{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,2]],"date-time":"2026-03-02T16:59:15Z","timestamp":1772470755025,"version":"3.50.1"},"reference-count":47,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2019,4,26]],"date-time":"2019-04-26T00:00:00Z","timestamp":1556236800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Lei Shi","award":["This article is supported financially by the Anyang Normal University, Anyan 455002, Henan, China."],"award-info":[{"award-number":["This article is supported financially by the Anyang Normal University, Anyan 455002, Henan, China."]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In the current article, we consider certain subfamilies     S  e  \u2217     and     C e     of univalent functions associated with exponential functions which are symmetric along real axis in the region of open unit disk. For these classes our aim is to find the bounds of Hankel determinant of order three. Further, the estimate of third Hankel determinant for the family     S  e  \u2217     in this work improve the bounds which was investigated recently. Moreover, the same bounds have been investigated for 2-fold symmetric and 3-fold symmetric functions.<\/jats:p>","DOI":"10.3390\/sym11050598","type":"journal-article","created":{"date-parts":[[2019,4,26]],"date-time":"2019-04-26T07:52:59Z","timestamp":1556265179000},"page":"598","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":74,"title":["An Investigation of the Third Hankel Determinant Problem for Certain Subfamilies of Univalent Functions Involving the Exponential Function"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3079-9944","authenticated-orcid":false,"given":"Lei","family":"Shi","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Anyang Normal University, Anyan 455002, Henan, China"}]},{"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"}]},{"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"}]},{"given":"Shehzad","family":"Hussain","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6417-1181","authenticated-orcid":false,"given":"Hassan","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, Pakistan"}]}],"member":"1968","published-online":{"date-parts":[[2019,4,26]]},"reference":[{"key":"ref_1","unstructured":"Bieberbach, L. (1916). \u00dcber di\u00e9 koeffizienten derjenigen Potenzreihen, welche eine Schlichte Abbildung des Einheitskreises vermitteln, Reimer in Komm."},{"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","first-page":"321","DOI":"10.1017\/S0004972700002410","article-title":"Some applications of differential subordination","volume":"32","author":"Padmanabhan","year":"1985","journal-title":"Bull. Aust. Math. Soc."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1155\/S0161171289000384","article-title":"Convolution and differential subordination","volume":"12","author":"Shanmugam","year":"1989","journal-title":"Int. J. Math. Math. Sci."},{"key":"ref_5","unstructured":"Ma, W., and Minda, D. (2011). A unified treatment of some special classes of univalent functions. Int. J. Math. Math. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"159","DOI":"10.4064\/ap-23-2-159-177","article-title":"Extremal problems for a family of functions with positive real part and for some related families","volume":"23","author":"Janowski","year":"1971","journal-title":"Ann. Polonici Math."},{"key":"ref_7","first-page":"101","article-title":"Radius of convexity of some subclasses of strongly starlike functions","volume":"19","author":"Stankiewicz","year":"1996","journal-title":"Zeszyty Naukowe\/Oficyna Wydawnicza al. Powsta\u0144c\u00f3w Warszawy"},{"key":"ref_8","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_9","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1007\/s40840-014-0026-8","article-title":"On a subclass of strongly starlike functions associated with exponential function","volume":"38","author":"Mendiratta","year":"2015","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1112\/jlms\/s1-41.1.111","article-title":"On the coefficients and Hankel determinants of univalent functions","volume":"1","author":"Pommerenke","year":"1966","journal-title":"J. Lond. Math. Soc."},{"key":"ref_11","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_12","unstructured":"Dienes, P. (1957). The Taylor Series: An Introduction to the Theory of Functions of a Complex Variable, NewYork-Dover."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"362","DOI":"10.1090\/S0002-9904-1963-10923-4","article-title":"Power series with integral coefficients","volume":"69","author":"Cantor","year":"1963","journal-title":"Bull. Am. Math. Soc.."},{"key":"ref_14","first-page":"20","article-title":"Sur les determinants recurrents et less singularities d\u2019une fonction donee por son developpement de Taylor","volume":"7","author":"Edrei","year":"1940","journal-title":"Comput. Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"259","DOI":"10.2140\/pjm.1958.8.295","article-title":"Remarks on de la Vallee Poussin means and convex conformal maps of the circle","volume":"8","author":"Polya","year":"1958","journal-title":"Pac. J. Math."},{"key":"ref_16","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. Inequal. Pure Appl. Math."},{"key":"ref_17","first-page":"619","article-title":"Hankel determinant for starlike and convex functions","volume":"1","author":"Janteng","year":"2007","journal-title":"Int. J. Math. Anal."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1063","DOI":"10.1016\/j.crma.2017.09.006","article-title":"Second Hankel determinant for close-to-convex functions","volume":"355","author":"Zaprawa","year":"2017","journal-title":"C. R. Math."},{"key":"ref_19","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_20","doi-asserted-by":"crossref","first-page":"503","DOI":"10.2298\/FIL1802503S","article-title":"Hankel determinant for a subclass of bi-univalent functions defined by using a symmetric q-derivative operator","volume":"32","author":"Srivastava","year":"2018","journal-title":"Filomath"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Srivastava, H.M., Ahmad, Q.Z., Khan, N., and Khan, B. (2019). Hankel and Toeplitz determinants for a subclass of q-starlike functions associated with a general conic domain. Mathematics, 7.","DOI":"10.3390\/math7020181"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"694","DOI":"10.3906\/mat-1602-25","article-title":"Second Hankel determinant for certain subclasses of bi-univalent functions","volume":"41","author":"Deniz","year":"2017","journal-title":"Turk. J. Math."},{"key":"ref_23","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_24","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1112\/plms\/s3-18.1.77","article-title":"On second Hankel determinant of mean univalent functions","volume":"3","author":"Hayman","year":"1968","journal-title":"Proc. Lond. Math. Soc."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Lee, S.K., Ravichandran, V., and Supramaniam, S. (2013). Bounds for the second Hankel determinant of certain univalent functions. J. Inequal. Appl.","DOI":"10.1186\/1029-242X-2013-281"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"4081","DOI":"10.1007\/s00009-016-0733-5","article-title":"Upper bound of second Hankel determinant for bi-Bazilevic functions","volume":"13","year":"2016","journal-title":"Mediterr. J. Math."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Liu, M.S., Xu, J.F., and Yang, M. (2014). Upper bound of second Hankel determinant for certain subclasses of analytic functions. Abstr. Appl. Anal.","DOI":"10.1155\/2014\/603180"},{"key":"ref_28","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_29","doi-asserted-by":"crossref","first-page":"679","DOI":"10.3906\/mat-1505-3","article-title":"Bounds for the second Hankel determinant of certain bi-univalent functions","volume":"40","author":"Orhan","year":"2016","journal-title":"Turk. J. Math."},{"key":"ref_30","first-page":"1","article-title":"On H3(1) Hankel determinant for some classes of univalent functions","volume":"6","author":"Babalola","year":"2010","journal-title":"Inequal. Theory Appl."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1186\/1029-242X-2012-22","article-title":"Hankel determinant problem of a subclass of analytic functions","volume":"2012","author":"Arif","year":"2012","journal-title":"J. Inequal. Appl."},{"key":"ref_32","first-page":"91","article-title":"Third Hankel determinant for Bazilevi\u010d functions","volume":"5","year":"2016","journal-title":"Adv. Math."},{"key":"ref_33","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_34","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1016\/j.jnnms.2015.03.001","article-title":"Third Hankel determinant for bounded turning functions of order alpha","volume":"34","author":"Krishna","year":"2015","journal-title":"J. Niger. Math. Soc."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"412","DOI":"10.1186\/1029-242X-2013-412","article-title":"Upper bound of third Hankel determinant for a class of analytic functions related with lemniscate of Bernoulli","volume":"2013","author":"Raza","year":"2013","journal-title":"J. Inequal. Appl."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"107","DOI":"10.56947\/gjom.v2i2.202","article-title":"Third Hankel determinant for \u03b1-starlike functions","volume":"2","author":"Shanmugam","year":"2014","journal-title":"Gulf J. Math."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"10","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_38","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":"2019","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Mahmood, S., Khan, I., Srivastava, H.M., and Malik, S.N. (2019). Inclusion relations for certain families of integral operators associated with conic regions. J. Inequal. Appl., 59.","DOI":"10.1186\/s13660-019-2015-9"},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Mahmood, S., Srivastava, H.M., and Malik, S.N. (2019). Some subclasses of uniformly univalent functions with respect to symmetric points. Symmetry, 11.","DOI":"10.3390\/sym11020287"},{"key":"ref_41","first-page":"1","article-title":"Some coefficient inequalities of q-starlike functions associated with conic domain defined by q-derivative","volume":"1","author":"Mahmood","year":"2018","journal-title":"J. Funct. Spaces"},{"key":"ref_42","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_43","doi-asserted-by":"crossref","unstructured":"Lecko, A., Sim, Y.J., and \u015amiarowska, B. (2018). The sharp bound of the Hankel determinant of the third kind for starlike functions of order 1\/2. Complex Anal. Oper. Theory.","DOI":"10.1007\/s11785-018-0819-0"},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Mahmood, S., Srivastava, H.M., Khan, N., Ahmad, Q.Z., Khan, B., and Ali, I. (2019). Upper bound of the third Hankel determinant for a subclass of q-starlike functions. Symmetry, 11.","DOI":"10.3390\/sym11030347"},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Zhang, H.-Y., Tang, H., and Niu, X.-M. (2018). Third-order Hankel determinant for certain class of analytic functions related with exponential function. Symmetry, 10.","DOI":"10.3390\/sym10100501"},{"key":"ref_46","unstructured":"Pommerenke, C., and Jensen, G. (1975). Univalent Functions, Vandenhoeck and Ruprecht."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"8","DOI":"10.1090\/S0002-9939-1969-0232926-9","article-title":"A coefficient inequality for certain subclasses of analytic functions","volume":"20","author":"Keough","year":"1969","journal-title":"Proc. Am. Math. Soc."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/5\/598\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:47:18Z","timestamp":1760186838000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/5\/598"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,4,26]]},"references-count":47,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2019,5]]}},"alternative-id":["sym11050598"],"URL":"https:\/\/doi.org\/10.3390\/sym11050598","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,4,26]]}}}