{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,8]],"date-time":"2026-04-08T19:52:54Z","timestamp":1775677974884,"version":"3.50.1"},"reference-count":42,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2021,11,7]],"date-time":"2021-11-07T00:00:00Z","timestamp":1636243200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Jiraporn Reunsumrit","award":["KMUTNB-64-KNOW-32"],"award-info":[{"award-number":["KMUTNB-64-KNOW-32"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The theory of convexity plays an important role in various branches of science and engineering. The objective of this paper is to introduce a new notion of preinvex functions by unifying the n-polynomial preinvex function with the s-type m\u2013preinvex function and to present inequalities of the Hermite\u2013Hadamard type in the setting of the generalized s-type m\u2013preinvex function. First, we give the definition and then investigate some of its algebraic properties and examples. We also present some refinements of the Hermite\u2013Hadamard-type inequality using H\u00f6lder\u2019s integral inequality, the improved power-mean integral inequality, and the H\u00f6lder-\u0130\u015fcan integral inequality. Finally, some results for special means are deduced. The results established in this paper can be considered as the generalization of many published results of inequalities and convexity theory.<\/jats:p>","DOI":"10.3390\/axioms10040296","type":"journal-article","created":{"date-parts":[[2021,11,7]],"date-time":"2021-11-07T20:41:14Z","timestamp":1636317674000},"page":"296","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["New Integral Inequalities via Generalized Preinvex Functions"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8372-2532","authenticated-orcid":false,"given":"Muhammad","family":"Tariq","sequence":"first","affiliation":[{"name":"Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology, Jamshoro 76062, Pakistan"}]},{"given":"Asif Ali","family":"Shaikh","sequence":"additional","affiliation":[{"name":"Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology, Jamshoro 76062, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4524-1951","authenticated-orcid":false,"given":"Soubhagya Kumar","family":"Sahoo","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Institute of Technical Education and Research, Siksha \u2018O\u2019 Anusandhan University, Bhubaneswar 751030, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5438-5407","authenticated-orcid":false,"given":"Hijaz","family":"Ahmad","sequence":"additional","affiliation":[{"name":"Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele II, 39, 00186 Roma, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8455-1402","authenticated-orcid":false,"given":"Thanin","family":"Sitthiwirattham","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science and Technology, Suan Dusit University, Bangkok 10300, Thailand"}]},{"given":"Jiraporn","family":"Reunsumrit","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Applied Science, King Mongkut\u2019s University of Technology North Bangkok, Bangkok 10800, Thailand"}]}],"member":"1968","published-online":{"date-parts":[[2021,11,7]]},"reference":[{"key":"ref_1","first-page":"1","article-title":"Some new Hermite\u2013Hadamard type integral inequalities for the s\u2013convex functions and theirs applications","volume":"201","year":"2019","journal-title":"J. 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