{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,5]],"date-time":"2026-01-05T22:05:25Z","timestamp":1767650725619,"version":"build-2065373602"},"reference-count":55,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2022,12,20]],"date-time":"2022-12-20T00:00:00Z","timestamp":1671494400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Taif University Researchers Supporting Project Number","award":["TURSP-2020\/345"],"award-info":[{"award-number":["TURSP-2020\/345"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The objective of the current paper is to incorporate the new class and concepts of convexity and Hermite\u2013Hadamard inequality with the fuzzy Riemann integral operators because almost all classical single-valued and interval-valued convex functions are special cases of fuzzy-number valued convex mappings. Therefore, a new class of nonconvex mapping in the fuzzy environment has been defined; up and down h-pre-invex fuzzy-number valued mappings (U.D\u00a0h-pre-invex F-N\u2219V\u2219Ms). With the help of this newly defined class, some new versions of Hermite\u2013Hadamard (HH) type inequalities have been also presented. Moreover, some related inequalities such as HH Fej\u00e9r- and Pachpatte-type inequalities for U\u2219D\u00a0h-pre-invex F-N\u2219V\u2219Ms are also introduced. Some exceptional cases have been discussed, which can be seen as applications of the main results. We have provided some nontrivial examples. Finally, we also discuss some future scopes.<\/jats:p>","DOI":"10.3390\/axioms12010001","type":"journal-article","created":{"date-parts":[[2022,12,20]],"date-time":"2022-12-20T04:21:56Z","timestamp":1671510116000},"page":"1","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Up and Down h-Pre-Invex Fuzzy-Number Valued Mappings and Some Certain Fuzzy Integral Inequalities"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7450-8067","authenticated-orcid":false,"given":"Muhammad Bilal","family":"Khan","sequence":"first","affiliation":[{"name":"Department of Mathematics, COMSATS University Islamabad, Islamabad 44000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9121-8552","authenticated-orcid":false,"given":"Hatim Ghazi","family":"Zaini","sequence":"additional","affiliation":[{"name":"Department of Computer Engineering, College of Computers and Information Technology, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7580-7533","authenticated-orcid":false,"given":"Jorge E.","family":"Mac\u00edas-D\u00edaz","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1ticas y F\u00edsica, Universidad Aut\u00f3noma de Aguascalientes, Avenida Universidad 940, Ciudad Universitaria, Aguascalientes 20131, Mexico"},{"name":"Department of Mathematics, School of Digital Technologies, Tallinn University, Narva Rd. 25, 10120 Tallinn, Estonia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9431-4195","authenticated-orcid":false,"given":"Mohamed S.","family":"Soliman","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, College of Engineering, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,20]]},"reference":[{"key":"ref_1","first-page":"775","article-title":"On Hadamard\u2019s inequality for convex functions on the co-ordinates in a rectangle from the plane","volume":"4","author":"Dragomir","year":"2001","journal-title":"Taiwan. J. Math."},{"key":"ref_2","first-page":"935","article-title":"Hermite\u2013Hadamard type inequalities for harmonically convex functions","volume":"43","author":"Iscan","year":"2014","journal-title":"Hacet. J. Math. Stat."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1007\/BF01836150","article-title":"On midpoint convex set-valued functions","volume":"33","author":"Nikodem","year":"1987","journal-title":"Aequ. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"95","DOI":"10.7153\/mia-2020-23-08","article-title":"On Hermite-Hadamard type inequalities for harmonical h-convex interval-valued functions","volume":"3","author":"Zhao","year":"2020","journal-title":"Math. Inequalities Appl."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Cristescu, G., and Lupsa, L. (2002). Non-connected Convexities and Applications, Applied Optimization, Kluwer Academic Publishers.","DOI":"10.1007\/978-1-4615-0003-2"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"901","DOI":"10.1006\/jmaa.1995.1057","article-title":"On invex sets and preinvex functions","volume":"189","author":"Mohan","year":"1995","journal-title":"J. Math. Anal. Appl."},{"key":"ref_7","first-page":"171","article-title":"\u00c9tude sur les propri\u00e9t\u00e9s des fonctions enti\u00e8res et en particulier d\u2019une fonction consid\u00e9r\u00e9e par Riemann","volume":"7","author":"Hadamard","year":"1893","journal-title":"J. Math\u00e9matiques Pures Appliqu\u00e9es"},{"key":"ref_8","first-page":"82","article-title":"Sur deux limites d\u2019une int\u00e9grale d\u00e9finie","volume":"3","author":"Hermite","year":"1883","journal-title":"Mathesis"},{"key":"ref_9","first-page":"369","article-title":"Uberdie Fourierreihen II","volume":"24","year":"1906","journal-title":"Math. Naturwise. Anz Ungar. Akad. Wiss."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2403","DOI":"10.1016\/j.mcm.2011.12.048","article-title":"Hermite- Hadamard\u2019s inequalities for fractional integrals and related fractional inequalities","volume":"57","author":"Sarikaya","year":"2013","journal-title":"Math. Comput. Model."},{"key":"ref_11","unstructured":"Dragomir, S.S., and Pearce, C.E.M. (2000). Selected Topics on Hermite\u2013Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University."},{"key":"ref_12","unstructured":"Moore, R.E. (1996). Interval Analysis, Prentice-Hall."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"332","DOI":"10.1007\/BF03322144","article-title":"Hadamard inequality and a refinement of jensen inequality for set valued functions","volume":"32","author":"Sadowska","year":"1997","journal-title":"Results Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"705","DOI":"10.1090\/proc\/14741","article-title":"Fractional Hermite-Hadamard-type inequalities for interval-valued functions","volume":"148","author":"Budak","year":"2020","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Khan, M.B., Mohammed, P.O., Noor, M.A., Baleanu, D., and Guirao, J.L.G. (2021). Some new fractional estimates of inequalities for LR-P-convex interval-valued functions by means of pseudo order relation. Axioms, 10.","DOI":"10.3390\/axioms10030175"},{"key":"ref_16","first-page":"1","article-title":"Fractional Hermite\u2013Hadamard type inequalities for interval\u2013valued functions","volume":"266","author":"Liu","year":"2019","journal-title":"J. Inequalities Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"222","DOI":"10.1186\/s13660-020-02488-5","article-title":"Hermite-Hadamard-type inequalities for the interval-valued approximately h-convex functions via generalized fractional integrals","volume":"222","author":"Zhao","year":"2020","journal-title":"J. Inequalities Appl."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"111846","DOI":"10.1016\/j.chaos.2022.111846","article-title":"On the fractional double integral inclusion relations having exponential kernels via interval-valued co-ordinated convex mappings","volume":"156","author":"Du","year":"2022","journal-title":"Chaos Solitons Fractals"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"104","DOI":"10.1186\/s13662-021-03261-8","article-title":"Weighted Hermite-Hadamard type inclusions for products of co-ordinated convex interval-valued functions, Adv","volume":"104","author":"Kara","year":"2021","journal-title":"Differ. Equ."},{"key":"ref_20","first-page":"17","article-title":"Some fractional Hermite-Hadamard-type inequalities for interval-valued coordinated functions","volume":"32","author":"Shi","year":"2021","journal-title":"Adv. Differ. Equ."},{"key":"ref_21","first-page":"1","article-title":"On the Hermite-Hadamard inequalities for interval-valued coordinated convex functions","volume":"570","author":"Zhao","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_22","first-page":"1","article-title":"Existence results and the dimension of the solution set for a nonlocal inclusions problem with mixed fractional derivatives and integrals","volume":"2020","author":"Alsaedi","year":"2020","journal-title":"J. Nonlinear Funct. Anal."},{"key":"ref_23","first-page":"353","article-title":"(e S-iterative techniques on Hadamard manifolds and applications","volume":"2","author":"Sahu","year":"2020","journal-title":"J. Appl. Numer. Optim."},{"key":"ref_24","first-page":"251","article-title":"On bounded solutions of semilinear fractional order differential inclusions in Hilbert spaces","volume":"5","author":"Kamenskii","year":"2021","journal-title":"J. Nonlinear Var. Anal."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"6","DOI":"10.1186\/s13662-021-03245-8","article-title":"New Hermite-Hadamard type inequalities for -convex fuzzy-interval-valued functions","volume":"2021","author":"Khan","year":"2021","journal-title":"Adv. Differ. Equ."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1809","DOI":"10.2991\/ijcis.d.210620.001","article-title":"Harmonically convex fuzzy-interval-valued functions and fuzzy-interval Riemann\u2013Liouville fractional integral inequalities","volume":"14","author":"Sana","year":"2021","journal-title":"Int. J. Comput. Intell. Syst."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Khan, M.B., Noor, M.A., Shah, N.A., Abualnaja, K.M., and Botmart, T. (2022). Some New Versions of Hermite\u2013Hadamard Integral Inequalities in Fuzzy Fractional Calculus for Generalized Pre-Invex Functions via Fuzzy-Interval-Valued Settings. Fractal Fract., 6.","DOI":"10.3390\/fractalfract6020083"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1403","DOI":"10.2991\/ijcis.d.210409.001","article-title":"Some new classes of preinvex fuzzy-interval-valued functions and inequalities","volume":"14","author":"Khan","year":"2021","journal-title":"Int. J. Comput. Intell. Syst."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1856","DOI":"10.2991\/ijcis.d.210616.001","article-title":"Higher-order strongly preinvex fuzzy mappings and fuzzy mixed variational-like inequalities","volume":"14","author":"Khan","year":"2021","journal-title":"Int. J. Comput. Intell. Syst."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"112692","DOI":"10.1016\/j.chaos.2022.112692","article-title":"Some new concepts related to fuzzy fractional calculus for up and down convex fuzzy-number valued functions and inequalities","volume":"164","author":"Khan","year":"2022","journal-title":"Chaos Solitons Fractals"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1016\/0165-0114(92)90256-4","article-title":"Convex fuzzy mappings","volume":"48","author":"Nanda","year":"1992","journal-title":"Fuzzy Sets Syst."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Diamond, P., and Kloeden, P.E. (1994). Metric Spaces of Fuzzy Sets: Theory and Applications, World Scientific.","DOI":"10.1142\/2326"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1016\/0165-0114(87)90029-7","article-title":"Fuzzy differential equations","volume":"24","author":"Kaleva","year":"1987","journal-title":"Fuzzy Sets Syst."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"110","DOI":"10.1016\/j.ins.2017.08.055","article-title":"Some integral inequalities for fuzzy-interval-valued functions","volume":"420","author":"Costa","year":"2017","journal-title":"Inf. Sci."},{"key":"ref_35","first-page":"39","article-title":"Continuity of generalized convex and generalized concave set\u2013valued functions","volume":"22","author":"Breckner","year":"1993","journal-title":"Rev. Anal Num\u00e9r. Th\u00e9or. Approx."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"655","DOI":"10.1515\/dema-2013-0483","article-title":"Hermite\u2013Hadamard inequalities for convex set-valued functions","volume":"46","author":"Mitroi","year":"2013","journal-title":"Demonstratio Mathematica"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Aubin, J.P., and Cellina, A. (1984). Differential Inclusions: Set-Valued Maps and Viability Theory, Grundlehren der Mathematischen Wissenschaften, Springer.","DOI":"10.1007\/978-3-642-69512-4"},{"key":"ref_38","unstructured":"Aubin, J.P., and Frankowska, H. (1990). Set-Valued Analysis, Birkh\u00e4user."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1016\/j.fss.2017.02.001","article-title":"Jensen\u2019s inequality type integral for fuzzy-interval-valued functions","volume":"327","author":"Costa","year":"2017","journal-title":"Fuzzy Sets Syst."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.fss.2019.06.002","article-title":"Jensen\u2019s inequalities for set-valued and fuzzy set-valued functions","volume":"2020","author":"Zhang","year":"2020","journal-title":"Fuzzy Sets Syst."},{"key":"ref_41","first-page":"52","article-title":"Inequalities for h-preinvex functions","volume":"234","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"15659","DOI":"10.3934\/math.2022857","article-title":"Some new Riemann-Liouville fractional integral inequalities for interval-valued mappings","volume":"7","author":"Khan","year":"2022","journal-title":"AIMS Math."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"15497","DOI":"10.3934\/math.2022849","article-title":"New class of convex interval-valued functions and Riemann Liouville fractional integral inequalities","volume":"7","author":"Khan","year":"2022","journal-title":"AIMS Math."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Khan, M.B., Zaini, H.G., Santos-Garc\u00eda, G., Noor, M.A., and Soliman, M.S. (2022). New Class Up and Down \u03bb-Convex Fuzzy-Number Valued Mappings and Related Fuzzy Fractional Inequalities. Fractal Fract., 6.","DOI":"10.3390\/fractalfract6110679"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1023\/B:REOM.0000032115.22510.b5","article-title":"Probability-possibility transformations, triangular fuzzy sets, and probabilistic inequalities","volume":"10","author":"Dubois","year":"2004","journal-title":"Reliab. Comput."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1016\/0165-0114(94)90097-3","article-title":"Interval number and fuzzy number linear programming\u2019s","volume":"66","author":"Shaocheng","year":"1994","journal-title":"Fuzzy Sets Syst."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"393","DOI":"10.1016\/j.ins.2021.06.053","article-title":"Dynamic output-feedback control for singular interval-valued fuzzy systems: Linear matrix inequality approach","volume":"576","author":"Park","year":"2021","journal-title":"Inf. Sci."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1016\/S0165-0114(98)00407-2","article-title":"Interpretation of inequality constraints involving interval coefficients and a solution to interval linear programming","volume":"119","author":"Sengupta","year":"2001","journal-title":"Fuzzy Sets Syst."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"5689","DOI":"10.3233\/JIFS-181408","article-title":"Calculation formulas and correlation inequalities for variance bounds and semi-variances of fuzzy intervals","volume":"37","author":"Gu","year":"2019","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"4645","DOI":"10.1016\/j.ins.2007.05.001","article-title":"Numerical methods for interval and fuzzy number comparison based on the probabilistic approach and Dempster\u2013Shafer theory","volume":"177","author":"Sevastianov","year":"2007","journal-title":"Inf. Sci."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1007\/BF00179828","article-title":"Fuzzy measurement of income inequality: A class of fuzzy inequality measures","volume":"12","author":"Ok","year":"1995","journal-title":"Soc. Choice Welf."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s13660-021-02623-w","article-title":"Hermite\u2013Hadamard-type inequalities for interval-valued preinvex functions via Riemann\u2013Liouville fractional integrals","volume":"2021","author":"Sharma","year":"2021","journal-title":"J. Inequal. Appl."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"373","DOI":"10.1016\/j.ejcon.2020.08.001","article-title":"Hybrid data-driven fuzzy active disturbance rejection control for tower crane systems","volume":"58","author":"Roman","year":"2021","journal-title":"Eur. J. Control."},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"Chi, R., Li, H., Shen, D., Hou, Z., and Huang, B. Enhanced P-type Control: Indirect Adaptive Learning from Set-point Updates. IEEE Trans. Autom. Control., 2022.","DOI":"10.1109\/TAC.2022.3154347"},{"key":"ref_55","doi-asserted-by":"crossref","unstructured":"Khan, M.B., Santos-Garc\u00eda, G., Trean\u021b\u01ce, S., and Soliman, M.S. (2022). New Class Up and Down Pre-Invex Fuzzy Number Valued Mappings and Related Inequalities via Fuzzy Riemann Integrals. Symmetry, 14.","DOI":"10.3390\/sym14112322"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/1\/1\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:44:32Z","timestamp":1760147072000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/1\/1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,20]]},"references-count":55,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,1]]}},"alternative-id":["axioms12010001"],"URL":"https:\/\/doi.org\/10.3390\/axioms12010001","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2022,12,20]]}}}