{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:46:02Z","timestamp":1760244362358,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2022,12,24]],"date-time":"2022-12-24T00:00:00Z","timestamp":1671840000000},"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 work, we obtain some new integral inequalities of the Hermite\u2013Hadamard\u2013Fej\u00e9r type for operator \u03c91,\u03c92-preinvex functions. The bounds for both left-hand and right-hand sides of the integral inequality are established for operator \u03c91,\u03c92-preinvex functions of the positive self-adjoint operator in the complex Hilbert spaces. We give the special cases to our results; thus, the established results are generalizations of earlier work. In the last section, we give applications for synchronous (asynchronous) functions.<\/jats:p>","DOI":"10.3390\/axioms12010016","type":"journal-article","created":{"date-parts":[[2022,12,27]],"date-time":"2022-12-27T02:55:07Z","timestamp":1672109707000},"page":"16","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Fej\u00e9r-Type Midpoint and Trapezoidal Inequalities for the Operator \u03c91,\u03c92-Preinvex Functions"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4042-4399","authenticated-orcid":false,"given":"Sikander","family":"Mehmood","sequence":"first","affiliation":[{"name":"Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan 60800, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9277-8092","authenticated-orcid":false,"given":"Hari Mohan","family":"Srivastava","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada"},{"name":"Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan"},{"name":"Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, AZ1007 Baku, Azerbaijan"},{"name":"Center for Converging Humanities, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul 02447, Republic of Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6837-8075","authenticated-orcid":false,"given":"Pshtiwan Othman","family":"Mohammed","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Education, University of Sulaimani, Sulaimani 46001, Iraq"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0223-4711","authenticated-orcid":false,"given":"Eman","family":"Al-Sarairah","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Khalifa University, Abu Dhabi P.O. Box 127788, United Arab Emirates"},{"name":"Department of Mathematics, Al-Hussein Bin Talal University, Ma\u2019an P.O. Box 33011, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0552-2783","authenticated-orcid":false,"given":"Fiza","family":"Zafar","sequence":"additional","affiliation":[{"name":"Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan 60800, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7469-5402","authenticated-orcid":false,"given":"Kamsing","family":"Nonlaopon","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,24]]},"reference":[{"key":"ref_1","first-page":"369","article-title":"\u00dcber die Fourierreihen","volume":"24","year":"1906","journal-title":"II. Math. Naturwiss Anz. Ungar. Akad. 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