h-convex fuzzy-interval-valued functions, Int. J. Comput. Intell. Syst., 14 (2021), 155. doi: 10.1007/s44196-021-00004-1. doi: 10.1007/s44196-021-00004-1
[29]	P. Liu, M. B. Khan, M. A. Noor, K. I. 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. doi: 10.1007/s40747-021-00379-w. doi: 10.1007/s40747-021-00379-w
[30]	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., 2021 (2021), 1809-1822. doi: 10.2991/ijcis.d.210620.001. doi: 10.2991/ijcis.d.210620.001
[31]	M. B. Khan, P. O. Mohammed, M. A. Noor, K. M. Abualnaja, Fuzzy integral inequalities on coordinates of convex fuzzy interval-valued functions, Math. Biosci. Eng., 18 (2021), 6552-6580. doi: 10.3934/mbe.2021325. doi: 10.3934/mbe.2021325
[32]	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. Inf. Sci., 15 (2021), 459-470. doi: 10.18576/amis/150408
[33]	M. B. Khan, P. O. Mohammed, M. A. Noor, D. Baleanu, J. L. G. Guirao, Some new fractional estimates of inequalities for LR-p-convex interval-valued functions by means of pseudo order relation, Axioms, 10 (2021), 175. doi: 10.3390/axioms10030175. doi: 10.3390/axioms10030175
[34]	P. Liu, M. B. Khan, M. A. Noor, K. I. Noor, On strongly generalized preinvex fuzzy mappings, J. Math., 2021 (2021), 6657602. doi: 10.1155/2021/6657602. doi: 10.1155/2021/6657602
[35]	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. doi: 10.1155/2021/6652930. doi: 10.1155/2021/6652930
[36]	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. Anal. Appl., 19 (2021), 518-541. doi: 10.28924/2291-8639-19-2021-518. doi: 10.28924/2291-8639-19-2021-518
[37]	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), 1856-1870. doi: 10.2991/ijcis.d.210616.001. doi: 10.2991/ijcis.d.210616.001
[38]	B. Bede, S. G. Gal, Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations, Fuzzy Sets Syst., 151 (2005), 581-599. doi: 10.1016/j.fss.2004.08.001. doi: 10.1016/j.fss.2004.08.001
[39]	R. Goetschel Jr., W. Voxman, Elementary fuzzy calculus, Fuzzy Sets Syst., 18 (1986), 31-43. doi: 10.1016/0165-0114(86)90026-6.
[40]	O. Kaleva, Fuzzy differential equations, Fuzzy Sets Syst., 24 (1987), 301-317. doi: 10.1016/0165-0114(87)90029-7.
[41]	U. W. Kulish, W. L. Miranker, Computer arithmetic in theory and practice, New York: Academic Press, 1981.
[42]	İ. İşcan, Hermite-Hadamard type inequalities for harmonically convex functions, Hacet. J. Math. Stat., 43 (2014), 935-942.
[43]	M. A. Noor, K. I. Noor, M. U. Awan, S. Costache, Some integral inequalities for harmonically h-convex functions, U.P.B. Sci. Bull., Seri. A, 77 (2015), 5-16.
[44]	M. B. Khan, M. A. Noor, P. O. Mohammed, J. L. G. Guirao, K. I. Noor, Some integral inequalities for generalized convex fuzzy-interval-valued functions via fuzzy Riemann integrals, Int. J. Comput. Intell. Syst., 14 (2021), 158. doi: 10.1007/s44196-021-00009-w. doi: 10.1007/s44196-021-00009-w
[45]	M. B. Khan, H. M. Srivastava, P. O. Mohammed, J. L. Guirao, Fuzzy mixed variational-like and integral inequalities for strongly preinvex fuzzy mappings, Symmetry, 13 (2021), 1816. doi: 10.3390/sym13101816. doi: 10.3390/sym13101816

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)