{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,4,29]],"date-time":"2025-04-29T10:34:06Z","timestamp":1745922846457,"version":"3.37.3"},"reference-count":40,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2021,5,13]],"date-time":"2021-05-13T00:00:00Z","timestamp":1620864000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,5,13]],"date-time":"2021-05-13T00:00:00Z","timestamp":1620864000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["71771140"],"award-info":[{"award-number":["71771140"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Complex Intell. Syst."],"published-print":{"date-parts":[[2022,2]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this paper, our aim is to consider the new class of log-convex fuzzy-interval-valued function known as log-<jats:italic>s<\/jats:italic>-convex fuzzy-interval-valued functions (log-<jats:italic>s<\/jats:italic>-convex fuzzy-IVFs). By this concept, we have introduced Hermite\u2013Hadamard inequalities (HH-inequalities) by means of fuzzy order relation. This fuzzy order relation is defined level-wise through Kulisch\u2013Miranker order relation defined on interval space. Moreover, some new Hermite\u2013Hadamard\u2013Fej\u00e9r inequalities (HH\u2013Fej\u00e9r-inequalities) and Jensen\u2019s inequalities via log-<jats:italic>s<\/jats:italic>-convex fuzzy-IVFs are also established and verified with the support of useful examples. Some special cases are also discussed which can be viewed as applications of fuzzy-interval HH-inequalities. The concepts and approaches of this paper may be the starting point for further research in this area.<\/jats:p>","DOI":"10.1007\/s40747-021-00379-w","type":"journal-article","created":{"date-parts":[[2021,5,13]],"date-time":"2021-05-13T08:03:00Z","timestamp":1620892980000},"page":"413-427","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":19,"title":["New Hermite\u2013Hadamard and Jensen inequalities for log-$$s$$-convex fuzzy-interval-valued functions in the second sense"],"prefix":"10.1007","volume":"8","author":[{"given":"Peide","family":"Liu","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7450-8067","authenticated-orcid":false,"given":"Muhammad Bilal","family":"Khan","sequence":"additional","affiliation":[]},{"given":"Muhammad Aslam","family":"Noor","sequence":"additional","affiliation":[]},{"given":"Khalida Inayat","family":"Noor","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,5,13]]},"reference":[{"key":"379_CR1","doi-asserted-by":"publisher","first-page":"1071","DOI":"10.1016\/j.aml.2010.04.038","volume":"23","author":"M Alomari","year":"2010","unstructured":"Alomari M, Darus M, Dragomir SS, Cerone P (2010) Ostrowski type inequalities for functions whose derivatives are s-convex in the second sense. Appl Math Lett 23:1071\u20131076","journal-title":"Appl Math Lett"},{"issue":"2","key":"379_CR2","doi-asserted-by":"publisher","first-page":"299","DOI":"10.1515\/dema-2004-0208","volume":"37","author":"P Cerone","year":"2004","unstructured":"Cerone P, Dragomir SS (2004) Ostrowski type inequalities for functions whose derivatives satisfy certain convexity assumptions. Demonstr Math 37(2):299\u2013308","journal-title":"Demonstr Math"},{"key":"379_CR3","doi-asserted-by":"publisher","first-page":"301","DOI":"10.1016\/0165-0114(87)90029-7","volume":"24","author":"O Kaleva","year":"1987","unstructured":"Kaleva O (1987) Fuzzy differential equations. Fuzzy Sets Syst 24:301\u2013317","journal-title":"Fuzzy Sets Syst"},{"key":"379_CR4","first-page":"171","volume":"7","author":"J Hadamard","year":"1893","unstructured":"Hadamard J (1893) \u00c9tude sur les propri\u00e9t\u00e9s des fonctions enti\u00e8res et en particulier d\u2019une fonction consid\u00e9r\u00e9e par Riemann. Journal de math\u00e9matiques pures et appliqu\u00e9es 7:171\u2013215","journal-title":"Journal de math\u00e9matiques pures et appliqu\u00e9es"},{"key":"379_CR5","first-page":"82","volume":"3","author":"C Hermite","year":"1883","unstructured":"Hermite C (1883) Sur deux limites d\u2019une int\u00e9grale d\u00e9finie. Mathesis 3:82\u201397","journal-title":"Mathesis"},{"issue":"1","key":"379_CR6","doi-asserted-by":"publisher","first-page":"303","DOI":"10.1016\/j.jmaa.2006.02.086","volume":"326","author":"S Varo\u0161anec","year":"2007","unstructured":"Varo\u0161anec S (2007) On h-convexity. J Math Anal Appl 326(1):303\u2013311","journal-title":"J Math Anal Appl"},{"issue":"3","key":"379_CR7","first-page":"515","volume":"35","author":"BY Xi","year":"2015","unstructured":"Xi BY, Qi F (2015) Some integral inequalities of Hermite\u2013Hadamard type for s-logarithmically convex functions. Acta Math Sci Ser A (Chin Ed) 35(3):515\u2013524","journal-title":"Acta Math Sci Ser A (Chin Ed)"},{"issue":"2","key":"379_CR8","first-page":"126","volume":"2","author":"MA Noor","year":"2007","unstructured":"Noor MA (2007) Hermite\u2013Hadamard integral inequalities for log-preinvex functions. J Math Anal Approx Theory 2(2):126\u2013131","journal-title":"J Math Anal Approx Theory"},{"key":"379_CR9","volume-title":"Convex functions, partial orderingsand statistical applications","author":"JE Pe\u010dari\u0107","year":"1991","unstructured":"Pe\u010dari\u0107 JE, Proschan F, Tong YL (1991) Convex functions, partial orderingsand statistical applications. Academic Press"},{"issue":"4","key":"379_CR10","doi-asserted-by":"publisher","first-page":"367","DOI":"10.1524\/anly.2013.1223","volume":"33","author":"MA Noor","year":"2013","unstructured":"Noor MA, Qi F, Awan MU (2013) Some Hermite\u2013Hadamard type inequalities for log-h-convex functions. Analysis 33(4):367\u2013375","journal-title":"Analysis"},{"issue":"5","key":"379_CR11","doi-asserted-by":"publisher","first-page":"4931","DOI":"10.3934\/math.2020315","volume":"5","author":"M Adil Khan","year":"2020","unstructured":"Adil Khan M, Pe\u010dari\u0107 J, Chu Y-M (2020) Refinements of Jensen\u2019s and McShane\u2019s inequalities with applications. AIMS Math 5(5):4931\u20134945","journal-title":"AIMS Math"},{"issue":"3","key":"379_CR12","first-page":"129","volume":"28","author":"SS Dragomir","year":"2001","unstructured":"Dragomir SS (2001) Refinements of the Hermite\u2013Hadamard integral inequality for log-convex functions. Aust Math Soc Gaz 28(3):129\u2013134","journal-title":"Aust Math Soc Gaz"},{"issue":"1","key":"379_CR13","first-page":"82","volume":"10","author":"SS Dragomir","year":"2011","unstructured":"Dragomir SS (2011) A survey of Jensen type inequalities for log-convex functions of selfadjoint operators in Hilbert spaces. Commun Math Anal 10(1):82\u2013104","journal-title":"Commun Math Anal"},{"issue":"2","key":"379_CR14","first-page":"354","volume":"31","author":"SS Dragomir","year":"1998","unstructured":"Dragomir SS, Mond B (1998) Integral inequalities of Hadamard type for log convexfunctions. Demonstr Math 31(2):354\u2013364","journal-title":"Demonstr Math"},{"issue":"2","key":"379_CR15","first-page":"98","volume":"3","author":"SS Dragomir","year":"2017","unstructured":"Dragomir SS (2017) New inequalities of Hermite\u2013Hadamard type for log convex functions. Khayyam J Math 3(2):98\u2013115","journal-title":"Khayyam J Math"},{"issue":"2","key":"379_CR16","doi-asserted-by":"publisher","first-page":"662","DOI":"10.1016\/j.na.2011.08.066","volume":"75","author":"CP Niculescu","year":"2012","unstructured":"Niculescu CP (2012) The Hermite\u2013Hadamard inequality for log convex functions. Nonlinear Anal 75(2):662\u2013669","journal-title":"Nonlinear Anal"},{"key":"379_CR17","volume-title":"Interval analysis","author":"RE Moore","year":"1966","unstructured":"Moore RE (1966) Interval analysis. Prentice Hall, Englewood Cliffs"},{"key":"379_CR18","volume-title":"Computer arithmetic in theory and practice","author":"UW Kulish","year":"1981","unstructured":"Kulish UW, Miranker W (1981) Computer arithmetic in theory and practice. Academic Press, New York"},{"key":"379_CR19","doi-asserted-by":"publisher","first-page":"86739","DOI":"10.1109\/ACCESS.2019.2925153","volume":"7","author":"Y Guo","year":"2019","unstructured":"Guo Y, Ye G, Zhao D, Liu W (2019) Some integral inequalities for log-h-convex interval-valued functions. IEEE Access 7:86739\u201386745","journal-title":"IEEE Access"},{"issue":"3","key":"379_CR20","first-page":"457","volume":"31","author":"Y Chalco-Cano","year":"2012","unstructured":"Chalco-Cano Y, Flores-Franuli\u010d A, Rom\u00e1n-Flores H (2012) Ostrowski type inequalities for interval-valued functions using generalized Hukuhara derivative. Comput Appl Math 31(3):457\u2013472","journal-title":"Comput Appl Math"},{"key":"379_CR21","doi-asserted-by":"publisher","first-page":"3293","DOI":"10.1007\/s00500-014-1483-6","volume":"19","author":"Y Chalco-Cano","year":"2015","unstructured":"Chalco-Cano Y, Lodwick WA, Condori-Equice W (2015) Ostrowski type inequalities and applications in numerical integration for interval-valued functions. Soft Comput 19:3293\u20133300","journal-title":"Soft Comput"},{"key":"379_CR22","doi-asserted-by":"publisher","first-page":"48","DOI":"10.1016\/j.fss.2018.04.012","volume":"358","author":"TM Costa","year":"2019","unstructured":"Costa TM, Rom\u00e1n-Flores H, Chalco-Cano Y (2019) Opial-type inequalities for interval-valued functions. Fuzzy Sets Syst 358:48\u201363","journal-title":"Fuzzy Sets Syst"},{"key":"379_CR23","volume-title":"Selected topics on Hermite\u2013Hadamard inequalities and applications","author":"SS Dragomir","year":"2000","unstructured":"Dragomir SS, Pearce CEM (2000) Selected topics on Hermite\u2013Hadamard inequalities and applications. RGMIA Monographs Victoria University"},{"issue":"2","key":"379_CR24","doi-asserted-by":"publisher","first-page":"1306","DOI":"10.1007\/s40314-016-0396-7","volume":"37","author":"H Rom\u00e1n-Flores","year":"2018","unstructured":"Rom\u00e1n-Flores H, Chalco-Cano Y, Lodwick WA (2018) Some integral inequalities for interval-valued functions. Comput Appl Math 37(2):1306\u20131318","journal-title":"Comput Appl Math"},{"key":"379_CR25","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1186\/s13660-017-1594-6","volume":"2018","author":"DF Zhao","year":"2018","unstructured":"Zhao DF, An TQ, Ye GJ, Liu W (2018) New Jensen and Hermite\u2013Hadamard type inequalities for h-convex interval-valued functions. J Inequal Appl 2018:1\u201314","journal-title":"J Inequal Appl"},{"issue":"3","key":"379_CR26","doi-asserted-by":"publisher","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","volume":"8","author":"LA Zadeh","year":"1965","unstructured":"Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338\u2013353","journal-title":"Inf Control"},{"key":"379_CR27","doi-asserted-by":"publisher","first-page":"129","DOI":"10.1016\/0165-0114(92)90256-4","volume":"48","author":"S Nanda","year":"1992","unstructured":"Nanda S, Kar K (1992) Convex fuzzy mappings. Fuzzy Sets Syst 48:129\u2013132","journal-title":"Fuzzy Sets Syst"},{"key":"379_CR28","volume-title":"Variational inequality and complementarity problems theory and applications","author":"SS Chang","year":"1991","unstructured":"Chang SS (1991) Variational inequality and complementarity problems theory and applications. Shanghai Scientific and Technological Literature Publishing House, Shanghai"},{"issue":"1","key":"379_CR29","doi-asserted-by":"publisher","first-page":"95","DOI":"10.1016\/0165-0114(94)90011-6","volume":"64","author":"MA Noor","year":"1994","unstructured":"Noor MA (1994) Fuzzy preinvex functions. Fuzzy Sets Syst 64(1):95\u2013104","journal-title":"Fuzzy Sets Syst"},{"issue":"3","key":"379_CR30","doi-asserted-by":"publisher","first-page":"581","DOI":"10.1016\/j.fss.2004.08.001","volume":"151","author":"B Bede","year":"2005","unstructured":"Bede B, Gal SG (2005) Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations. Fuzzy Sets Syst 151(3):581\u2013599","journal-title":"Fuzzy Sets Syst"},{"key":"379_CR31","doi-asserted-by":"crossref","unstructured":"Bede B (2013) Mathematics of fuzzy sets and fuzzy logic. In: Studies in fuzziness and soft computing, vol. 295. Springer","DOI":"10.1007\/978-3-642-35221-8"},{"key":"379_CR32","doi-asserted-by":"crossref","unstructured":"Cervelati J, Jim\u00e9nez-Gamero MD, Vilca-Labra F, Rojas-Medar MA (2004) Continuity for s-convex fuzzy processes. In: Soft methodology and random information systems, vol 7, pp 653\u2013660","DOI":"10.1007\/978-3-540-44465-7_81"},{"issue":"10\u201311","key":"379_CR33","doi-asserted-by":"publisher","first-page":"1117","DOI":"10.1016\/S0898-1221(00)00226-1","volume":"40","author":"Y Chalco-Cano","year":"2000","unstructured":"Chalco-Cano Y, Rojas-Medar MA, Rom\u00e1n-Flores H (2000) M-convex fuzzy mappings and fuzzy integral mean. Comput Math Appl 40(10\u201311):1117\u20131126","journal-title":"Comput Math Appl"},{"key":"379_CR34","doi-asserted-by":"publisher","first-page":"6","DOI":"10.1186\/s13662-020-03166-y","volume":"2021","author":"MB Khan","year":"2021","unstructured":"Khan MB, Noor MA, Noor KI, Chu Y-M (2021) New Hermite\u2013Hadamard type inequalities for (h1, h2)-convex fuzzy-interval-valued functions. Adv Differ Equ 2021:6\u201320","journal-title":"Adv Differ Equ"},{"key":"379_CR35","doi-asserted-by":"crossref","unstructured":"Puri ML, Ralescu DA (1986) Fuzzy random variables. In: Readings in fuzzy sets for intelligent systems, vol 114, pp 409\u2013422","DOI":"10.1016\/0022-247X(86)90093-4"},{"key":"379_CR36","doi-asserted-by":"crossref","unstructured":"Osuna-G\u00f3mez R, Jim\u00e9nez-Gamero MD, Chalco-Cano Y, Rojas-Medar MA (2004) Hadamard and Jensen inequalities for s-convex fuzzy processes. In: Soft methodology and random information systems. Advances in soft computing, vol l26. Springer, Berlin, pp 1\u201315","DOI":"10.1007\/978-3-540-44465-7_80"},{"key":"379_CR37","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.fss.2017.02.001","volume":"327","author":"TM Costa","year":"2017","unstructured":"Costa TM (2017) Jensen\u2019s inequality type integral for fuzzy-interval-valued functions. Fuzzy Sets Syst 327:1\u201347","journal-title":"Fuzzy Sets Syst"},{"key":"379_CR38","doi-asserted-by":"publisher","first-page":"31","DOI":"10.1016\/0165-0114(86)90026-6","volume":"18","author":"R Goetschel Jr","year":"1986","unstructured":"Goetschel R Jr, Voxman W (1986) Elementary fuzzy calculus. Fuzzy Sets Syst 18:31\u201343","journal-title":"Fuzzy Sets Syst"},{"key":"379_CR39","doi-asserted-by":"publisher","DOI":"10.1142\/2326","volume-title":"Metric spaces of fuzzy sets: theory and applications","author":"P Diamond","year":"1994","unstructured":"Diamond P, Kloeden PE (1994) Metric spaces of fuzzy sets: theory and applications. World Scientific"},{"key":"379_CR40","doi-asserted-by":"publisher","first-page":"110","DOI":"10.1016\/j.ins.2017.08.055","volume":"420","author":"TM Costa","year":"2017","unstructured":"Costa TM, Roman-Flores H (2017) Some integral inequalities for fuzzy-interval-valued functions. Inf Sci 420:110\u2013125","journal-title":"Inf Sci"}],"container-title":["Complex &amp; Intelligent Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40747-021-00379-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40747-021-00379-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40747-021-00379-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,3,3]],"date-time":"2022-03-03T12:35:04Z","timestamp":1646310904000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40747-021-00379-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,5,13]]},"references-count":40,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2022,2]]}},"alternative-id":["379"],"URL":"https:\/\/doi.org\/10.1007\/s40747-021-00379-w","relation":{},"ISSN":["2199-4536","2198-6053"],"issn-type":[{"type":"print","value":"2199-4536"},{"type":"electronic","value":"2198-6053"}],"subject":[],"published":{"date-parts":[[2021,5,13]]},"assertion":[{"value":"12 January 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 April 2021","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 May 2021","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}