{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,19]],"date-time":"2026-03-19T02:33:33Z","timestamp":1773887613411,"version":"3.50.1"},"reference-count":16,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2022,7,8]],"date-time":"2022-07-08T00:00:00Z","timestamp":1657238400000},"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>By making use of some families of integral and derivative operators, many distinct subclasses of analytic, starlike functions, and symmetric starlike functions have already been defined and investigated from numerous perspectives. In this article, with the help of the one-parameter Bernardi integral operator, we investigate several majorization results for the class of normalized starlike functions, which are associated with the Janowski functions. We also give some particular cases of our main results. Finally, we direct the interested readers to the possibility of examining the fundamental or quantum (or q-) extensions of the findings provided in this work in the concluding section. However, the (p,q)-variations of the suggested q-results will provide relatively minor and inconsequential developments because the additional (rather forced-in) parameter p is obviously redundant.<\/jats:p>","DOI":"10.3390\/sym14071404","type":"journal-article","created":{"date-parts":[[2022,7,11]],"date-time":"2022-07-11T00:06:21Z","timestamp":1657497981000},"page":"1404","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Majorization Results Based upon the Bernardi Integral Operator"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5287-4656","authenticated-orcid":false,"given":"Isra","family":"Al-Shbeil","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, the University of Jordan, Amman 11942, Jordan"}],"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":"Section of Mathematics, International Telematic University Uninettuno, I-00186 Rome, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1484-7643","authenticated-orcid":false,"given":"Muhammad","family":"Arif","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mirajul","family":"Haq","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1123-8578","authenticated-orcid":false,"given":"Nazar","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Abbottabad University of Science and Technology, Abbottabad 22010, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2427-2003","authenticated-orcid":false,"given":"Bilal","family":"Khan","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences and Shanghai Key Laboratory of PMMP, East China Normal University, 500 Dongchuan Road, Shanghai 200241, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,7,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1090\/S0002-9904-1970-12356-4","article-title":"Quasi-subordination and coefficient conjectures","volume":"76","author":"Roberston","year":"1970","journal-title":"Bull. 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