{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,16]],"date-time":"2026-04-16T01:04:41Z","timestamp":1776301481902,"version":"3.50.1"},"reference-count":52,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2025,2,24]],"date-time":"2025-02-24T00:00:00Z","timestamp":1740355200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The research on coefficient inequalities in various classes of univalent holomorphic functions focuses on interpreting their coefficients through the coefficients associated with Carath\u00e9odory functions. Therefore, researchers can investigate the behavior of coefficient functionals by applying the known inequalities for Carath\u00e9odory functions. This study will explore various coefficient inequalities employing the techniques developed for the previously discussed family of functions. These coefficient inequalities include the Krushkal, Zalcman, and Fekete-Szeg\u00f6 inequalities, along with the second and third Hankel determinants. The class of symmetric starlike functions linked with a petal-shaped domain is the primary focus of our study.<\/jats:p>","DOI":"10.3390\/axioms14030165","type":"journal-article","created":{"date-parts":[[2025,2,24]],"date-time":"2025-02-24T10:04:28Z","timestamp":1740391468000},"page":"165","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["On Coefficient Inequalities for Functions of Symmetric Starlike Related to a Petal-Shaped Domain"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0009-0001-9962-2146","authenticated-orcid":false,"given":"Muhammad","family":"Abbas","sequence":"first","affiliation":[{"name":"Department of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9344-2008","authenticated-orcid":false,"given":"Reem K.","family":"Alhefthi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Sciences, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0095-1346","authenticated-orcid":false,"given":"Daniel","family":"Breaz","sequence":"additional","affiliation":[{"name":"Department of Mathematics, \u201c1 Decembrie 1918\u201d University of Alba Iulia, 510009 Alba Iulia, Romania"}]},{"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"}]}],"member":"1968","published-online":{"date-parts":[[2025,2,24]]},"reference":[{"key":"ref_1","first-page":"940","article-title":"\u00dcber die Koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Eintheitskreises vermitteln","volume":"138","author":"Bieberbach","year":"1916","journal-title":"Sitzungsber. Preuss. Akad. Wiss. Phys. Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1007\/BF01448091","article-title":"Untersuchungen iiber schlichte konforme Abbildungen des Einheitskreises","volume":"89","year":"1923","journal-title":"Math. Ann."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"611","DOI":"10.1215\/S0012-7094-43-01056-7","article-title":"The coefficients of schlicht functions","volume":"10","author":"Schaeffer","year":"1943","journal-title":"Duke Math. J."},{"key":"ref_4","unstructured":"Jenkins, J.A. (2015). On certain coefficients of univalent functions. Analytic Functions, Princeton University Press."},{"key":"ref_5","first-page":"428","article-title":"A proof of the Bieberbach conjecture for the fourth coefficient","volume":"4","author":"Garabedian","year":"1955","journal-title":"J. Rational Mech. Anal."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1007\/BF00281531","article-title":"A proof of the Bieberbach conjecture for the fifth coefficient","volume":"45","author":"Pederson","year":"1972","journal-title":"Arch. Rational Mech. Anal."},{"key":"ref_7","unstructured":"Pommerenke, C. (1975). Univalent functions. Vandenhoeck and Ruprecht G\u00f6ttingen, Springer Science & Business Media."},{"key":"ref_8","first-page":"97","article-title":"On the Bieberbach conjecture for the sixth coefficient","volume":"21","author":"Ozawa","year":"1969","journal-title":"Kodai Math. Sem. Rep."},{"key":"ref_9","first-page":"129","article-title":"An elementary proof of the Bieberbach conjecture for the sixth coefficient","volume":"21","author":"Ozawa","year":"1969","journal-title":"Kodai Math. Sem. Rep."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1007\/BF00251415","article-title":"A proof of the Bieberbach conjecture for the sixth coefficient","volume":"31","author":"Pederson","year":"1968","journal-title":"Arch. Rational Mech. Anal."},{"key":"ref_11","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_12","unstructured":"Ma, W.C., and Minda, D. (1992). A unified treatment of some special classes of univalent functions. Proceedings of the Conference on Complex Analysis, International Press Inc."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1637","DOI":"10.15672\/hujms.1019973","article-title":"Starlike functions associated with an Epicycloid","volume":"51","author":"Gandhi","year":"2022","journal-title":"Hacet. J. Math. Stat."},{"key":"ref_14","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. Iranian Math. Soc."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Alotaibi, A., Arif, M., Alghamdi, M.A., and Hussain, S. (2020). Starlikness associated with cosine hyperbolic function. Mathematics, 8.","DOI":"10.3390\/math8071118"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"923","DOI":"10.1007\/s13370-015-0387-7","article-title":"Starlike functions associated with a cardioid","volume":"27","author":"Sharma","year":"2016","journal-title":"Afr. Mat."},{"key":"ref_17","first-page":"1147","article-title":"Inclusion relations and radius problems for a subclass of starlike functions","volume":"58","author":"Gupta","year":"2021","journal-title":"J. Korean Math. Soc."},{"key":"ref_18","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_19","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_20","doi-asserted-by":"crossref","first-page":"2423","DOI":"10.1080\/17476933.2021.1931149","article-title":"The second Hankel determinant for starlike and convex functions of order alpha","volume":"67","author":"Sim","year":"2022","journal-title":"Complex Var. Elliptic Equ."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Abbas, M., Alhefthi, R.K., Ritelli, D., and Arif, M. (2024). Sharp second-order Hankel determinants bounds for Alpha-convex functions connected with modifed sigmoid functions. Axioms, 13.","DOI":"10.3390\/axioms13120844"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"128","DOI":"10.1007\/s13398-020-00871-x","article-title":"Coefficient bounds and differential subordinations for analytic functions associated with starlike functions","volume":"114","author":"Ebadian","year":"2020","journal-title":"Rev. R. Acad. Cienc. Exactas F\u00eds. Nat. Ser. A Mat. RACSAM"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Mamon, M.A., Halouani, B., Elshazly, I.S., Murugusundaramoorthy, G., and El-Qadeem, A.H. (2024). Second Hankel determinant bound application to certain family of bi-univalent functions. Axioms, 13.","DOI":"10.3390\/axioms13120819"},{"key":"ref_24","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":"Turkish J. Math."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1007\/s00009-017-1031-6","article-title":"An unified approach to second Hankel determinant of bi-subordinate functions","volume":"14","author":"Kanas","year":"2017","journal-title":"Mediterr. J. Math."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2515","DOI":"10.1007\/s10231-021-01089-3","article-title":"The second Hankel determinant for strongly convex and Ozaki close-to-convex functions","volume":"200","author":"Sim","year":"2021","journal-title":"Ann. Mat."},{"key":"ref_27","first-page":"1","article-title":"On D3,1 Hankel determinant for some classes of univalent functions","volume":"6","author":"Babalola","year":"2010","journal-title":"Inequal. Theory Appl."},{"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":"14","author":"Zaprawa","year":"2017","journal-title":"Mediterr. J. Math."},{"key":"ref_29","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":"2019","journal-title":"Complex Anal. Oper. Theory"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Arif, M., Barukab, O.M., Khan, S.A., 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_31","doi-asserted-by":"crossref","first-page":"1163","DOI":"10.1007\/s11253-024-02379-8","article-title":"The sharp bound of the third Hankel determinant for certain subfamilies of analytic functions","volume":"76","author":"Rath","year":"2024","journal-title":"Ukr. Math. J."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Shakir, Q.A., Tayyah, A.S., Breaz, D., Cot\u00eerl\u0103, L.-I., Rapeanu, E., and Sakar, F.M. (2024). Upper bounds of the third Hankel determinant for bi-univalent functions in crescent-shaped domains. Symmetry, 16.","DOI":"10.3390\/sym16101281"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Tang, H., Arif, M., Abbas, M., Tawfiq, F.M.O., and Malik, S.N. (2023). Analysis of coefficient-related problems for starlike functions with symmetric points connected with a three-leaf-shaped domain. Symmetry, 15.","DOI":"10.3390\/sym15101837"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"2109","DOI":"10.2989\/16073606.2024.2352873","article-title":"Third-order Hankel determinants for q-analogue analytic functions defined by a modified q-Bernardi integral operator","volume":"47","author":"Hadi","year":"2024","journal-title":"Quaest. Math."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"191","DOI":"10.7153\/jmi-2023-17-14","article-title":"The sharp bound of the third Hankel determinant of convex functions of order \u22121\/2","volume":"17","author":"Kowalczyk","year":"2023","journal-title":"J. Math. Inequal."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"72","DOI":"10.2969\/jmsj\/01110072","article-title":"On a certain univalent mapping","volume":"11","author":"Sakaguchi","year":"1959","journal-title":"J. Math. Soc. Jpn."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"62","DOI":"10.1007\/s40590-021-00370-y","article-title":"Initial logarithmic coefficients for functions starlike with respect to symmetric points","volume":"27","author":"Zaprawa","year":"2021","journal-title":"Bol. Soc. Mat. Mex."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1007\/s40590-022-00409-8","article-title":"On coefficient problems for functions starlike with respect to symmetric points","volume":"28","author":"Zaprawa","year":"2022","journal-title":"Bol. Soc. Mat. Mex."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Faisal, M.I., Al-Shbeil, I., Abbas, M., Arif, M., and Alhefthi, R.K. (2023). Problems concerning coefficients of symmetric starlike functions connected with the sigmoid function. Symmetry, 15.","DOI":"10.3390\/sym15071292"},{"key":"ref_40","first-page":"993","article-title":"Starlike functions associated with a petal shaped domain","volume":"59","author":"Arora","year":"2022","journal-title":"Bull. Korean Math. Soc."},{"key":"ref_41","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_42","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_43","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_44","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_45","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_46","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 Funct. Theory"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1112\/jlms\/s1-8.2.85","article-title":"Eine bemerkung \u00fcber ungerade schlichte funktionen","volume":"8","author":"Fekete","year":"1933","journal-title":"J. Lond. Math. Soc."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"663","DOI":"10.1515\/gmj.2010.043","article-title":"Proof of the Zalcman conjecture for initial coefficients","volume":"17","author":"Krushkal","year":"2010","journal-title":"Georgian Math. J."},{"key":"ref_49","unstructured":"Krushkal, S.L. (2014). A short geometric proof of the Zalcman and Bieberbach conjectures. arXiv."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"41666","DOI":"10.1016\/j.heliyon.2025.e41666","article-title":"Third-order Hankel determinant sharp estimates for the inverse of complex valued holomorphic functions","volume":"11","author":"Abbas","year":"2025","journal-title":"Heliyon"},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Wanas, A.K., Sakar, F.M., Oros, G.I., and Cot\u00eerl\u0103, L. (2023). Toeplitz determinants for a certain family of analytic functions endowed with borel distribution. Symmetry, 15.","DOI":"10.3390\/sym15020262"},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Jabeen, K., Saliu, A., Gong, J., and Hussain, S. (2024). Majorization problem for q-general family of functions with bounded radius Rotations. Mathematics, 12.","DOI":"10.3390\/math12172605"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/3\/165\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T16:41:36Z","timestamp":1760028096000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/3\/165"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,24]]},"references-count":52,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2025,3]]}},"alternative-id":["axioms14030165"],"URL":"https:\/\/doi.org\/10.3390\/axioms14030165","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,2,24]]}}}