{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:30:35Z","timestamp":1760243435082,"version":"build-2065373602"},"reference-count":42,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,9,27]],"date-time":"2022-09-27T00:00:00Z","timestamp":1664236800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Fundamental Fund of Khon Kaen University"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The main objective of this article is to introduce a new notion of convexity, i.e., modified exponential type convex function, and establish related fractional inequalities. To strengthen the argument of the paper, we introduce two new lemmas as auxiliary results and discuss some algebraic properties of the proposed notion. Considering a generalized fractional integral operator and differentiable mappings, whose initial absolute derivative at a given power is a modified exponential type convex, various improvements of the Hermite\u2013Hadamard inequality are presented. Thanks to the main results, some generalizations about the earlier findings in the literature are recovered.<\/jats:p>","DOI":"10.3390\/axioms11100508","type":"journal-article","created":{"date-parts":[[2022,9,27]],"date-time":"2022-09-27T21:23:27Z","timestamp":1664313807000},"page":"508","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Some New Refinements of Trapezium-Type Integral Inequalities in Connection with Generalized Fractional Integrals"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8372-2532","authenticated-orcid":false,"given":"Muhammad","family":"Tariq","sequence":"first","affiliation":[{"name":"Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology, Jamshoro 76062, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4524-1951","authenticated-orcid":false,"given":"Soubhagya Kumar","family":"Sahoo","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Institute of Technical Education and Research, Siksha O Anusandhan University, Bhubaneswar 751030, India"},{"name":"Department of Mathematics, Aryan Institute of Engineering and Technology, Bhubaneswar 752050, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7695-2118","authenticated-orcid":false,"given":"Sotiris K.","family":"Ntouyas","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece"},{"name":"Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7530-6264","authenticated-orcid":false,"given":"Omar Mutab","family":"Alsalami","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, College of Engineering, Taif University, Taif 21944, Saudi Arabia"}]},{"given":"Asif Ali","family":"Shaikh","sequence":"additional","affiliation":[{"name":"Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology, Jamshoro 76062, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7469-5402","authenticated-orcid":false,"given":"Kamsing","family":"Nonlaopon","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1016\/S1573-4382(81)01005-9","article-title":"Mathematical analysis and convexity with applications to economics","volume":"Volume 1","author":"Green","year":"1981","journal-title":"Handbook of Mathematical Economics"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"429","DOI":"10.1007\/s007800200072","article-title":"Convex measures of risk and trading constraints","volume":"6","author":"Schied","year":"2002","journal-title":"Financ. 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