{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T11:30:43Z","timestamp":1774611043772,"version":"3.50.1"},"reference-count":36,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2023,8,18]],"date-time":"2023-08-18T00:00:00Z","timestamp":1692316800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research, King Faisal University","award":["GRANT3816"],"award-info":[{"award-number":["GRANT3816"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, new Fej\u00e9r-type inequalities for harmonically convex functions are obtained. Some mappings related to the Fej\u00e9r-type inequalities for harmonically convex are defined. Properties of these mappings are discussed and, as a consequence, we obtain refinements of some known results.<\/jats:p>","DOI":"10.3390\/sym15081602","type":"journal-article","created":{"date-parts":[[2023,8,21]],"date-time":"2023-08-21T01:33:11Z","timestamp":1692581591000},"page":"1602","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Fej\u00e9r-Type Inequalities for Harmonically Convex Functions and Related Results"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2349-3445","authenticated-orcid":false,"given":"Muhammad","family":"Amer Latif","sequence":"first","affiliation":[{"name":"Department of Basic Sciences, Deanship of Preparatory Year, King Faisal University, Hofuf 31982, Al-Hasa, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,8,18]]},"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":"82","article-title":"Sur deux limites d\u2019une int\u00e9grale d\u00e9finie","volume":"3","author":"Hermite","year":"1883","journal-title":"Mathesis"},{"key":"ref_3","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_4","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_5","first-page":"369","article-title":"\u00dcber die Fourierreihen, II","volume":"24","year":"1906","journal-title":"Math. Naturwiss. Anz. Ungar. Akad. Wiss."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"560","DOI":"10.15672\/hujms.1096357","article-title":"Convexity and double-sided Taylor\u2019s approximations","volume":"52","author":"Bagul","year":"2023","journal-title":"Hacet. J. Math. Stat."},{"key":"ref_7","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_8","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_9","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":"3","author":"Dragomir","year":"2003","journal-title":"Tamkang J. Math."},{"key":"ref_10","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_11","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_12","first-page":"215","article-title":"On Hadamard\u2019s inequality for convex functions","volume":"6","author":"Dragomir","year":"1992","journal-title":"Mat. Balk."},{"key":"ref_13","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_14","first-page":"2","article-title":"On Hadamard\u2019s inequality on a disk","volume":"1","author":"Dragomir","year":"2000","journal-title":"J. Ineq. Pure Appl. Maht."},{"key":"ref_15","first-page":"21","article-title":"On some integral inequalities for convex functions","volume":"18","author":"Dragomir","year":"1996","journal-title":"Zb.-Rad."},{"key":"ref_16","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_17","first-page":"9","article-title":"Fej\u00e9r inequalities for Wright-convex functions","volume":"8","author":"Ho","year":"2007","journal-title":"JIPAM J. Inequal. Pure Appl. Math."},{"key":"ref_18","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_19","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-Oxford J. Math. Sci."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Latif, M.A. (2022). Weighted Integral Inequalities for Harmonic Convex Functions in Connection with Fej\u00e9r\u2019s Result. Axioms, 11.","DOI":"10.3390\/axioms11100564"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Latif, M.A. (2022). On Symmetrized Stochastic Harmonically Convexity and Hermite\u2013Hadamard Type Inequalities. Axioms, 11.","DOI":"10.3390\/axioms11100570"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Latif, M.A. (2022). Some Companions of Fej\u00e9r-Type Inequalities for Harmonically Convex Functions. Symmetry, 14.","DOI":"10.3390\/sym14112268"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"665","DOI":"10.52280\/pujm.2022.541101","article-title":"Mappings related to Hermite-Hadamard type inequalities for harmonically convex functions","volume":"54","author":"Latif","year":"2022","journal-title":"Punjab Univ. J. Math."},{"key":"ref_24","first-page":"795","article-title":"Fej\u00e9r type inequalities for harmonically-convex functions with applications","volume":"7","author":"Latif","year":"2017","journal-title":"J. Appl. Anal. Comput."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"386806","DOI":"10.1155\/2014\/386806","article-title":"Fej\u00e9r and Hermite-Hadamard type inequalities for harmonically convex functions","volume":"2014","author":"Chen","year":"2014","journal-title":"J. Appl. Math."},{"key":"ref_26","first-page":"103","article-title":"Inequalities of Jensen type for HA-convex functions","volume":"XXVII","author":"Dragomir","year":"2020","journal-title":"Analele Univ. Oradea Fasc. Mat."},{"key":"ref_27","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":"Moroccan J. Pure Appl. Anal."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"15234","DOI":"10.3934\/math.2022835","article-title":"Fej\u00e9r type inequalities for harmonically convex functions","volume":"7","author":"Latif","year":"2022","journal-title":"AIMS Math."},{"key":"ref_29","first-page":"51","article-title":"On some new inequalities of Hermite-Hadamard- Fej\u00e9r type involving convex functions","volume":"XL","author":"Tseng","year":"2007","journal-title":"Demonstr. Math."},{"key":"ref_30","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. Slovaca"},{"key":"ref_31","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_32","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_33","first-page":"131","article-title":"On certain multiple integral inequalities related to Hermite-Hadamard inequalities","volume":"62","author":"Yang","year":"2002","journal-title":"Utilitas Math."},{"key":"ref_34","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_35","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_36","first-page":"935","article-title":"Hermite-Hadamard type inequalities for harmonically convex functions","volume":"43","year":"2014","journal-title":"Hacet. J. Math. Stat."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/8\/1602\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:37:15Z","timestamp":1760128635000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/8\/1602"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,8,18]]},"references-count":36,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2023,8]]}},"alternative-id":["sym15081602"],"URL":"https:\/\/doi.org\/10.3390\/sym15081602","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,8,18]]}}}