{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,13]],"date-time":"2026-01-13T22:33:25Z","timestamp":1768343605885,"version":"3.49.0"},"reference-count":46,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2025,5,4]],"date-time":"2025-05-04T00:00:00Z","timestamp":1746316800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper investigates the analytical properties of multiplicative generalized proportional \u03c3-Riemann\u2013Liouville fractional integrals and the corresponding Hermite\u2013Hadamard-type inequalities. Central to our study are two key notions: multiplicative \u03c3-convex functions and multiplicative generalized proportional \u03c3-Riemann\u2013Liouville fractional integrals, both of which serve as the foundational framework for our analysis. We first introduce and examine several fundamental properties of the newly defined fractional integral operator, including continuity, commutativity, semigroup behavior, and boundedness. Building on these results, we derive a novel identity involving this operator, which forms the basis for establishing new Hermite\u2013Hadamard-type inequalities within the multiplicative setting. To validate the theoretical results, we provide multiple illustrative examples and perform graphical visualizations. These examples not only demonstrate the correctness of the derived inequalities but also highlight the practical relevance and potential applications of the proposed framework.<\/jats:p>","DOI":"10.3390\/sym17050702","type":"journal-article","created":{"date-parts":[[2025,5,4]],"date-time":"2025-05-04T20:42:37Z","timestamp":1746391357000},"page":"702","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Analytical Properties and Hermite\u2013Hadamard Type Inequalities Derived from Multiplicative Generalized Proportional \u03c3-Riemann\u2013Liouville Fractional Integrals"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3076-9390","authenticated-orcid":false,"given":"Fuxiang","family":"Liu","sequence":"first","affiliation":[{"name":"Department of Mathematics & Three Gorges Mathematical Research Center, College of Mathematics and Physics, China Three Gorges University, Yichang 443002, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0005-6809-8467","authenticated-orcid":false,"given":"Jielan","family":"Li","sequence":"additional","affiliation":[{"name":"Department of Mathematics & Three Gorges Mathematical Research Center, College of Mathematics and Physics, China Three Gorges University, Yichang 443002, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,5,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Nail, B., Atoussi, M.A., Saadi, S., Tibermacine, I.E., and Napoli, C. 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