{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,16]],"date-time":"2026-04-16T01:04:40Z","timestamp":1776301480168,"version":"3.50.1"},"reference-count":40,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,9,29]],"date-time":"2022-09-29T00:00:00Z","timestamp":1664409600000},"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 our present study, two subclasses of starlike functions which are symmetric about the origin are considered. These two classes are defined with the use of the sigmoid function and the trigonometric function, respectively. We estimate the first four initial logarithmic coefficients, the Zalcman functional, the Fekete\u2013Szeg\u00f6 functional, and the bounds of second-order Hankel determinants with logarithmic coefficients for the first class Sseg* and improve the obtained estimate of the existing second-order Hankel determinant of logarithmic coefficients for the second class Ssin*. All the bounds that we obtain in this article are proven to be sharp.<\/jats:p>","DOI":"10.3390\/sym14102039","type":"journal-article","created":{"date-parts":[[2022,9,30]],"date-time":"2022-09-30T01:43:01Z","timestamp":1664502181000},"page":"2039","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Estimation of the Second-Order Hankel Determinant of Logarithmic Coefficients for Two Subclasses of Starlike Functions"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5210-8505","authenticated-orcid":false,"given":"Pongsakorn","family":"Sunthrayuth","sequence":"first","affiliation":[{"name":"Department of Mathematics and Computer Science, Faculty of Science and Technology, Rajamangala University of Technology Thanyaburi (RMUTT) Thanyaburi, Pathumthani 12110, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ibtisam","family":"Aldawish","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, IMSIU (Imam Mohammad Ibn Saud Islamic University), Riyadh 11564, Saudi Arabia"}],"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":"Muhammad","family":"Abbas","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4052-391X","authenticated-orcid":false,"given":"Sheza","family":"El-Deeb","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Damietta University, New Damietta 34517, Egypt"},{"name":"Department of Mathematics, College of Science and Arts, Al-Badaya, Qassim University, Buraidah 52571, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1007\/s00205-011-0483-2","article-title":"Harmonic maps and ideal fluid flows","volume":"204","author":"Aleman","year":"2012","journal-title":"Arch. Ration. Mech. Anal."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1007\/BF02392821","article-title":"A proof of the Bieberbach conjecture","volume":"154","year":"1985","journal-title":"Acta Math."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Avkhadiev, F.G., and Wirths, K.J. (2009). Schwarz-Pick Type Inequalities, Springer Science & Business Media.","DOI":"10.1007\/978-3-0346-0000-2"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"683","DOI":"10.1090\/S0002-9947-1985-0792819-9","article-title":"The de Branges theorem on univalent functions","volume":"290","author":"FitzGerald","year":"1985","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_5","first-page":"21","article-title":"A theorem of de Branges on univalent functions","volume":"13","author":"FitzGerald","year":"1987","journal-title":"Serdica"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"498","DOI":"10.1007\/s11006-005-0149-1","article-title":"On Brennan\u2019s conjecture for a special class of functions","volume":"78","author":"Kayumov","year":"2005","journal-title":"Math. Notes"},{"key":"ref_7","first-page":"6690027","article-title":"Logarithmic coefficient bounds and coefficient conjectures for classes associated with convex functions","volume":"2021","author":"Alimohammadi","year":"2021","journal-title":"J. Funct. Spaces"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"721","DOI":"10.1512\/iumj.1988.37.37035","article-title":"Inequalities for logarithmic coefficients of univalent functions and their derivatives","volume":"37","author":"Andreev","year":"1988","journal-title":"Indiana Univ. Math. J."},{"key":"ref_9","first-page":"5889","article-title":"On the logarithmic coefficients of Bazilevic functions","volume":"217","author":"Deng","year":"2011","journal-title":"Appl. Math. Comput."},{"key":"ref_10","first-page":"337","article-title":"Logarithmic coefficients of univalent functions","volume":"Volume 25","author":"Girela","year":"2000","journal-title":"Annales-Academiae Scientiarum Fennicae Series A1 Mathematica"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1007\/s00605-017-1024-3","article-title":"Logarithmic coefficients and a coefficient conjecture for univalent functions","volume":"185","author":"Ponnusamy","year":"2018","journal-title":"Monatshefte F\u00dcR Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2051","DOI":"10.1090\/S0002-9939-07-08660-1","article-title":"A sharp inequality for the logarithmic coefficients of univalent functions","volume":"135","author":"Roth","year":"2007","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Shi, L., Arif, M., Raza, M., and Abbas, M. (2022). Hankel determinant containing logarithmic coefficients for bounded turning functions connected to a three-leaf-shaped domain. Mathematics, 10.","DOI":"10.3390\/math10162924"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Shi, L., Arif, M., Rafiq, A., Abbas, M., and Iqbal, J. (2022). Sharp bounds of Hankel determinant on logarithmic coefficients for functions of bounded turning associated with petal-shaped domain. Mathematics, 10.","DOI":"10.3390\/math10111939"},{"key":"ref_15","first-page":"445","article-title":"The logarithmic coefficients of close-to-convex functions","volume":"3","author":"Ye","year":"2008","journal-title":"Bull. Inst. Math. Acad. Sin."},{"key":"ref_16","first-page":"111","article-title":"On the coefficients and Hankel determinants of univalent functions","volume":"1","author":"Pommerenke","year":"1996","journal-title":"J. Lond. Math. Soc."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"108","DOI":"10.1112\/S002557930000807X","article-title":"On the Hankel determinants of univalent functions","volume":"14","author":"Pommerenke","year":"1967","journal-title":"Mathematika"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Arif, M., Barukab, O.M., Afzal khan, S., and Abbas, M. (2022). The sharp bounds of Hankel determinants for the families of three-leaf-type analytic functions. Fractal Fract., 6.","DOI":"10.3390\/fractalfract6060291"},{"key":"ref_19","first-page":"5535629","article-title":"Sharp bounds of the coefficient results for the family of bounded turning functions associated with petal shaped domain","volume":"2021","author":"Barukab","year":"2021","journal-title":"J. Funct. Spaces"},{"key":"ref_20","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_21","doi-asserted-by":"crossref","first-page":"767","DOI":"10.1007\/s40840-018-0683-0","article-title":"The bound of the Hankel determinant of the third kind for starlike functions","volume":"42","author":"Kwon","year":"2018","journal-title":"Bull. Of The Malays. Math. Sci. Soc."},{"key":"ref_22","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 1\/2","volume":"13","author":"Lecko","year":"2018","journal-title":"Complex Anal. Oper. Theory"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1007\/s11139-020-00338-y","article-title":"Coefficient estimates for a certain family of analytic functions involving q-derivative operator","volume":"55","author":"Raza","year":"2021","journal-title":"Ramanujan J."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Shi, L., Shutaywi, M., Alreshidi, N., Arif, M., and Ghufran, S.M. (2022). The sharp bounds of the third-order Hankel determinant for certain analytic functions associated with an eight-shaped domain. Fractal Fractional, 6.","DOI":"10.3390\/fractalfract6040223"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Shi, L., Arif, M., Ullah, K., Alreshidi, N., and Shutaywi, M. (2022). On sharp estimate of third Hankel determinant for a subclass of starlike functions. Fractal Fract., 6.","DOI":"10.3390\/fractalfract6080437"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"194","DOI":"10.1186\/s13660-021-02729-1","article-title":"A study of sharp coefficient bounds for a new subfamily of starlike functions","volume":"2021","author":"Ullah","year":"2021","journal-title":"J. Inequalities Appl."},{"key":"ref_27","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_28","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":"141","author":"Zaprawa","year":"2017","journal-title":"Mediterr. J. Math."},{"key":"ref_29","first-page":"139","article-title":"Certain class of bi-Bazilevic functions with bounded boundary rotation involving Sal\u0103ge\u0103n operator","volume":"3","author":"Aouf","year":"2020","journal-title":"Constr. Math. Anal."},{"key":"ref_30","first-page":"36","article-title":"Fekete-Szeg\u00f6 problem for certain subclass of analytic functions with complex order defined by q-analogue of Ruscheweyh operator","volume":"3","author":"Seoudy","year":"2020","journal-title":"Constr. Math. Anal."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"458","DOI":"10.1017\/S0004972721000836","article-title":"Second Hankel determinant of logarithmic coefficients of convex and starlike functions","volume":"105","author":"Kowalczyk","year":"2021","journal-title":"Bull. Aust. Math. Soc."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"727","DOI":"10.1007\/s40840-021-01217-5","article-title":"Second Hankel Determinant of logarithmic coefficients of convex and starlike functions of order alpha","volume":"45","author":"Kowalczyk","year":"2022","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"957","DOI":"10.1007\/s40840-019-00784-y","article-title":"Certain class of starlike functions associated with Modified sigmoid function","volume":"43","author":"Goel","year":"2019","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1007\/s41980-018-0127-5","article-title":"Radius problems for starlike functions associated with the sine function","volume":"45","author":"Cho","year":"2019","journal-title":"Bull. Iran. Math. Soc."},{"key":"ref_35","unstructured":"Pommerenke, C. (1975). Univalent Function, Vanderhoeck & Ruprecht."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1090\/S0002-9939-1982-0652447-5","article-title":"Early coefficients of the inverse of a regular convex function","volume":"85","author":"Libera","year":"1982","journal-title":"Proc. Am. Soc."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1007\/s40315-017-0229-8","article-title":"On the fourth coefficient of functions in the Carath\u00e9odory class","volume":"18","author":"Kwon","year":"2018","journal-title":"Comput. Methods And Function Theory"},{"key":"ref_38","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. Amer. Math. Soc."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1007\/BF03014795","article-title":"\u00dcber den Variabilit\u00e4tsbereich der Fourier\u2019schen Konstanten von position harmonischen Funktionen","volume":"32","year":"1911","journal-title":"Rend. Del Circ. Mat. Palermo"},{"key":"ref_40","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":"C. R. Math. Acad. Sci. Paris"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/10\/2039\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:42:12Z","timestamp":1760143332000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/10\/2039"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,29]]},"references-count":40,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2022,10]]}},"alternative-id":["sym14102039"],"URL":"https:\/\/doi.org\/10.3390\/sym14102039","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,9,29]]}}}