{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,22]],"date-time":"2026-02-22T10:49:37Z","timestamp":1771757377121,"version":"3.50.1"},"reference-count":36,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2024,9,23]],"date-time":"2024-09-23T00:00:00Z","timestamp":1727049600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this study, we introduce a novel local fractional integral identity related to the Gaussian two-point left Radau rule. Based on this identity, we establish some new fractal inequalities for functions whose first-order local fractional derivatives are generalized convex and concave. The obtained results not only represent an extension of certain previously established findings to fractal sets but also a refinement of these when the fractal dimension \u03bc is equal to one. Finally, to support our findings, we present a practical application to demonstrate the effectiveness of our results.<\/jats:p>","DOI":"10.3390\/axioms13090653","type":"journal-article","created":{"date-parts":[[2024,9,24]],"date-time":"2024-09-24T08:56:06Z","timestamp":1727168166000},"page":"653","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["An Extension of Left Radau Type Inequalities to Fractal Spaces and Applications"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2160-4159","authenticated-orcid":false,"given":"Bandar","family":"Bin-Mohsin","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2943-2678","authenticated-orcid":false,"given":"Abdelghani","family":"Lakhdari","sequence":"additional","affiliation":[{"name":"Laboratory of Energy Systems Technology, National Higher School of Technology and Engineering, Annaba 23005, Algeria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5307-7090","authenticated-orcid":false,"given":"Nour El Islem","family":"Karabadji","sequence":"additional","affiliation":[{"name":"Laboratory of Energy Systems Technology, National Higher School of Technology and Engineering, Annaba 23005, Algeria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1019-9485","authenticated-orcid":false,"given":"Muhammad Uzair","family":"Awan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Government College University Faisalabad, Faisalabad 38000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7142-7026","authenticated-orcid":false,"given":"Abdellatif Ben","family":"Makhlouf","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Sciences, University of Sfax, BP 1171, Sfax 3000, Tunisia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0156-7864","authenticated-orcid":false,"given":"Badreddine","family":"Meftah","sequence":"additional","affiliation":[{"name":"Laboratory of Analysis and Control of Differential Equations \u2018ACED\u2019, Department of Mathematics, 8 May 1945 University, Guelma 24000, Algeria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2902-6805","authenticated-orcid":false,"given":"Silvestru Sever","family":"Dragomir","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Engineering & Science, Victoria University, P.O. Box 14428, Melbourne City, VIC 8001, Australia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,9,23]]},"reference":[{"key":"ref_1","first-page":"155","article-title":"Convexity according to the geometric mean","volume":"3","author":"Niculescu","year":"2000","journal-title":"Math. Inequal. Appl."},{"key":"ref_2","unstructured":"Pe\u010dari\u0107, J.E., Proschan, F., and Tong, Y.L. (1992). Convex Functions, Partial Orderings, and Statistical Applications, Academic Press, Inc.. Mathematics in Science and Engineering, 187."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"330","DOI":"10.1007\/s10598-023-09576-3","article-title":"Some new Hermite-Hadamard type integral inequalities for twice differentiable s-convex functions","volume":"33","author":"Meftah","year":"2022","journal-title":"Comput. Math. 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