New Hermite–Hadamard Inequalities for Convex Fuzzy-Number-Valued Mappings via Fuzzy Riemann Integrals
Abstract
:1. Introduction
2. Preliminaries
- (1)
- is normal, i.e., there exists such that
- (2)
- is upper semi-continuous, i.e., for given for every there exist and there exist such that for all with
- (3)
- is fuzzy convex, i.e., and ;
- (4)
- is compactly supported, i.e., is compact.
- (1)
- is a non-decreasing mapping.
- (2)
- is a non-increasing mapping.
- (3)
- .
- (4)
- and are bounded, left-continuous on , and right-continuous at .
- Convex on if
- Concave on if inequality (10) is reversed; and
- Affine convex on if
3. Fuzzy Hermite–Hadamard Inequalities
- 1.
- If with , then Theorems 11 and 12 reduce to classical first and second 𝐻𝐻-Fejér inequality for convex mapping, see [10].
- 2.
- If , then, combining Theorems 10 and 11, we get Theorem 7.
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Alomari, M.; Darus, M.; Dragomir, S.S.; Cerone, P. Ostrowski type inequalities for mappings whose derivatives are s-convex in the second sense. Appl. Math. Lett. 2010, 23, 1071–1076. [Google Scholar] [CrossRef]
- Anderson, G.D.; Vamanamurthy, M.K.; Vuorinen, M. Generalized convexity and inequalities. J. Math. Anal. Appl. 2007, 335, 1294–1308. [Google Scholar] [CrossRef]
- Avci, M.; Kavurmaci, H.; Ozdemir, M.E. New inequalities of Hermite–Hadamard type via s-convex mappings in the second sense with applications. Appl. Math. Comput. 2011, 217, 5171–5176. [Google Scholar] [CrossRef]
- Awan, M.U.; Noor, M.A.; Noor, K.I. Hermite–Hadamard inequalities for exponentially convex mappings. Appl. Math. Inf. Sci. 2018, 12, 405–409. [Google Scholar] [CrossRef]
- Zhao, T.-H.; Castillo, O.; Jahanshahi, H.; Yusuf, A.; Alassafi, M.O.; Alsaadi, F.E.; Chu, Y.-M. A fuzzy-based strategy to suppress the novel coronavirus (2019-NCOV) massive outbreak. Appl. Comput. Math. 2021, 20, 160–176. [Google Scholar]
- Zhao, T.-H.; Wang, M.-K.; Chu, Y.-M. On the bounds of the perimeter of an ellipse. Acta Math. Sci. 2022, 42B, 491–501. [Google Scholar] [CrossRef]
- Zhao, T.-H.; Wang, M.-K.; Hai, G.-J.; Chu, Y.-M. Landen inequalities for Gaussian hypergeometric function. Rev. Real Acad. Cienc. Exactas Físicas Naturales. Ser. A Matemáticas 2022, 116, 1–23. [Google Scholar] [CrossRef]
- Hadamard, J. Étude sur les propriétés des fonctions entières et en particulier d’une fonction considérée par Riemann. J. Math. Pures Appl. 1893, 7, 171–215. [Google Scholar]
- Hermite, C. Sur deux limites d’une intégrale définie. Mathesis 1883, 3, 82–97. [Google Scholar]
- Fejér, L. Uberdie Fourierreihen II. Math. Naturwise. Anz. Ungar. Akad. Wiss. 1906, 24, 369–390. [Google Scholar]
- Moore, R.E. Interval Analysis; Prentice Hall: Englewood Cliffs, NJ, USA, 1966. [Google Scholar]
- Kulisch, U.; Miranker, W. Computer Arithmetic in Theory and Practice; Academic Press: New York, NY, USA, 2014. [Google Scholar]
- Zhao, D.F.; An, T.Q.; Ye, G.J.; Liu, W. New Jensen and Hermite-Hadamard type inequalities for h-convex interval-valued mappings. J. Inequalities Appl. 2018, 2018, 302. [Google Scholar] [CrossRef]
- Bede, B. Studies in Fuzziness and Soft Computing. In Mathematics of Fuzzy Sets and Fuzzy Logic; Springer: Berlin/Heidelberg, Germany, 2013; Volume 295. [Google Scholar]
- Chalco-Cano, Y.; Flores-Franulič, A.; Román-Flores, H. Ostrowski type inequalities for interval-valued mappings using generalized Hukuhara derivative. Comput. Appl. Math. 2012, 31, 457–472. [Google Scholar]
- Costa, T.M.; Román-Flores, H.; Chalco-Cano, Y. Opial-type inequalities for interval-valued mappings. Fuzzy Sets Syst. 2019, 358, 48–63. [Google Scholar] [CrossRef]
- Diamond, P.; Kloeden, P.E. Metric Spaces of Fuzzy Sets: Theory and Applications; World Scientific: Singapore, 1994; Volume 38, p. 188. [Google Scholar]
- Wang, M.-K.; Hong, M.-Y.; Xu, Y.-F.; Shen, Z.-H.; Chu, Y.-M. Inequalities for generalized trigonometric and hyperbolic functions with one parameter. J. Math. Inequal. 2020, 14, 1–21. [Google Scholar] [CrossRef]
- Zhao, T.-H.; Qian, W.-M.; Chu, Y.-M. Sharp power mean bounds for the tangent and hyperbolic sine means. J. Math. Inequal. 2021, 15, 1459–1472. [Google Scholar] [CrossRef]
- Hajiseyedazizi, S.N.; Samei, M.E.; Alzabut, J.; Chu, Y.-M. On multi-step methods for singular fractional q-integro-differential equations. Open Math. 2021, 19, 1378–1405. [Google Scholar] [CrossRef]
- Jin, F.; Qian, Z.-S.; Chu, Y.-M.; Rahman, M. On nonlinear evolution model for drinking behavior under Caputo-Fabrizio derivative. J. Appl. Anal. Comput. 2022, 12, 790–806. [Google Scholar] [CrossRef]
- Wang, F.-Z.; Khan, M.N.; Ahmad, I.; Ahmad, H.; Abu-Zinadah, H.; Chu, Y.-M. Numerical solution of traveling waves in chemical kinetics: Time-fractional fishers equations. Fractals 2022, 30, 2240051-34. [Google Scholar] [CrossRef]
- Zhao, T.-H.; Bhayo, B.A.; Chu, Y.-M. Inequalities for generalized Grötzsch ring function. Comput. Methods Funct. Theory 2022, 22, 559–574. [Google Scholar] [CrossRef]
- Zadeh, L.A. Fuzzy sets. Inf. Control. 1965, 8, 338–353. [Google Scholar] [CrossRef]
- Nanda, S.; Kar, K. Convex fuzzy mappings. Fuzzy Sets Syst. 1992, 48, 129–132. [Google Scholar] [CrossRef]
- Chang, S.S.; Zhu, Y.G. On variational inequalities for fuzzy mappings. Fuzzy Sets Syst. 1989, 32, 359–367. [Google Scholar] [CrossRef]
- Noor, M.A. Fuzzy preinvex mappings. Fuzzy Sets Syst. 1994, 64, 95–104. [Google Scholar] [CrossRef]
- Bede, B.; Gal, S.G. Generalizations of the differentiability of fuzzy-number-valued mappings with applications to fuzzy differential equations. Fuzzy Sets Syst. 2005, 151, 581–599. [Google Scholar] [CrossRef]
- Ben-Israel, A.; Mond, B. What is invexity? ANZIAM J. 1986, 28, 1–9. [Google Scholar] [CrossRef] [Green Version]
- Chalco-Cano, Y.; Lodwick, W.A.; Condori-Equice, W. Ostrowski type inequalities and applications in numerical integration for interval-valued mappings. Soft Comput. 2015, 19, 3293–3300. [Google Scholar] [CrossRef]
- Chalco-Cano, Y.; Rojas-Medar, M.A.; Román-Flores, H. M-convex fuzzy mappings and fuzzy integral mean. Comput. Math. Appl. 2000, 40, 1117–1126. [Google Scholar] [CrossRef]
- Mohan, M.S.; Neogy, S.K. On invex sets and preinvex mappings. J. Math. Anal. Appl. 1995, 189, 901–908. [Google Scholar] [CrossRef]
- Zhao, T.-H.; He, Z.-Y.; Chu, Y.-M. Sharp bounds for the weighted Hölder mean of the zero-balanced generalized complete elliptic integrals. Comput. Methods Funct. Theory 2021, 21, 413–426. [Google Scholar] [CrossRef]
- Zhao, T.-H.; Wang, M.-K.; Chu, Y.-M. Concavity and bounds involving generalized elliptic integral of the first kind. J. Math. Inequal. 2021, 15, 701–724. [Google Scholar] [CrossRef]
- Zhao, T.-H.; Wang, M.-K.; Chu, Y.-M. Monotonicity and convexity involving generalized elliptic integral of the first kind. Rev. Real Acad. Cienc. Exactas Fis. Naturales. Ser. A Mat. 2021, 115, 1–13. [Google Scholar] [CrossRef]
- Chu, H.-H.; Zhao, T.-H.; Chu, Y.-M. Sharp bounds for the Toader mean of order 3 in terms of arithmetic, quadratic and contra harmonic means. Math. Slovaca 2020, 70, 1097–1112. [Google Scholar] [CrossRef]
- Zhao, T.-H.; He, Z.-Y.; Chu, Y.-M. On some refinements for inequalities involving zero-balanced hyper geometric function. AIMS Math. 2020, 5, 6479–6495. [Google Scholar] [CrossRef]
- Zhao, T.-H.; Wang, M.-K.; Chu, Y.-M. A sharp double inequality involving generalized complete elliptic integral of the first kind. AIMS Math. 2020, 5, 4512–4528. [Google Scholar] [CrossRef]
- Zhao, T.-H.; Shi, L.; Chu, Y.-M. Convexity and concavity of the modified Bessel functions of the first kind with respect to Hölder means. Rev. Real Acad. Cienc. Exactas Fis. Y Naturales. Ser. A Mat. 2020, 114, 1–14. [Google Scholar] [CrossRef]
- Zhao, T.-H.; Zhou, B.-C.; Wang, M.-K.; Chu, Y.-M. On approximating the quasi-arithmetic mean. J. Inequal. Appl. 2019, 2019, 1–12. [Google Scholar] [CrossRef]
- Zhao, T.-H.; Wang, M.-K.; Zhang, W.; Chu, Y.-M. Quadratic transformation inequalities for Gaussian hyper geometric function. J. Inequal. Appl. 2018, 2018, 1–15. [Google Scholar] [CrossRef] [PubMed]
- Chu, Y.-M.; Zhao, T.-H. Concavity of the error function with respect to Hölder means. Math. Inequal. Appl. 2016, 19, 589–595. [Google Scholar] [CrossRef]
- Osuna-Gómez, R.; Jiménez-Gamero, M.D.; Chalco-Cano, Y.; Rojas-Medar, M.A. Hadamard and Jensen Inequalities for s−Convex Fuzzy Processes. In Soft Methodology and Random Information Systems; Advances in Soft Computing; Springer: Berlin/Heidelberg, Germany, 2004; Volume l26, pp. 645–652. [Google Scholar]
- Costa, T.M. Jensen’s inequality type integral for fuzzy-interval-valued mappings. Fuzzy Sets Syst. 2017, 327, 31–47. [Google Scholar] [CrossRef]
- Costa, T.M.; Roman-Flores, H. Some integral inequalities for fuzzy-interval-valued mappings. Inf. Sci. 2017, 420, 110–125. [Google Scholar] [CrossRef]
- Qian, W.-M.; Chu, H.-H.; Wang, M.-K.; Chu, Y.-M. Sharp inequalities for the Toader mean of order −1 in terms of other bivariate means. J. Math. Inequal. 2022, 16, 127–141. [Google Scholar] [CrossRef]
- Zhao, T.-H.; Chu, H.-H.; Chu, Y.-M. Optimal Lehmer mean bounds for the nth power-type Toader mean of n = −1, 1, 3. J. Math. Inequal. 2022, 16, 157–168. [Google Scholar] [CrossRef]
- Khan, M.B.; Treanțǎ, S.; Alrweili, H.; Saeed, T.; Soliman, M.S. Some new Riemann-Liouville fractional integral inequalities for interval-valued mappings. AIMS Math. 2022, 7, 15659–15679. [Google Scholar] [CrossRef]
- Khan, M.B.; Alsalami, O.M.; Treanțǎ, S.; Saeed, T.; Nonlaopon, K. New class of convex interval-valued functions and Riemann Liouville fractional integral inequalities. AIMS Math. 2022, 7, 15497–15519. [Google Scholar] [CrossRef]
- Saeed, T.; Khan, M.B.; Treanțǎ, S.; Alsulami, H.H.; Alhodaly, M.S. Interval Fejér-Type Inequalities for Left and Right-λ-Preinvex Functions in Interval-Valued Settings. Axioms 2022, 11, 368. [Google Scholar] [CrossRef]
- Khan, M.B.; Cătaş, A.; Alsalami, O.M. Some New Estimates on Coordinates of Generalized Convex Interval-Valued Functions. Fractal Fract. 2022, 6, 415. [Google Scholar] [CrossRef]
- Zhao, T.-H.; Wang, M.-K.; Dai, Y.-Q.; Chu, Y.-M. On the generalized power-type Toader mean. J. Math. Inequal. 2022, 16, 247–264. [Google Scholar] [CrossRef]
- Iqbal, S.A.; Hafez, M.G.; Chu, Y.-M.; Park, C. Dynamical Analysis of nonautonomous RLC circuit with the absence and presence of Atangana-Baleanu fractional derivative. J. Appl. Anal. Comput. 2022, 12, 770–789. [Google Scholar] [CrossRef]
- Huang, T.-R.; Chen, L.; Chu, Y.-M. Asymptotically sharp bounds for the complete p-elliptic integral of the first kind. Hokkaido Math. J. 2022, 51, 189–210. [Google Scholar]
- Zhao, T.-H.; Qian, W.-M.; Chu, Y.-M. On approximating the arc lemniscate functions. Indian J. Pure Appl. Math. 2022, 53, 316–329. [Google Scholar] [CrossRef]
- Santos-García, G.; Khan, M.B.; Alrweili, H.; Alahmadi, A.A.; Ghoneim, S.S. Hermite–Hadamard and Pachpatte type inequalities for coordinated preinvex fuzzy-interval-valued functions pertaining to a fuzzy-interval double integral operator. Mathematics 2022, 10, 2756. [Google Scholar] [CrossRef]
- Macías-Díaz, J.E.; Khan, M.B.; Alrweili, H.; Soliman, M.S. Some Fuzzy Inequalities for Harmonically s-Convex Fuzzy Number Valued Functions in the Second Sense Integral. Symmetry 2022, 14, 1639. [Google Scholar] [CrossRef]
- Khan, M.B.; Noor, M.A.; Macías-Díaz, J.E.; Soliman, M.S.; Zaini, H.G. Some integral inequalities for generalized left and right log convex interval-valued functions based upon the pseudo-order relation. Demonstr. Math. 2022, 55, 387–403. [Google Scholar] [CrossRef]
- Khan, M.B.; Noor, M.A.; Zaini, H.G.; Santos-García, G.; Soliman, M.S. The New Versions of Hermite–Hadamard Inequalities for Pre-invex Fuzzy-Interval-Valued Mappings via Fuzzy Riemann Integrals. Int. J. Comput. Intell. Syst. 2022, 15, 66. [Google Scholar] [CrossRef]
- Goetschel, R., Jr.; Voxman, W. Elementary fuzzy calculus. Fuzzy Sets Syst. 1986, 18, 31–43. [Google Scholar] [CrossRef]
- Khan, M.B.; Noor, M.A.; Al-Shomrani, M.M.; Abdullah, L. Some Novel Inequalities for LR-h-Convex Interval-Valued Functions by Means of Pseudo Order Relation. Math. Meth. Appl. Sci. 2022, 45, 1310–1340. [Google Scholar] [CrossRef]
- Khan, M.B.; Noor, M.A.; Noor, K.I.; Nisar, K.S.; Ismail, K.A.; Elfasakhany, A. Some Inequalities for LR-(h1,h2)-Convex Interval-Valued Functions by Means of Pseudo Order Relation. Int. J. Comput. Intell. Syst. 2021, 14, 1–15. [Google Scholar] [CrossRef]
- Khan, M.B.; Noor, M.A.; Abdullah, L.; Chu, Y.M. Some new classes of preinvex fuzzy-interval-valued functions and inequalities. Int. J. Comput. Intell. Syst. 2021, 14, 1403–1418. [Google Scholar] [CrossRef]
- Liu, P.; Khan, M.B.; Noor, M.A.; Noor, K.I. 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. [Google Scholar] [CrossRef]
- Kaleva, O. Fuzzy differential equations. Fuzzy Sets Syst. 1987, 24, 301–317. [Google Scholar] [CrossRef]
- Puri, M.L.; Ralescu, D.A. Fuzzy Random Variables. J. Math. Anal. Appl. 1986, 114, 409–422. [Google Scholar] [CrossRef]
- Sana, G.; Khan, M.B.; Noor, M.A.; Mohammed, P.O.; Chu, Y.M. Harmonically convex fuzzy-interval-valued functions and fuzzy-interval Riemann–Liouville fractional integral inequalities. Int. J. Comput. Intell. Syst. 2021, 14, 1809–1822. [Google Scholar] [CrossRef]
- Khan, M.B.; Noor, M.A.; Noor, K.I.; Chu, Y.M. New Hermite-Hadamard type inequalities for -convex fuzzy-interval-valued functions. Adv. Differ. Equ. 2021, 2021, 6–20. [Google Scholar] [CrossRef]
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Khan, M.B.; Santos-García, G.; Noor, M.A.; Soliman, M.S. New Hermite–Hadamard Inequalities for Convex Fuzzy-Number-Valued Mappings via Fuzzy Riemann Integrals. Mathematics 2022, 10, 3251. https://doi.org/10.3390/math10183251
Khan MB, Santos-García G, Noor MA, Soliman MS. New Hermite–Hadamard Inequalities for Convex Fuzzy-Number-Valued Mappings via Fuzzy Riemann Integrals. Mathematics. 2022; 10(18):3251. https://doi.org/10.3390/math10183251
Chicago/Turabian StyleKhan, Muhammad Bilal, Gustavo Santos-García, Muhammad Aslam Noor, and Mohamed S. Soliman. 2022. "New Hermite–Hadamard Inequalities for Convex Fuzzy-Number-Valued Mappings via Fuzzy Riemann Integrals" Mathematics 10, no. 18: 3251. https://doi.org/10.3390/math10183251
APA StyleKhan, M. B., Santos-García, G., Noor, M. A., & Soliman, M. S. (2022). New Hermite–Hadamard Inequalities for Convex Fuzzy-Number-Valued Mappings via Fuzzy Riemann Integrals. Mathematics, 10(18), 3251. https://doi.org/10.3390/math10183251