{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T04:30:44Z","timestamp":1772253044236,"version":"3.50.1"},"reference-count":28,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2018,10,13]],"date-time":"2018-10-13T00:00:00Z","timestamp":1539388800000},"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>A class of function called sub-b-s-preinvex function is defined as a generalization of s-convex and b-preinvex functions, and some of its basic properties are presented here. The sufficient conditions of optimality for unconstrainded and inquality constrained programming are discussed under the sub-b-s-preinvexity. Moreover, some new inequalities of the Hermite\u2014Hadamard type for differentiable sub-b-s-preinvex functions are presented. Examples of applications of these inequalities are shown.<\/jats:p>","DOI":"10.3390\/sym10100493","type":"journal-article","created":{"date-parts":[[2018,10,15]],"date-time":"2018-10-15T03:43:01Z","timestamp":1539574981000},"page":"493","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Generalized Preinvex Functions and Their Applications"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1217-963X","authenticated-orcid":false,"given":"Adem","family":"Kili\u00e7man","sequence":"first","affiliation":[{"name":"Department of Mathematics, Putra University of Malaysia (UPM), Serdang 43400, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wedad","family":"Saleh","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Taibah University, Al-Medina 20012, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,10,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1007\/BF00939919","article-title":"B-vex functions","volume":"71","author":"Bector","year":"1991","journal-title":"J. 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