{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,8]],"date-time":"2026-04-08T19:51:39Z","timestamp":1775677899255,"version":"3.50.1"},"reference-count":21,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2022,7,13]],"date-time":"2022-07-13T00:00:00Z","timestamp":1657670400000},"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>The main purpose of this research is to concentrate on the development of new definitions for the weighted geometric fractional integrals of the left-hand side and right-hand side of the function \u2135 with regard to an increasing function used as an integral kernel. Moreover, the newly developed class of left-hand side and right-hand side weighted geometric fractional integrals of a function \u2135, by applying an additional increasing function, identifies a variety of novel classes as special cases. This is a development of the previously established fractional integrals by making use of the class of geometrically convex functions. Geometrically convex functions in weighted fractional integrals of a function \u2135 in the form of another rising function yield the Hermite\u2013Hadamard inequality type. We also establish a novel midpoint identity and the associated inequalities for a class of weighted fractional integral functions known as geometrically convex with respect to an increasing function and symmetric with respect to the geometric mean of the endpoints of the interval. In order to demonstrate the validity of our research, we present examples. Moreover, fractional inequalities and their solutions are applied in many symmetrical domains.<\/jats:p>","DOI":"10.3390\/sym14071440","type":"journal-article","created":{"date-parts":[[2022,7,14]],"date-time":"2022-07-14T00:12:40Z","timestamp":1657757560000},"page":"1440","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["New Hermite\u2013Hadamard Integral Inequalities for Geometrically Convex Functions via Generalized Weighted Fractional Operator"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5835-3349","authenticated-orcid":false,"given":"Humaira","family":"Kalsoom","sequence":"first","affiliation":[{"name":"Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2349-3445","authenticated-orcid":false,"given":"Muhammad Amer","family":"Latif","sequence":"additional","affiliation":[{"name":"Department of Basic Sciences, Deanship of Preparatory Year, King Faisal University, Hofuf 31982, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8377-0208","authenticated-orcid":false,"given":"Zareen A.","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, College of Science, Princess Nourah Bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}]},{"given":"Areej A.","family":"Al-Moneef","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, College of Science, Princess Nourah Bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2022,7,13]]},"reference":[{"key":"ref_1","first-page":"82","article-title":"Sur deux limites d\u2019une int\u00e9grale d\u00e9 finie","volume":"3","author":"Hermite","year":"1883","journal-title":"Mathesis"},{"key":"ref_2","first-page":"171","article-title":"\u00c9tude sur les propri\u00e9t\u00e9s des fonctions enti\u00e9res en particulier d\u2019une function consid\u00e9r\u00e9 par Riemann","volume":"58","author":"Hadamard","year":"1893","journal-title":"J. Math. Pures Appl."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1016\/S0893-9659(98)00086-X","article-title":"Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula","volume":"11","author":"Dragomir","year":"1998","journal-title":"Appl. Math. Lett."},{"key":"ref_4","first-page":"65","article-title":"Some Hermite-Hadamard type integral inequalities whose n-times differentiable functions are s-logarithmically convex functions","volume":"2019","author":"Kalsoom","year":"2019","journal-title":"Punjab Univ. J. Math."},{"key":"ref_5","first-page":"7","article-title":"New inequaities of Hermite-Hadamard\u2019s type","volume":"12","author":"Sarikaya","year":"2009","journal-title":"Res. Rep. Collect."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2403","DOI":"10.1016\/j.mcm.2011.12.048","article-title":"Hermite-Hadamard\u2019s inequalities for fractional integrals and related fractional inequalities","volume":"57","author":"Sarikaya","year":"2013","journal-title":"Math. Comput. Model."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"112740","DOI":"10.1016\/j.cam.2020.112740","article-title":"On generalized fractional integral inequalities for twice differentiable convex functions","volume":"372","author":"Mohammed","year":"2020","journal-title":"J. Comput. Appl. Math."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Kalsoom, H., Vivas-Cortez, M., Amer, L.M., and Ahmad, H. (2021). Weighted Midpoint Hermite-Hadamard-Fej\u00e9r Type Inequalities in Fractional Calculus for Harmonically Convex Functions. Fractal Fract., 5.","DOI":"10.3390\/fractalfract5040252"},{"key":"ref_9","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_10","doi-asserted-by":"crossref","unstructured":"Kalsoom, H., Latif, M.A., Khan, Z.A., and Vivas-Cortez, M. (2021). Some New Hermite-Hadamard-Fej\u00e9r Fractional Type Inequalities for h-Convex and Harmonically h-Convex Interval-Valued Functions. Mathematics, 10.","DOI":"10.3390\/math10010074"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1049","DOI":"10.18514\/MMN.2017.1197","article-title":"On Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals","volume":"17","author":"Sarikaya","year":"2017","journal-title":"Miskolc Math. Notes"},{"key":"ref_12","first-page":"355","article-title":"Hermite-Hadamard-Fej\u00e9r type inequalities for convex functions via fractional integrals","volume":"60","year":"2015","journal-title":"Stud. Univ. Babes Bolyai Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"55","DOI":"10.36753\/mathenot.207633","article-title":"On generalization of different type integral inequalities for s-convex functions via fractional integrals","volume":"2","year":"2014","journal-title":"Math. Sci. Appl. E-Notes"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Kalsoom, H., and Khan, Z.A. (2022). Hermite-Hadamard-Fej\u00e9r Type Inequalities with Generalized K-Fractional Conformable Integrals and Their Applications. Mathematics, 10.","DOI":"10.3390\/math10030483"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1142\/S0218348X20400113","article-title":"On the weighted fractional operators of a function with respect to another function","volume":"28","author":"Jarad","year":"2020","journal-title":"Fractals"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1153","DOI":"10.1515\/math-2021-0072","article-title":"Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals","volume":"19","author":"Kalsoom","year":"2021","journal-title":"Open Math."},{"key":"ref_17","first-page":"155","article-title":"Convexity according to the geometric mean","volume":"3","author":"Niculescu","year":"2000","journal-title":"Math. Inequal. Appl."},{"key":"ref_18","first-page":"51","article-title":"Fej\u00e9r type integral inequalities related with geometrically-arithmetically convex functions with applications","volume":"23","author":"Dragomir","year":"2019","journal-title":"Acta Comment. Univ. Tartu. Math."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"491","DOI":"10.1186\/1029-242X-2013-491","article-title":"New general integral inequalities for quasi-geometrically convex functions via fractional integrals","volume":"2013","year":"2013","journal-title":"J. Inequalities Appl."},{"key":"ref_20","first-page":"1","article-title":"Fractional Hermite\u2013Hadamard\u2013Fej\u00e9r type inequalities for GA-convex functions","volume":"2","author":"Kunt","year":"2018","journal-title":"Turk. J. Inequal"},{"key":"ref_21","unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/7\/1440\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:49:38Z","timestamp":1760140178000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/7\/1440"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,13]]},"references-count":21,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2022,7]]}},"alternative-id":["sym14071440"],"URL":"https:\/\/doi.org\/10.3390\/sym14071440","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,7,13]]}}}