{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T00:34:01Z","timestamp":1775522041713,"version":"3.50.1"},"reference-count":28,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2023,2,28]],"date-time":"2023-02-28T00:00:00Z","timestamp":1677542400000},"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>In the present paper, we first prove a new integral identity. Using that identity, we establish some fractional weighted midpoint-type inequalities for functions whose first derivatives are extended s-convex. Some special cases are discussed. Finally, to prove the effectiveness of our main results, we provide some applications to numerical integration as well as special means.<\/jats:p>","DOI":"10.3390\/sym15030612","type":"journal-article","created":{"date-parts":[[2023,2,28]],"date-time":"2023-02-28T06:09:52Z","timestamp":1677564592000},"page":"612","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Fractional Weighted Midpoint-Type Inequalities for s-Convex Functions"],"prefix":"10.3390","volume":"15","author":[{"given":"Nassima","family":"Nasri","sequence":"first","affiliation":[{"name":"Departement de Math\u00e9matiques, Universit\u00e9 20 Ao\u00fbt 1955, Skikda Bp 26 Route El-Hadaiek, Skikda 21000, Algeria"}]},{"given":"Fatima","family":"Aissaoui","sequence":"additional","affiliation":[{"name":"D\u00e9partement de Math\u00e9matiques, Facult\u00e9 des Math\u00e9matiques, de l\u2019informatique et des Sciences de la Mati\u00e8re, Universit\u00e9 8 Mai 1945 Guelma, Guelma 24000, Algeria"}]},{"given":"Keltoum","family":"Bouhali","sequence":"additional","affiliation":[{"name":"Departement de Math\u00e9matiques, Universit\u00e9 20 Ao\u00fbt 1955, Skikda Bp 26 Route El-Hadaiek, Skikda 21000, Algeria"},{"name":"Department of Mathematics, College of Science and Arts, Qassim University, Ar-Rass 51452, Saudi Arabia"}]},{"given":"Assia","family":"Frioui","sequence":"additional","affiliation":[{"name":"D\u00e9partement de Math\u00e9matiques, Facult\u00e9 des Math\u00e9matiques, de l\u2019informatique et des Sciences de la Mati\u00e8re, Universit\u00e9 8 Mai 1945 Guelma, Guelma 24000, Algeria"}]},{"given":"Badreddine","family":"Meftah","sequence":"additional","affiliation":[{"name":"D\u00e9partement de Math\u00e9matiques, Facult\u00e9 des Math\u00e9matiques, de l\u2019informatique et des Sciences de la Mati\u00e8re, Universit\u00e9 8 Mai 1945 Guelma, Guelma 24000, Algeria"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7895-4168","authenticated-orcid":false,"given":"Khaled","family":"Zennir","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science and Arts, Qassim University, Ar-Rass 51452, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1466-8821","authenticated-orcid":false,"given":"Taha","family":"Radwan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science and Arts, Qassim University, Ar-Rass 51452, Saudi Arabia"},{"name":"Department of Mathematics and Statistics, Faculty of Management Technology and Information Systems, Port Said University, Port Said 42511, Egypt"}]}],"member":"1968","published-online":{"date-parts":[[2023,2,28]]},"reference":[{"key":"ref_1","first-page":"335","article-title":"Some inequalities of Hadamard type","volume":"21","author":"Dragomir","year":"1995","journal-title":"Soochow J. Math."},{"key":"ref_2","first-page":"403","article-title":"Hardy-type inequalities via convexity","volume":"8","author":"Kaijser","year":"2005","journal-title":"Math. Inequal. Appl."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"75","DOI":"10.48185\/jfcns.v1i1.150","article-title":"Some weighted Simpson type inequalities for differentiable s-convex functions and their applications","volume":"1","author":"Kashuri","year":"2020","journal-title":"J. Fract. Calc. Nonlinear Syst."},{"key":"ref_4","first-page":"145","article-title":"Opial and P\u00f3lya type inequalities via convexity","volume":"60","author":"Saker","year":"2018","journal-title":"Fasc. Math."},{"key":"ref_5","first-page":"1936461","article-title":"Simpson\u2019s integral inequalities for twice differentiable convex functions","volume":"2020","author":"Abdeljawad","year":"2020","journal-title":"Math. Probl. Eng."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1049","DOI":"10.18514\/MMN.2017.1197","article-title":"On Hermite-Hadamard type inequalities for Riemann\u2013Liouville fractional integrals","volume":"17","author":"Sarikaya","year":"2016","journal-title":"Miskolc Math. Notes"},{"key":"ref_7","first-page":"655","article-title":"Hermite-Hadamard inequalities for convex set-valued functions","volume":"46","author":"Mitroi","year":"2013","journal-title":"Demonstratio Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/S0893-9659(99)00164-0","article-title":"Inequalities for differentiable mappings with application to special means and quadrature formul\u00e6","volume":"13","author":"Pearce","year":"2000","journal-title":"Appl. Math. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1016\/S0096-3003(02)00657-4","article-title":"Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula","volume":"147","author":"Kirmaci","year":"2004","journal-title":"Appl. Math. Comput."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"510","DOI":"10.26637\/mjm204\/019","article-title":"Some new general integral inequalities for P-functions","volume":"2","author":"Set","year":"2014","journal-title":"Malaya J. Mat."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Gorenflo, R., and Mainardi, F. (1997). Fractional Calculus: Integral and Differential Equations of Fractional Order, Springer. Fractals and Fractional Calculus in Continuum Mechanics (Udine, 1996), 223\u2013276, CISM Courses and Lect., 378.","DOI":"10.1007\/978-3-7091-2664-6_5"},{"key":"ref_12","unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier Science B.V.. North-Holland Mathematics Studies 204."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Eskandari, Z., Avazzadeh, Z., Ghaziani, R.K., and Li, B. Dynamics and bifurcations of a discrete-time Lotka-Volterra model using nonstandard finite difference discretization method. Math. Meth. Appl. Sci., 2022.","DOI":"10.1002\/mma.8859"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"115089","DOI":"10.1016\/j.cam.2023.115089","article-title":"Bifurcation analysis and complex dynamics of a Kopel triopoly model","volume":"426","author":"Li","year":"2023","journal-title":"J. Comp. Appl. Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"110856","DOI":"10.1016\/j.chaos.2021.110856","article-title":"Multiple and generic bifurcation analysis of a discrete Hindmarsh-Rose model","volume":"146","author":"Li","year":"2021","journal-title":"Chaos Solitons Fractals"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"111860","DOI":"10.1016\/j.chaos.2022.111860","article-title":"Complex dynamics of Kopel model with nonsymmetric response between oligopolists","volume":"156","author":"Li","year":"2022","journal-title":"Chaos Solitons Fractals"},{"key":"ref_17","first-page":"994","article-title":"On fractional-order symmetric oscillator withoffset-boosting control","volume":"27","author":"Xu","year":"2022","journal-title":"Nonlinear Anal. Model. Control."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"4211","DOI":"10.3934\/math.2022234","article-title":"Dynamics of fractional order delay model of coronavirus disease","volume":"7","author":"Zhang","year":"2022","journal-title":"AIMS Math."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"252","DOI":"10.3390\/fractalfract5040252","article-title":"Weighted midpoint Hermite-Hadamard-Fej\u00e9r type inequalities in fractional calculus for harmonically convex functions","volume":"5","author":"Kalsoom","year":"2021","journal-title":"Fractal Fract."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"73","DOI":"10.2478\/jamsi-2022-0006","article-title":"Fractional Simpson like type inequalities for differentiable s-convex functions","volume":"18","author":"Kamouche","year":"2022","journal-title":"J. Appl. Math. Stat. Inform."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Kashuri, A., Meftah, B., Mohammed, P.O., Lupa, A.A., Abdalla, B., Hamed, Y.S., and Abdeljawad, T. (2021). Fractional weighted Ostrowski type inequalities and their applications. Symmetry, 13.","DOI":"10.3390\/sym13060968"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"6689779","DOI":"10.1155\/2021\/6689779","article-title":"Some properties of fractional Boas transforms of wavelets","volume":"2021","author":"Khanna","year":"2021","journal-title":"J. Math."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1186\/s13662-020-2541-2","article-title":"Modification of certain fractional integral inequalities for convex functions","volume":"2020","author":"Mohammed","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Mohammed, P.O., Aydi, H., Kashuri, A., Hamed, Y.S., and Abualnaja, K.M. (2021). Midpoint inequalities in fractional calculus defined using positive weighted symmetry function kernels. Symmetry, 13.","DOI":"10.3390\/sym13040550"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"454","DOI":"10.1186\/s13662-019-2381-0","article-title":"Certain inequalities via generalized proportional Hadamard fractional integral operators","volume":"2019","author":"Rahman","year":"2019","journal-title":"Adv. Difference Equ."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2150016","DOI":"10.1142\/S1793557121500169","article-title":"A study on Hermite\u2013Hadamard-type inequalities via new fractional conformable integrals","volume":"14","author":"Set","year":"2021","journal-title":"Asian-Eur. J. Math."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"3","DOI":"10.15393\/j3.art.2016.3071","article-title":"On inequalities of Hermite-Hadamard type involving an s-convex function with applications","volume":"5","author":"Liu","year":"2016","journal-title":"Issues Anal."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"373","DOI":"10.1080\/09720502.2021.1932858","article-title":"Some weighted integral inequalities for differentiable beta-convex functions","volume":"25","author":"Azzouza","year":"2022","journal-title":"J. Interdiscip. Math."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/3\/612\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:44:01Z","timestamp":1760121841000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/3\/612"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,28]]},"references-count":28,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2023,3]]}},"alternative-id":["sym15030612"],"URL":"https:\/\/doi.org\/10.3390\/sym15030612","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,2,28]]}}}