{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T04:09:29Z","timestamp":1777435769887,"version":"3.51.4"},"reference-count":39,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2024,9,29]],"date-time":"2024-09-29T00:00:00Z","timestamp":1727568000000},"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>This paper investigates the third Hankel determinant, denoted H3(1), for functions within the subclass RS\u2211*(\u03bb) of bi-univalent functions associated with crescent-shaped regions \u03c6\u2985z=z+1+z2. The primary aim of this study is to establish upper bounds for H3(1). By analyzing functions within this specific geometric context, we derive precise constraints on the determinant, thereby enhancing our understanding of its behavior. Our results and examples provide valuable insights into the properties of bi-univalent functions in crescent-shaped domains and contribute to the broader theory of analytic functions.<\/jats:p>","DOI":"10.3390\/sym16101281","type":"journal-article","created":{"date-parts":[[2024,9,30]],"date-time":"2024-09-30T03:12:28Z","timestamp":1727665948000},"page":"1281","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Upper Bounds of the Third Hankel Determinant for Bi-Univalent Functions in Crescent-Shaped Domains"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0009-0002-2734-9432","authenticated-orcid":false,"given":"Qasim Ali","family":"Shakir","sequence":"first","affiliation":[{"name":"Department of Computer Science, College of Computer Science and Information Technology, University of Al-Qadisiyah, Diwaniyah 58002, Iraq"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4380-1428","authenticated-orcid":false,"given":"Adel Salim","family":"Tayyah","sequence":"additional","affiliation":[{"name":"Department of Computer Science, College of Computer Science and Information Technology, University of Al-Qadisiyah, Diwaniyah 58002, Iraq"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Daniel","family":"Breaz","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Alba Iulia, 510009 Alba-Iulia, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0269-0688","authenticated-orcid":false,"given":"Luminita-Ioana","family":"Cot\u00eerl\u0103","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Tehnical University of Cluj-Napoca, 400114 Cluj-Napoca, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0007-1212-4245","authenticated-orcid":false,"given":"Eleonora","family":"Rapeanu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, \u201cMircea cel Batran\u201d, Naval Academy, 900218 Constanta, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3884-3957","authenticated-orcid":false,"given":"Fethiye M\u00fcge","family":"Sakar","sequence":"additional","affiliation":[{"name":"Department of Management, Dicle University, Diyarbakir 21280, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,9,29]]},"reference":[{"key":"ref_1","first-page":"70","article-title":"On some classes of bi-univalent functions","volume":"31","author":"Brannan","year":"1986","journal-title":"Studia Univ. 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