Some integral inequalities for coordinated log-$ h $-convex interval-valued functions

Fangfang Shi; Guoju Ye; Dafang Zhao; Wei Liu; Fangfang Shi; Guoju Ye; Dafang Zhao; Wei Liu

doi:10.3934/math.2022009

AIMS Mathematics

2022, Volume 7, Issue 1: 156-170. doi: 10.3934/math.2022009

Previous Article Next Article

Research article

Some integral inequalities for coordinated log-$ h $-convex interval-valued functions

1.
College of Science, Hohai University, 210098 Nanjing, China
2.
School of Mathematics and Statistics, Hubei Normal University, 435002 Huangshi, China

Received: 26 July 2021 Accepted: 25 September 2021 Published: 09 October 2021
MSC : 26D15, 26E25, 26B25, 28B20

We introduce and investigate the coordinated log-$ h $-convexity for interval-valued functions. Also, we prove some new Jensen type inequalities and Hermite-Hadamard type inequalities, which generalize some known results in the literature. Moreover, some examples are given to illustrate our results.
- interval-valued functions,
- coordinated $ \log $-$ h $-convex,
- Jensen type inequalities,
- Hermite-Hadamard type inequalities
Citation: Fangfang Shi, Guoju Ye, Dafang Zhao, Wei Liu. Some integral inequalities for coordinated log-$ h $-convex interval-valued functions[J]. AIMS Mathematics, 2022, 7(1): 156-170. doi: 10.3934/math.2022009

Related Papers:

Abstract

We introduce and investigate the coordinated log-$ h $-convexity for interval-valued functions. Also, we prove some new Jensen type inequalities and Hermite-Hadamard type inequalities, which generalize some known results in the literature. Moreover, some examples are given to illustrate our results.

References

[1]	M. Z. Sarikaya, A. Saglam, H. Yildirim, On some Hadamard-type inequalities for $h$-convex functions, J. Math. Inequal., 2 (2008), 335–341. doi: 10.7153/jmi-02-30. doi: 10.7153/jmi-02-30
[2]	M. Bombardelli, S. Varošanec, Properties of $h$-convex functions related to the Hermite-Hadamard-Fejér inequalities, Comput. Math. Appl., 58 (2009), 1869–1877. doi: 10.1016/j.camwa.2009.07.073. doi: 10.1016/j.camwa.2009.07.073
[3]	M. Noor, K. Noor, M. Awan, A new Hermite-Hadamard type inequality for $h$-convex functions, Creat. Math. Inform., 24 (2015), 191–197.
[4]	D. Zhao, T. An, G. Ye, W. Liu, New Jensen and Hermite-Hadamard type inequalities for $h$-convex interval-valued functions, J. Inequal. Appl., 302 (2018). doi: 10.1186/s13660-018-1896-3.
[5]	D. Zhao, G. Zhao, G. Ye, W. Liu, S. Dragomir, On Hermite-Hadamard-type inequalities for coordinated $h$-convex interval-valued functions, Mathematics, 9 (2021), 2352. doi: 10.3390/math9192352. doi: 10.3390/math9192352
[6]	S. Dragomir, B. Mond, Integral inequalities of Hadamard type for log-convex functions, Demonstr. Math., 31 (1998), 354–364. doi: 10.1515/dema-1998-0214. doi: 10.1515/dema-1998-0214
[7]	S. Dragomir, Refinements of the Hermite-Hadamard integral inequality for log-convex functions, Aust. Math. Sco. Gaz., 28 (2000), 129–134.
[8]	S. Dragomir, A survey of Jensen type inequalities for log-convex functions of selfadjoint operators in Hilbert spaces, Commun. Math. Anal., 10 (2011), 82–104.
[9]	C. Niculescu, The Hermite-Hadamard inequality for log-convex functions, Nonlinear Anal-Theor., 75 (2012), 662–669. doi: 10.1016/j.na.2011.08.066. doi: 10.1016/j.na.2011.08.066
[10]	S. Dragomir, New inequalities of Hermite-Hadamard type for log-convex functions, Khayyam J. Math., 3 (2017), 98–115.
[11]	M. Noor, F. Qi, M. Awan, Some Hermite-Hadamard type inequalities for log-$h$-convex functions, Analysis, 33 (2013), 367–375. doi: 10.1524/anly.2013.1223. doi: 10.1524/anly.2013.1223
[12]	S. Dragomir, On the Hadamard's inequality for convex functions on the co-ordinates in a rectangle from the plane, Taiwanese J. Math., 5 (2001), 775–788. doi: 10.11650/twjm/1500574995. doi: 10.11650/twjm/1500574995
[13]	M. Alomari, M. Darus, On the Hadamard's inequality for log-convex functions on the coordinates, J. Inequal. Appl., 2009 (2009), 283147. doi: 10.1155/2009/283147. doi: 10.1155/2009/283147
[14]	M. Sarikaya, On the Hermite-Hadamard-type inequalities for co-ordinated convex function via fractional integrals, Integr. Transf. Spec. F., 25 (2014), 134–147. doi: 10.1080/10652469.2013.824436. doi: 10.1080/10652469.2013.824436
[15]	M. Özdemir, E. Set, M. Sarikaya, Some new Hadamard type inequalities for co-ordinated m-convex and ($\alpha$, m)-convex functions, Hacet. J. Math. Stat., 40 (2011), 219–229. doi: 10.1007/s10883-011-9120-5. doi: 10.1007/s10883-011-9120-5
[16]	M. Alomari, M. Darus, Co-ordinated s-convex function in the first sense with some Hadamard-type inequalities, Int. J. Contemp. Math. Sci., 3 (2008), 1557–1567.
[17]	M. Latif, M. Alomari, Hadamard-type inequalities for product two convex functions on the co-ordinates, Int. Math. Forum, 4 (2009), 2327–2338.
[18]	D. Singh, B. A. Dar, D. S. Kim, Sufficiency and duality in non-smooth interval valued programming problems, J. Ind. Manag. Optim., 15 (2019), 647–665. doi: 10.3934/jimo.2018063. doi: 10.3934/jimo.2018063
[19]	I. Ahmad, D. Singh, B. A. Dar, Optimality conditions in multiobjective programming problems with interval valued objective functions, Control Cybern., 44 (2015), 19–45.
[20]	T. Costa, H. Román-Flores, Y. Chalco-Cano, Opial-type inequalities for interval-valued functions, Fuzzy Set. Syst., 358 (2019), 48–63. doi: 10.1016/j.fss.2018.04.012. doi: 10.1016/j.fss.2018.04.012
[21]	Y. Chalco-Cano, W. Lodwick, W. Condori-Equice, Ostrowski type inequalities for interval-valued functions using generalized Hukuhara derivative, Comput. Appl. Math., 31 (2012), 475–472. doi: 10.1007/s40314-016-0396-7. doi: 10.1007/s40314-016-0396-7
[22]	A. Flores-Franulič, Y. Chalco-Cano, H. Román-Flores, An Ostrowski type inequality for interval-valued functions, In: IFSA World Congress and NAFIPS Annual Meeting IEEE, 35 (2013), 1459–1462. doi: 10.1109/ifsa-nafips.2013.6608617. doi: 10.1109/ifsa-nafips.2013.6608617
[23]	H. Román-Flores, Y. Chalco-Cano, W. Lodwick, Some integral inequalities for interval-valued functions, Comput. Appl. Math., 37 (2016), 1306–1318. doi: 10.1007/s40314-016-0396-7. doi: 10.1007/s40314-016-0396-7
[24]	D. Zhao, M. Ali, G. Murtaza, On the Hermite-Hadamard inequalities for interval-valued coordinated convex functions, Adv. Differ. Equ., 570 (2020). doi: 10.1186/s13662-020-03028-7.
[25]	D. Zhao, T. An, G. Ye, W. Liu, Chebyshev type inequalities for interval-valued functions, Fuzzy Set. Syst., 396 (2020), 82–101. doi: 10.1016/j.fss.2019.10.006. doi: 10.1016/j.fss.2019.10.006
[26]	Y. Guo, G. Ye, D. Zhao, W. Liu, Some integral inequalities for log-$h$-convex interval-valued functions, IEEE Access, 7 (2019), 86739–86745. doi: 10.1109/access.2019.2925153. doi: 10.1109/access.2019.2925153
[27]	Z. Zhang, M. Ali, H. Budak, M. Sarikaya, On Hermite-Hadamard type inequalities for interval-valued multiplicative integrals, Commun. Fac. Sci. Univ., 69 (2020), 1428–1448. doi: 10.31801/cfsuasmas.754842. doi: 10.31801/cfsuasmas.754842

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)