{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:39:32Z","timestamp":1760150372810,"version":"build-2065373602"},"reference-count":41,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2023,11,10]],"date-time":"2023-11-10T00:00:00Z","timestamp":1699574400000},"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>In this research, we define a few subclasses of analytic functions which are connected to sine functions and the cardioid domain in the unit disk. We investigate initial coefficient bounds, the Fekete\u2013Szego problem and second and third Hankel determinants for the functions f belonging to these newly defined classes. We also define the class of m-fold symmetric functions related with the sine function and then investigate the bounds of the third Hankel determinant for twofold symmetric and threefold symmetric functions.<\/jats:p>","DOI":"10.3390\/sym15112039","type":"journal-article","created":{"date-parts":[[2023,11,10]],"date-time":"2023-11-10T01:17:51Z","timestamp":1699579071000},"page":"2039","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Third Hankel Determinant for Subclasses of Analytic and m-Fold Symmetric Functions Involving Cardioid Domain and Sine Function"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5898-3797","authenticated-orcid":false,"given":"Ayman","family":"Alahmade","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science and Art, AlUla Branch, Taibah University, Medina 42353, Saudi Arabia"}]},{"given":"Zeeshan","family":"Mujahid","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Abbottabad University of Science and Technology, Abbottabad 22500, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1655-5185","authenticated-orcid":false,"given":"Ferdous M. O.","family":"Tawfiq","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, King Saud University, Riyadh 22452, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2427-2003","authenticated-orcid":false,"given":"Bilal","family":"Khan","sequence":"additional","affiliation":[{"name":"Shanghai Key Laboratory of PMMP, School of Mathematical Sciences, East China Normal University, 500 Dongchuan Road, Shanghai 200241, China"}]},{"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 22500, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7855-508X","authenticated-orcid":false,"given":"Fairouz","family":"Tchier","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, King Saud University, Riyadh 22452, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2023,11,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1090\/S0002-9939-1977-0425097-1","article-title":"New subclasses of the class of close-to-convex functions","volume":"62","author":"Chichra","year":"1977","journal-title":"Proc. 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