{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:06:13Z","timestamp":1760144773938,"version":"build-2065373602"},"reference-count":66,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2024,5,22]],"date-time":"2024-05-22T00:00:00Z","timestamp":1716336000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Taif University, Saudi Arabia","award":["TU-DSPP-2024-87"],"award-info":[{"award-number":["TU-DSPP-2024-87"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The relationship between convexity and symmetry is widely recognized. In fuzzy theory, both concepts exhibit similar behavior. It is crucial to remember that real and interval-valued mappings are special instances of fuzzy-number-valued mappings (F-N-V-Ms), as fuzzy theory relies on the unit interval, which is crucial to resolving problems with interval analysis and fuzzy number theory. In this paper, a new harmonic convexities class of fuzzy numbers has been introduced via up and down relation. We show several Hermite\u2013Hadamard (H\u22c5H) and Fej\u00e9r-type inequalities by the implementation of fuzzy Aumann integrals using the newly defined class of convexities. Some nontrivial examples are also presented to validate the main outcomes.<\/jats:p>","DOI":"10.3390\/axioms13060344","type":"journal-article","created":{"date-parts":[[2024,5,22]],"date-time":"2024-05-22T10:00:11Z","timestamp":1716372011000},"page":"344","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["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"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6076-625X","authenticated-orcid":false,"given":"Ali","family":"Althobaiti","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Saad","family":"Althobaiti","sequence":"additional","affiliation":[{"name":"Department of Sciences and Technology, Ranyah University Collage, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1567-0264","authenticated-orcid":false,"given":"Miguel","family":"Vivas Cortez","sequence":"additional","affiliation":[{"name":"Escuela de Ciencias F\u00edsicas y Matem\u00e1ticas, Facultad de Ciencias Exactas y Naturales, Pontificia Universidad Cat\u00f3lica del Ecuador, Av. 12 de Octubre 1076, Apartado, Quito 17-01-2184, Ecuador"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,5,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1186\/s13662-017-1126-1","article-title":"Monotonicity results for fractional difference operators with discrete exponential kernels","volume":"2017","author":"Abdeljawad","year":"2017","journal-title":"Adv. Differ. 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