{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:16:12Z","timestamp":1760145372498,"version":"build-2065373602"},"reference-count":52,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2024,7,10]],"date-time":"2024-07-10T00:00:00Z","timestamp":1720569600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Foundation of Anhui Province Higher School","award":["2023AH051697","2023AH051682","2022jyxm481","gxgnfx2020122","TU-DSPP-2024-47"],"award-info":[{"award-number":["2023AH051697","2023AH051682","2022jyxm481","gxgnfx2020122","TU-DSPP-2024-47"]}]},{"name":"Taif University, Saudi Arabia","award":["2023AH051697","2023AH051682","2022jyxm481","gxgnfx2020122","TU-DSPP-2024-47"],"award-info":[{"award-number":["2023AH051697","2023AH051682","2022jyxm481","gxgnfx2020122","TU-DSPP-2024-47"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we establish several Milne-type inequalities for fuzzy number mappings and investigate their relationships with other inequalities. Specifically, we utilize Aumann\u2019s integral and the fuzzy Kulisch\u2013Miranker order, as well as the newly defined class, \u0127-Godunova\u2013Levin convex fuzzy number mappings, to derive Ostrowski\u2019s and Hermite\u2013Hadamard-type inequalities for fuzzy number mappings. Using the fuzzy Kulisch\u2013Miranker order, we also establish connections with Hermite\u2013Hadamard-type inequalities. Furthermore, we explore novel ideas and results based on Hermite\u2013Hadamard\u2013Fej\u00e9r and provide examples and applications to illustrate our findings. Some very interesting examples are also provided to discuss the validation of the main results. Additionally, some new exceptional and classical outcomes have been obtained, which can be considered as applications of our main results.<\/jats:p>","DOI":"10.3390\/axioms13070465","type":"journal-article","created":{"date-parts":[[2024,7,10]],"date-time":"2024-07-10T15:22:05Z","timestamp":1720624925000},"page":"465","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Fuzzy Milne, Ostrowski, and Hermite\u2013Hadamard-Type Inequalities for \u0127-Godunova\u2013Levin Convexity and Their Applications"],"prefix":"10.3390","volume":"13","author":[{"given":"Juan","family":"Wang","sequence":"first","affiliation":[{"name":"Faculty of Engineering, Anhui Sanlian University, Hefei 230601, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Valer-Daniel","family":"Breaz","sequence":"additional","affiliation":[{"name":"Department of Computing, Mathematics and Electronics, \u201c1 Decembrie 1918\u201d University of Alba Iulia, 510009 Alba Iulia, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0365-0282","authenticated-orcid":false,"given":"Yasser Salah","family":"Hamed","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0269-0688","authenticated-orcid":false,"given":"Luminita-Ioana","family":"Cotirla","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Technical University of Cluj-Napoca, 400114 Cluj-Napoca, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xuewu","family":"Zuo","sequence":"additional","affiliation":[{"name":"General Education Department, Anhui Xinhua University, Hefei 230088, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,7,10]]},"reference":[{"key":"ref_1","unstructured":"Pe\u010dric\u0107, J.E., Proechan, F., and Tong, Y.L. (1991). Convex Functions, Partial Ordering and Statistical Applications, Academic Press."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"140202","DOI":"10.1007\/s11432-023-3982-y","article-title":"Online Pareto optimal control of mean-field stochastic multi-player systems using policy iteration","volume":"67","author":"Jiang","year":"2024","journal-title":"Sci. China Inf. Sci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"155417","DOI":"10.1103\/PhysRevB.109.155417","article-title":"Valley quantum interference modulated by hyperbolic shear polaritons","volume":"109","author":"Jia","year":"2023","journal-title":"Phys. Rev. B"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"283147","DOI":"10.1155\/2009\/283147","article-title":"On the Hadamard\u2019s inequality for log-convex functions on the coordinates","volume":"13","author":"Alomari","year":"2009","journal-title":"J. Inequal. Appl."},{"key":"ref_5","first-page":"7","article-title":"Convex functions and the Hadamard inequality","volume":"28","author":"Azpeitia","year":"1994","journal-title":"Rev. Colomb. Mat."},{"key":"ref_6","first-page":"96","article-title":"Hadamard tpye inequalities for m-convex and (\u03b1, m)-convex functions","volume":"9","author":"Bakula","year":"2008","journal-title":"J. Ineq. Pure Appl. Math."},{"key":"ref_7","first-page":"74","article-title":"Note on some Hadamard-type inequalities","volume":"5","author":"Bakula","year":"2004","journal-title":"J. Ineq. Pure Appl. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1016\/S0893-9659(98)00086-X","article-title":"Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula","volume":"11","author":"Dragomir","year":"1998","journal-title":"Appl. Math. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"979","DOI":"10.1093\/mnras\/85.9.979","article-title":"Note on Rosseland\u2019s integral for the stellar absorption coefficient","volume":"85","author":"Milne","year":"1925","journal-title":"Mon. Not. R Astron. Soc."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"226","DOI":"10.1007\/BF01214290","article-title":"\u00dcber die absolutabweichung einer differentiebaren funktion van ihrem integralmitte wert","volume":"10","author":"Ostrowski","year":"1938","journal-title":"Comment Math. Helv."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Mitrinovi\u0107, D.S., and Vasi\u0107, P.M. (1970). Analytic Inequalities. Die Grundlehren der mathematischen Wissenschaften, Band 165, Springer.","DOI":"10.1007\/978-3-642-99970-3"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"151","DOI":"10.2298\/AADM200609054M","article-title":"Ostrowski type inequalities and some selected quadrature formulae","volume":"15","year":"2021","journal-title":"Appl. Anal. Discrete Math."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Irshad, N., Khan, A.R., Mehmood, F., and Pe\u010dari\u0107, J. (2022). New Perspectives on the Theory of Inequalities for Integral and Sum, Springer.","DOI":"10.1007\/978-3-030-90563-7"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Pachpatte, B.G. (2012). Analytic Inequalities: Recent Advances, Atlantis Press.","DOI":"10.2991\/978-94-91216-44-2"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1016\/j.camwa.2003.09.026","article-title":"Sharp inequalities of Simpson type and Ostrowski type","volume":"48","year":"2004","journal-title":"Comput. Math. Appl."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2027","DOI":"10.2298\/FIL1509027A","article-title":"Improvement of Gr\u00fcss and Ostrowski type inequalities","volume":"29","author":"Acu","year":"2015","journal-title":"Filomat"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1016\/j.ins.2020.04.037","article-title":"Ostrowski\u2013type inequalities for fuzzy valued functions and its applications in quadrature theory","volume":"529","author":"Costa","year":"2020","journal-title":"Inf. Sci."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"167","DOI":"10.2298\/FIL1401167L","article-title":"New bounds for the companion of Ostrowski\u2019s inequality and applications","volume":"28","author":"Liu","year":"2014","journal-title":"Filomat"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1142\/S0219530513500309","article-title":"A generalization of the Ostrowski-Gr\u00fcss inequality","volume":"12","author":"Dragomir","year":"2014","journal-title":"Anal. Appl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1441","DOI":"10.2298\/FIL1606441Q","article-title":"Improvement of Ostrowski integral type inequalities with application","volume":"30","author":"Qayyum","year":"2016","journal-title":"Filomat"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"224","DOI":"10.22436\/jmcs.028.03.02","article-title":"Fuzzy Ostrowski type inequalities via \u03c6-\u03bb-convex functions","volume":"28","author":"Hassan","year":"2023","journal-title":"J. Math. Comput. Sci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"148","DOI":"10.1016\/j.ins.2016.10.006","article-title":"Jensen-type inequalities for Sugeno integral","volume":"376","author":"Abbaszadeh","year":"2017","journal-title":"Inf. Sci."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"708","DOI":"10.1016\/j.fss.2009.10.007","article-title":"General Minkowski type inequalities for Sugeno integrals","volume":"161","author":"Agahi","year":"2010","journal-title":"Fuzzy Sets Syst."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"543","DOI":"10.1016\/j.ins.2009.10.014","article-title":"Generalization of the Jensen inequality for pseudo-integral","volume":"180","author":"Pap","year":"2010","journal-title":"Inf. Sci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1007\/s12190-009-0358-y","article-title":"Some inequalities and convergence theorems for Choquet integral","volume":"35","author":"Wang","year":"2011","journal-title":"J. Appl. Math. Comput."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0022-247X(65)90049-1","article-title":"Integrals of set-valued functions","volume":"12","author":"Aumann","year":"1965","journal-title":"J. Math. Anal. Appl."},{"key":"ref_27","unstructured":"Klein, E., and Thompson, A.C. (1984). Theory of Correspondences, A Wiley-Interscience Publication."},{"key":"ref_28","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_29","doi-asserted-by":"crossref","unstructured":"Khan, M.B., Zaini, H.G., Treana, S., Soliman, M.S., and Nonlaopon, K. (2022). Riemann\u2013Liouville fractional integral inequalities for generalized pre-invex functions of interval-valued settings based upon pseudo order relation. Mathematics, 10.","DOI":"10.3390\/math10020204"},{"key":"ref_30","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":"2019","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1016\/j.fss.2019.10.006","article-title":"Chebyshev type inequalities for interval-valued functions","volume":"396","author":"Zhao","year":"2020","journal-title":"Fuzzy Sets Syst."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"128","DOI":"10.1016\/j.fss.2021.03.017","article-title":"Some generalizations of Opial type inequalities for interval-valued functions","volume":"436","author":"Zhao","year":"2021","journal-title":"Fuzzy Sets Syst."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Zhao, D., Zhao, G., Ye, G., Liu, W., and Dragomir, S.S. (2021). On Hermite\u2013Hadamard-type inequalities for coordinated h-convex interval-valued functions. Mathematics, 9.","DOI":"10.3390\/math9192352"},{"key":"ref_34","first-page":"2531","article-title":"Fractional Ostrowski type inequalities for interval valued functions","volume":"36","author":"Budak","year":"2022","journal-title":"Mathematics"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Diamond, P., and Kloeden, P. (1994). Metric Spaces of Fuzzy Sets: Theory and Applications, World Scientific.","DOI":"10.1142\/2326"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1016\/0165-0114(86)90026-6","article-title":"Elementary fuzzy calculus","volume":"18","author":"Goetschel","year":"1986","journal-title":"Fuzzy Sets Syst."},{"key":"ref_37","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_38","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_39","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_40","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1007\/s44196-021-00004-1","article-title":"New Hermite\u2013Hadamard and Jensen inequalities for log-h-convex fuzzy interval valued functions","volume":"14","author":"Khan","year":"2021","journal-title":"Int. J. Comput. Intell. Syst."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1186\/s13662-021-03245-8","article-title":"New Hermite\u2013Hadamard\u2013type inequalities for (h1,h2)\u2013convex fuzzy-interval-valued functions","volume":"2021","author":"Khan","year":"2021","journal-title":"Adv. Differ. Equ."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Khan, M.B., Trean\u021b\u01ce, S., Soliman, M.S., Nonlaopon, K., and Zaini, H.G. (2022). Some Hadamard\u2013Fej\u00e9r Type Inequalities for LR-Convex Interval-Valued Functions. Fractal Fract., 6.","DOI":"10.3390\/fractalfract6040178"},{"key":"ref_43","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_44","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_45","unstructured":"Moore, R.E. (1966). Interval Analysis, Prentice Hall."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"178","DOI":"10.1016\/j.fss.2020.06.003","article-title":"Jensen\u2019s inequalities for set-valued and fuzzy set-valued functions","volume":"404","author":"Zhang","year":"2020","journal-title":"Fuzzy Sets Syst."},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Stojiljkovi\u0107, V., Ramaswamy, R., Ashour Abdelnaby, O.A., and Radenovi\u0107, S. (2022). Riemann-Liouville fractional inclusions for convex functions using interval valued setting. Mathematics, 10.","DOI":"10.3390\/math10193491"},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Kalsoom, H., Latif, M.A., Khan, Z.A., and Vivas-Cortez, M. (2021). Some New Hermite-Hadamard-Feje\u00b4r fractional type inequalities for h-convex and harmonically h-Convex interval-valued Functions. Mathematics, 10.","DOI":"10.3390\/math10010074"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"302","DOI":"10.1186\/s13660-018-1896-3","article-title":"New Jensen and Hermite\u2013Hadamard type inequalities for h-convex interval-valued functions","volume":"2018","author":"Zhao","year":"2018","journal-title":"J. Inequal. Appl."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"335","DOI":"10.7153\/jmi-02-30","article-title":"On some Hadamard-type inequalities for h-convex functions","volume":"2","author":"Sarikaya","year":"2008","journal-title":"J. Math. Inequal."},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Althobaiti, A., Althobaiti, S., and Vivas Cortez, M. (2024). The Estimation of Different Kinds of Integral Inequalities for a Generalized Class of Convex Mapping and a Harmonic Set via Fuzzy Inclusion Relations and Their Applications in Quadrature Theory. Axioms, 13.","DOI":"10.3390\/axioms13060344"},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Rakhmangulov, A., Aljohani, A.F., Mubaraki, A., and Althobaiti, S. (2024). A New Class of Coordinated Non-Convex Fuzzy-Number-Valued Mappings with Related Inequalities and Their Applications. Axioms, 13.","DOI":"10.3390\/axioms13060404"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/7\/465\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:12:55Z","timestamp":1760109175000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/7\/465"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,7,10]]},"references-count":52,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2024,7]]}},"alternative-id":["axioms13070465"],"URL":"https:\/\/doi.org\/10.3390\/axioms13070465","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,7,10]]}}}