{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,3]],"date-time":"2026-06-03T07:37:20Z","timestamp":1780472240543,"version":"3.54.1"},"reference-count":27,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2022,3,24]],"date-time":"2022-03-24T00:00:00Z","timestamp":1648080000000},"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 recent years, the Hankel determinant bounds for different subclasses of analytic, starlike and symmetric starlike functions have been discussed and studied by the many well-known authors. In this paper, we first consider a new subclass of analytic function and then we derive the fourth Hankel determinant bound for this class.<\/jats:p>","DOI":"10.3390\/sym14040663","type":"journal-article","created":{"date-parts":[[2022,3,24]],"date-time":"2022-03-24T23:31:43Z","timestamp":1648164703000},"page":"663","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Fourth Hankel Determinant Problem Based on Certain Analytic Functions"],"prefix":"10.3390","volume":"14","author":[{"given":"Huo","family":"Tang","sequence":"first","affiliation":[{"name":"School of Mathematics and Computer Sciences, Chifeng University, Chifeng 024000, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"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":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mirajul","family":"Haq","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"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":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mustaqeem","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Khurshid","family":"Ahmad","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Government Post Graduate Collage, Dargai 23060, Pakistan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"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":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,3,24]]},"reference":[{"key":"ref_1","first-page":"337","article-title":"On the second Hankel determinant of areally mean p-valent functions","volume":"223","author":"Noonan","year":"1976","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_2","first-page":"1","article-title":"Coefficient inequality for a function whose derivative has a positive real part","volume":"7","author":"Janteng","year":"2006","journal-title":"J. Inequalities Pure Appl. Math."},{"key":"ref_3","first-page":"619","article-title":"Hankel determinant for starlike and convex functions","volume":"13","author":"Janteng","year":"2007","journal-title":"Int. J. Math. Anal."},{"key":"ref_4","first-page":"413","article-title":"Second Hankel determinant for the class of Bazilevic functions","volume":"60","author":"Krishna","year":"2015","journal-title":"Stud. Univ. Babes-Bolyai Math."},{"key":"ref_5","first-page":"1","article-title":"On H3(1) Hankel determinant for some classes of univalent functions","volume":"6","author":"Babalola","year":"2007","journal-title":"Inequal. Theory Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1007\/s00009-016-0829-y","article-title":"Third Hankel determinants for subclasses of univalent functions","volume":"14","author":"Zaprawa","year":"2017","journal-title":"Mediterr. J. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1615","DOI":"10.1515\/math-2019-0132","article-title":"Hankel determinant of order three for familiar subsets of analytic functions related with sine function","volume":"17","author":"Arif","year":"2019","journal-title":"Open Math."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Shi, L., Ali, I., Arif, M., Cho, N.E., Hussain, S., and Khan, H. (2019). A study of third Hankel determinant problem for certain subfamilies of analytic functions involving cardioid domain. Mathematics, 7.","DOI":"10.3390\/math7050418"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"435","DOI":"10.1017\/S0004972717001125","article-title":"The sharp bound of the Hankel determinant of the third kind for convex functions","volume":"97","author":"Kowalczyk","year":"2018","journal-title":"Bull. Aust. Math. Soc."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2231","DOI":"10.1007\/s11785-018-0819-0","article-title":"The sharp bound of the Hankel determinant of the third kind for starlike functions of order 12","volume":"13","author":"Lecko","year":"2019","journal-title":"Complex Anal. Oper. Theory"},{"key":"ref_11","first-page":"511","article-title":"Estimates of the fourth Hankel determinant for a class of analytic functions with bounded turnings involving cardioid domains","volume":"22","author":"Srivastava","year":"2021","journal-title":"J. Nonlinear Convex Anal."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1007\/s40840-021-01195-8","article-title":"On the third and fourth Hankel determinants for a subclass of analytic functions","volume":"45","author":"Wang","year":"2022","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"ref_13","first-page":"75","article-title":"On the Upper Bound of the Third Hankel Determinant for Certain Class of Analytic Functions Related with Exponential Function","volume":"30","author":"Breaz","year":"2022","journal-title":"An. \u0218t. Univ. Ovidius Constan\u021ba"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1007\/s13370-021-00957-8","article-title":"New Concept on Fourth Hankel Determinant of a Certain Subclass of Analytic Functions","volume":"33","author":"Rahman","year":"2022","journal-title":"Afr. Mat."},{"key":"ref_15","first-page":"1703","article-title":"Fourth Hankel determinant for the family of functions with bounded turning","volume":"55","author":"Arif","year":"2018","journal-title":"Bull. Korean Math. Soc."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"53","DOI":"10.29020\/nybg.ejpam.v14i1.3902","article-title":"Coefficient problems in a class of functions with bounded turning associated with sine function","volume":"14","author":"Khan","year":"2021","journal-title":"Eur. J. Pure Appl. Math."},{"key":"ref_17","first-page":"251","article-title":"Coefficient bounds for the inverse of a function with derivative in P","volume":"87","author":"Libera","year":"1983","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"505","DOI":"10.1016\/j.crma.2015.03.003","article-title":"Bound for the fifth coefficient of certain starlike functions","volume":"353","author":"Ravichandran","year":"2015","journal-title":"Comptes Rendus Math."},{"key":"ref_19","unstructured":"Pommerenke, C. (1975). Univalent functions. Mathematik, Lehrbucher, Vandenhoeck and Ruprecht, Vandenhoeck & Ruprecht."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1016\/j.aml.2012.04.002","article-title":"Upper bound of second Hankel determinant for a new class of analytic functions","volume":"26","author":"Bansal","year":"2013","journal-title":"Appl. Math. Lett."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1139","DOI":"10.4134\/JKMS.2015.52.6.1139","article-title":"Third order Hankel determinant for certain univalent functions","volume":"52","author":"Bansal","year":"2015","journal-title":"J. Korean Math. Soc."},{"key":"ref_22","first-page":"729","article-title":"Bounds for the second Hankel determinant of certain univalent functions","volume":"2013","author":"Lee","year":"2013","journal-title":"J. Inequalities Appl."},{"key":"ref_23","first-page":"6116172","article-title":"Fourth Hankel Determinant for a Subclass of Starlike Functions Based on Modified Sigmoid","volume":"2021","author":"Mashwani","year":"2021","journal-title":"J. Funct. Spaces"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Murugusundaramoorthy, G., and Bulboac\u0103, T. (2020). Hankel determinants for new subclasses of analytic functions related to a shell shaped region. Mathematics, 8.","DOI":"10.3390\/math8061041"},{"key":"ref_25","first-page":"2378","article-title":"Upper bound of the third Hankel determinant for a class of analytic functions related with lemniscate of Bernoulli","volume":"2013","author":"Raza","year":"2013","journal-title":"J. Inequalities Appl."},{"key":"ref_26","first-page":"49","article-title":"Third Hankel determinant for univalent starlike functions","volume":"115","author":"Zaprawa","year":"2021","journal-title":"Rev. Real Acad. Cienc. Exactas Fis. Nat. Mat."},{"key":"ref_27","first-page":"9991460","article-title":"A Study of Fourth-Order Hankel Determinants for Starlike Functions Connected with the Sine Function","volume":"2021","author":"Zhang","year":"2021","journal-title":"J. Funct. Spaces"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/4\/663\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:42:14Z","timestamp":1760136134000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/4\/663"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,3,24]]},"references-count":27,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2022,4]]}},"alternative-id":["sym14040663"],"URL":"https:\/\/doi.org\/10.3390\/sym14040663","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,3,24]]}}}