{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:43:30Z","timestamp":1760060610757,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2025,9,13]],"date-time":"2025-09-13T00:00:00Z","timestamp":1757721600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the University of Oradea"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In the present work, we define certain families, M\u03a3\u03bc,\u03a5,\u2137,q;\u00a0x and N\u03a3\u03bc,\u03a5,\u2137,q;\u00a0x, of normalized holomorphic and bi-univalent functions associated with Bazilevi\u010d functions and \u2137-pseudo functions involving the q-Bernoulli polynomial, which is defined by the symmetric nature of quantum calculus in the open unit disk U. We determine the upper bounds for the initial symmetry Taylor\u2013Maclaurin coefficients and the Fekete\u2013Szeg\u00f6-type inequalities of functions in the families we have introduced here. In addition, we indicate certain special cases and consequences for our results.<\/jats:p>","DOI":"10.3390\/sym17091532","type":"journal-article","created":{"date-parts":[[2025,9,15]],"date-time":"2025-09-15T07:56:51Z","timestamp":1757923011000},"page":"1532","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Coefficient Estimates and Symmetry Analysis for Certain Families of Bi-Univalent Functions Defined by the q-Bernoulli Polynomial"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5838-7365","authenticated-orcid":false,"given":"Abbas Kareem","family":"Wanas","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Education for Women, University of Al-Qadisiyah, Al Diwaniyah 58001, Iraq"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0002-2734-9432","authenticated-orcid":false,"given":"Qasim Ali","family":"Shakir","sequence":"additional","affiliation":[{"name":"Department of Computer Science, College of Computer Science and Information Technology, University of Al-Qadisiyah, Al Diwaniyah 58001, Iraq"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1000-7375","authenticated-orcid":false,"given":"Adriana","family":"Catas","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, University of Oradea, 1 University Street, 410087 Oradea, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,9,13]]},"reference":[{"key":"ref_1","first-page":"261","article-title":"On Bazilevi\u010d functions","volume":"38","author":"Singh","year":"1973","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"137","DOI":"10.7153\/jca-03-12","article-title":"On \u03bb-pseudo-starlike functions","volume":"3","author":"Babalola","year":"2013","journal-title":"J. Class. Anal."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"522","DOI":"10.1016\/j.joems.2016.03.007","article-title":"On some subclasses of bi-univalent functions associated with pseudo-starlike functions","volume":"24","author":"Joshi","year":"2016","journal-title":"J. Egypt. Math. Soc."},{"key":"ref_4","first-page":"22","article-title":"Coefficient bounds for certain subclasses of analytic function","volume":"4","author":"Prema","year":"2013","journal-title":"J. Math. Anal."},{"key":"ref_5","first-page":"105","article-title":"Coefficient bounds and Fekete\u2013Szeg\u00f6 inequalities for new families of bi-starlike and bi-convex functions associated with the q-Bernoulli polynomials","volume":"25","author":"Wanas","year":"2025","journal-title":"Appl. Math. E-Notes"},{"key":"ref_6","unstructured":"Duren, P.L. (1983). Univalent Functions, Springer."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"102942","DOI":"10.1016\/j.bulsci.2020.102942","article-title":"Upper bound of the third Hankel determinant for a subclass of q-starlike functions associated with the q-exponential function","volume":"167","author":"Srivastava","year":"2021","journal-title":"Bull. Sci. Math."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Richter, W.-D. (2023). Deterministic and random generalized complex numbers related to a class of positively homogeneous functionals. Axioms, 12.","DOI":"10.3390\/axioms12010060"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Richter, W.-D. (2021). Complex numbers related to semi-antinorms, ellipses or matrix homogeneous functionals. Axioms, 10.","DOI":"10.3390\/axioms10040340"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"080061","DOI":"10.1063\/5.0161524","article-title":"Initial coefficient estimates for a new subclass of bi-univalent functions based on Horadam polynomials","volume":"2834","author":"Ramadhan","year":"2023","journal-title":"AIP Conf. Proc."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Srivastava, H.M., Motamednezhad, A., and Adegani, E.A. (2020). Faber polynomial coefficient estimates for bi-univalent functions defined by using differential subordination and a certain fractional derivative operator. Mathematics, 8.","DOI":"10.3390\/math8020172"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1188","DOI":"10.1016\/j.aml.2010.05.009","article-title":"Certain subclasses of analytic and bi-univalent functions","volume":"23","author":"Srivastava","year":"2010","journal-title":"Appl. Math. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"3563","DOI":"10.1007\/s13398-019-00713-5","article-title":"Fekete\u2013Szeg\u00f6 inequality for classes of (p,q)-starlike and (p,q)-convex functions","volume":"113","author":"Srivastava","year":"2019","journal-title":"Rev. Real Acad. Cienc. Exactas F\u00edsicas Nat. Ser. A Matem\u00e1ticas"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Adegani, A., Jafari, E., Bulboac\u0103, T., and Zaprawa, P. (2023). Coefficient bounds for some families of bi-univalent functions with missing coefficients. Axioms, 12.","DOI":"10.3390\/axioms12121071"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1112\/jlms\/s1-8.2.85","article-title":"Eine Bemerkung \u00fcber ungerade schlichte Funktionen","volume":"2","author":"Fekete","year":"1933","journal-title":"J. Lond. Math. Soc."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Miller, S.S., and Mocanu, P.T. (2000). Differential Subordinations: Theory and Applications, Marcel Dekker.","DOI":"10.1201\/9781482289817"},{"key":"ref_17","first-page":"193","article-title":"On q-definite integrals","volume":"41","author":"Jackson","year":"1910","journal-title":"Q. J. Pure Appl. Math."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1017\/S0080456800002751","article-title":"On q-functions and a certain difference operator","volume":"46","author":"Jackson","year":"1908","journal-title":"Trans. R. Soc. Edinb."},{"key":"ref_19","unstructured":"Exton, H. (1983). q-Hypergeometric Functions and Applications, Ellis Horwood."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Gasper, G., and Rahman, M. (2004). Basic Hypergeometric Series, Cambridge Univ. Press. [2nd ed.].","DOI":"10.1017\/CBO9780511526251"},{"key":"ref_21","first-page":"1617","article-title":"q-Derivative of basic hypergeometric series with respect to parameters","volume":"3","author":"Ghany","year":"2009","journal-title":"Int. J. Math. Anal."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"574","DOI":"10.1063\/1.531323","article-title":"q-exponential and q-gamma functions, II. q-gamma functions","volume":"36","author":"McAnally","year":"1995","journal-title":"J. Math. Phys."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1024","DOI":"10.3934\/math.2021061","article-title":"Applications of a certain q-integral operator to the subclasses of analytic and bi-univalent functions","volume":"6","author":"Khan","year":"2021","journal-title":"AIMS Math."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1089","DOI":"10.55730\/1300-0098.3144","article-title":"Application of Gegenbauer polynomials for bi-univalent functions defined by subordination","volume":"46","author":"Sakar","year":"2022","journal-title":"Turk. J. Math."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1007\/s40995-019-00815-0","article-title":"Operators of basic (or q-) calculus and fractional q-calculus and their applications in geometric function theory of complex analysis","volume":"44","author":"Srivastava","year":"2020","journal-title":"Iran. J. Sci. Technol. Trans. A Sci."},{"key":"ref_26","first-page":"239","article-title":"q-Bernoulli numbers and polynomials","volume":"17","year":"1959","journal-title":"Math. Nachr."},{"key":"ref_27","first-page":"539","article-title":"Difference equations of q-Appell polynomials","volume":"245","author":"Mahmudov","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_28","first-page":"070013","article-title":"Exploring new subclasses of bi-univalent functions using Gegenbauer polynomials: Coefficient bounds and Fekete\u2013Szeg\u00f6 problems","volume":"3169","year":"2025","journal-title":"AIP Conf. Proc."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Koshy, T. (2001). Fibonacci and Lucas Numbers with Applications, Wiley.","DOI":"10.1002\/9781118033067"},{"key":"ref_30","first-page":"2","article-title":"A Guide of Fibonacci and Lucas Polynomials","volume":"7","author":"Lupas","year":"1999","journal-title":"Octagon Math. Mag."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"2707","DOI":"10.3906\/mat-1907-41","article-title":"Study on the q-analogue of a certain family of linear operators","volume":"43","author":"Shah","year":"2019","journal-title":"Turk. J. Math."},{"key":"ref_32","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":"Filomat"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s40590-022-00411-0","article-title":"A comprehensive family of bi-univalent functions defined by (m,n)-Lucas polynomials","volume":"28","author":"Swamy","year":"2022","journal-title":"Bol. Soc. Mat. Mex."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"135","DOI":"10.7153\/jmi-10-11","article-title":"Coefficient estimates of new classes of q-starlike and q-convex functions of complex order","volume":"10","author":"Seoudy","year":"2016","journal-title":"J. Math. Inequal."},{"key":"ref_35","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"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/9\/1532\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:45:14Z","timestamp":1760035514000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/9\/1532"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,13]]},"references-count":35,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2025,9]]}},"alternative-id":["sym17091532"],"URL":"https:\/\/doi.org\/10.3390\/sym17091532","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2025,9,13]]}}}