{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,2]],"date-time":"2025-11-02T08:27:59Z","timestamp":1762072079311,"version":"build-2065373602"},"reference-count":34,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,10,18]],"date-time":"2022-10-18T00:00:00Z","timestamp":1666051200000},"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, on the subject of harmonic convex functions, we introduce some new functionals linked with weighted integral inequalities for harmonic convex functions. In addition, certain new inequalities of the Fej\u00e9r type are discovered.<\/jats:p>","DOI":"10.3390\/axioms11100564","type":"journal-article","created":{"date-parts":[[2022,10,18]],"date-time":"2022-10-18T21:18:02Z","timestamp":1666127882000},"page":"564","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Weighted Integral Inequalities for Harmonic Convex Functions in Connection with Fej\u00e9r\u2019s Result"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2349-3445","authenticated-orcid":false,"given":"Muhammad Amer","family":"Latif","sequence":"first","affiliation":[{"name":"Department of Basic Sciences, Deanship of Preparatory Year, King Faisal University, Al-Hofuf 31982, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,18]]},"reference":[{"key":"ref_1","first-page":"171","article-title":"\u00c9tude sur les propri\u00e9t\u00e9s des fonctions enti\u00e8res en particulier d\u2019une function consid\u00e9r\u00e9e par Riemann","volume":"58","author":"Hadamard","year":"1983","journal-title":"J. Math. Pures Appl."},{"key":"ref_2","first-page":"386806","article-title":"Fej\u00e9r and Hermite-Hadamard type inequalities for harmonicallyally convex functions","volume":"6","author":"Chen","year":"2014","journal-title":"J. Appl. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"775","DOI":"10.11650\/twjm\/1500574995","article-title":"On the Hadamard\u2019s inequality for convex on the co-ordinates in a rectangle from the plane","volume":"5","author":"Dragomir","year":"2001","journal-title":"Taiwan. J. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1006\/jmaa.2000.6769","article-title":"Inequalities of Hadamard\u2019s type for Lipschitzian mappings and their applications","volume":"245","author":"Dragomir","year":"2000","journal-title":"J. Math. Anal. Appl."},{"key":"ref_5","first-page":"3","article-title":"On some refinements of Hadamard\u2019s inequalities and applications","volume":"4","author":"Dragomir","year":"1993","journal-title":"Univ. Belgrad. Publ. Elek. Fak. Sci. Math."},{"key":"ref_6","first-page":"103","article-title":"Inequalities of Jensen type for HA-convex functions. Analele Universit\u0103\u0163ii Oradea Fasc","volume":"27","author":"Dragomir","year":"2000","journal-title":"Matematica"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1515\/mjpaa-2017-0008","article-title":"Inequalities of Hermite-Hadamard type for HA-convex functions","volume":"3","author":"Dragomir","year":"2017","journal-title":"Moroc. J. Pure Appl. Anal."},{"key":"ref_8","first-page":"215","article-title":"On Hadamard\u2019s inequality for convex functions","volume":"6","author":"Dragomir","year":"1992","journal-title":"Math. Balkan."},{"key":"ref_9","first-page":"177","article-title":"On Hadamard\u2019s inequality for the convex mappings defined on a ball in the space and applications","volume":"3","author":"Dragomir","year":"2000","journal-title":"Math. Ineq. Appl."},{"key":"ref_10","first-page":"2","article-title":"On Hadamard\u2019s inequality on a disk","volume":"1","author":"Dragomir","year":"2000","journal-title":"J. Ineq. Pure Appl. Math."},{"key":"ref_11","first-page":"21","article-title":"On some integral inequalities for convex functions","volume":"18","author":"Dragomir","year":"1996","journal-title":"Z. Rad."},{"key":"ref_12","first-page":"9","article-title":"Fejer inequalities for Wright-convex functions","volume":"8","author":"Ho","year":"2007","journal-title":"JIPAM J. Inequal. Pure Appl. Math."},{"key":"ref_13","first-page":"935","article-title":"Hermite-Hadamard type inequalities for harmonicallyally convex functions","volume":"43","year":"2014","journal-title":"Hacet. J. Math. Stat."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"63","DOI":"10.11650\/twjm\/1500404635","article-title":"Some Hadamard\u2019s inequalities for co-ordinated convex functions in a rectangle from the plane","volume":"11","author":"Hwang","year":"2007","journal-title":"Taiwan. J. Math."},{"key":"ref_15","first-page":"91","article-title":"On a weighted generalization of Hadamard\u2019s inequality for Gconvex functions","volume":"16","author":"Lee","year":"2000","journal-title":"Tamsui-Oxf. Math. Sci."},{"key":"ref_16","first-page":"795","article-title":"Fej\u00e9r type inequalities for harmonicallyally-convex functions with applications","volume":"7","author":"Latif","year":"2017","journal-title":"J. Appl. Anal. Comput."},{"key":"ref_17","first-page":"51","article-title":"On some new inequalities of Hermite-Hadamard-Fej\u00e9r type involving convex functions","volume":"40","author":"Tseng","year":"2007","journal-title":"Demonstr. Math."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"33","DOI":"10.5556\/j.tkjm.28.1997.4331","article-title":"A note on Hadamard\u2019s inequality","volume":"28","author":"Yang","year":"1997","journal-title":"Tamkang. J. Math."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"180","DOI":"10.1006\/jmaa.1999.6506","article-title":"On certain integral inequalities related to Hermite-Hadamard inequalities","volume":"239","author":"Yang","year":"1999","journal-title":"J. Math. Anal. Appl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"230","DOI":"10.1006\/jmaa.2000.7460","article-title":"Inequalities of Hadamard\u2019s type for Lipschitzian mappings","volume":"260","author":"Yang","year":"2001","journal-title":"J. Math. Anal. Appl."},{"key":"ref_21","first-page":"131","article-title":"On certain multiple integral inequalities related to Hermite-Hadamard inequalities","volume":"62","author":"Yang","year":"2002","journal-title":"Util. Math."},{"key":"ref_22","first-page":"433","article-title":"Inequalities of Hermite-Hadamard-Fej\u00e9r type for convex functions and Lipschitzian functions","volume":"7","author":"Yang","year":"2003","journal-title":"Taiwan. J. Math."},{"key":"ref_23","first-page":"369","article-title":"\u00dcber die Fourierreihen, II","volume":"24","year":"1906","journal-title":"Math. Natur. Ungar. Akad. Wiss."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1016\/0022-247X(92)90233-4","article-title":"Two mappings in connection to Hadamard\u2019s inequalities","volume":"167","author":"Dragomir","year":"1992","journal-title":"J. Math. Anal. Appl."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"101","DOI":"10.5556\/j.tkjm.24.1993.4479","article-title":"A refinement of Hadamard\u2019s inequality for isotonic linear functionals","volume":"24","author":"Dragomir","year":"1993","journal-title":"Tamkang. J. Math."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"45","DOI":"10.5556\/j.tkjm.34.2003.271","article-title":"Further properties of some mapping associated with Hermite-Hadamard inequalities","volume":"34","author":"Dragomir","year":"2003","journal-title":"Tamkang. J. Math."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1016\/S0893-9659(98)00030-5","article-title":"Two new mappings associated with Hadamard\u2019s inequalities for convex functions","volume":"11","author":"Dragomir","year":"1998","journal-title":"Appl. Math. Lett."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Latif, M.A. Mappings related to Hermite-Hadamard type inequalities for harmonicallyally convex functions. Punjab Univ. Math. J., 2022. in press.","DOI":"10.3390\/math10091398"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"15234","DOI":"10.3934\/math.2022835","article-title":"Fej\u00e9r type inequalities for harmonicallyally convex functions","volume":"7","author":"Latif","year":"2022","journal-title":"AIMS Math."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Latif, M.A. (2022). Fej\u00e9r Type Inequalities for Harmonically Convex Functions and Related Results. Submitted.","DOI":"10.3390\/sym14112268"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"531976","DOI":"10.1155\/2010\/531976","article-title":"Fej\u00e9r-Type Inequalities (I)","volume":"2010","author":"Tseng","year":"2010","journal-title":"J. Inequal. Appl."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"645","DOI":"10.1007\/s13398-014-0206-2","article-title":"Some companions of Fej\u00e9r\u2019s inequality for convex functions","volume":"109","author":"Tseng","year":"2015","journal-title":"Rev. Acad. Ciencia Sci. A Math."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1007\/s10998-012-3980-7","article-title":"Refinements of Fej\u00e9r\u2019s for convex functions","volume":"65","author":"Tseng","year":"2012","journal-title":"Period. Math. Hung."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1515\/ms-2016-0252","article-title":"Fej\u00e9r-type Inequalities (II)","volume":"67","author":"Tseng","year":"2017","journal-title":"Math. Slov."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/10\/564\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:56:25Z","timestamp":1760144185000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/10\/564"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,18]]},"references-count":34,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2022,10]]}},"alternative-id":["axioms11100564"],"URL":"https:\/\/doi.org\/10.3390\/axioms11100564","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2022,10,18]]}}}