{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T23:30:17Z","timestamp":1773271817203,"version":"3.50.1"},"reference-count":28,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,1,28]],"date-time":"2023-01-28T00:00:00Z","timestamp":1674864000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Umm Al-Qura University","award":["22UQU4320576DSR01"],"award-info":[{"award-number":["22UQU4320576DSR01"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Three subclasses of analytic and bi-univalent functions are introduced through the use of q\u2212Gegenbauer polynomials, which are a generalization of Gegenbauer polynomials. For functions falling within these subclasses, coefficient bounds a2 and a3 as well as Fekete\u2013Szeg\u00f6 inequalities are derived. Specializing the parameters used in our main results leads to a number of new results.<\/jats:p>","DOI":"10.3390\/axioms12020128","type":"journal-article","created":{"date-parts":[[2023,1,30]],"date-time":"2023-01-30T02:01:18Z","timestamp":1675044078000},"page":"128","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":31,"title":["A Generalization of Gegenbauer Polynomials and Bi-Univalent Functions"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9287-7704","authenticated-orcid":false,"given":"Ala","family":"Amourah","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science and Technology, Irbid National University, Irbid 21110, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5183-2654","authenticated-orcid":false,"given":"Abdullah","family":"Alsoboh","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Al-Leith University College, Umm Al-Qura University, Mecca 24231, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2370-6332","authenticated-orcid":false,"given":"Osama","family":"Ogilat","sequence":"additional","affiliation":[{"name":"Department of Basic Sciences, Faculty of Arts and Science, Al-Ahliyya Amman University, Amman 19328, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8645-1029","authenticated-orcid":false,"given":"Gharib Mousa","family":"Gharib","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Zarqa University, Zarqa 13110, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6394-1452","authenticated-orcid":false,"given":"Rania","family":"Saadeh","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Zarqa University, Zarqa 13110, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Maha","family":"Al Soudi","sequence":"additional","affiliation":[{"name":"Department of Basic Scientific Sciences, Applied Science Private University, Amman 11931, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,28]]},"reference":[{"key":"ref_1","unstructured":"Legendre, A. (2022, December 29). Available online: http:\/\/visualiseur.bnf.fr\/Visualiseur?O=NUMM-3221."},{"key":"ref_2","unstructured":"Bateman, H. (1953). Higher Transcendental Functions, McGraw-Hill."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1016\/S0377-0427(02)00642-8","article-title":"The Gegenbauer polynomials and typically real functions","volume":"153","author":"Kiepiela","year":"2003","journal-title":"J. Comput. Appl. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"545","DOI":"10.1023\/B:IJTP.0000028885.42890.f5","article-title":"Disentangling q-Exponentials: A General Approach","volume":"43","author":"Quesne","year":"2004","journal-title":"Int. J. Theor. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1112\/jlms\/s1-8.2.85","article-title":"Eine Bemerkung \u00c3ber ungerade schlichte Funktionen","volume":"1","author":"Fekete","year":"1933","journal-title":"J. Lond. Math. Soc."},{"key":"ref_6","unstructured":"Askey, R., and Ismail, M.E.H. (1983). Studies in Pure Mathematics, Birkh\u00e4user."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"12371","DOI":"10.1088\/0305-4470\/39\/40\/006","article-title":"New connection formulae for the q-orthogonal polynomials via a series expansion of the q-exponential","volume":"39","author":"Chakrabarti","year":"2006","journal-title":"J. Phys. Math. Gen."},{"key":"ref_8","first-page":"5574673","article-title":"Fekete-Szeg\u00f6 inequality for analytic and bi-univalent functions subordinate to Gegenbauer polynomials","volume":"2021","author":"Amourah","year":"2021","journal-title":"J. Funct. Spaces"},{"key":"ref_9","first-page":"625","article-title":"Gegenbauer polynomials and bi-univalent functions","volume":"10","author":"Amourah","year":"2021","journal-title":"Palest. J. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s13370-021-00881-x","article-title":"Application of Chebyshev polynomials to certain class of bi-Bazilevi\u010d functions of order \u03b1 + i\u03b2","volume":"32","author":"Amourah","year":"2021","journal-title":"Afr. Mat."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Aldawish, I., Al-Hawary, T., and Frasin, B.A. (2020). Subclasses of bi-univalent functions defined by Frasin differential operator. Mathematics, 8.","DOI":"10.3390\/math8050783"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1007\/s40590-019-00245-3","article-title":"A comprehensive subclass of bi-univalent functions associated with Chebyshev polynomials of the second kind","volume":"26","author":"Yousef","year":"2020","journal-title":"Bolet\u00edn Soc. Mat. Mex."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1007\/s13324-021-00491-7","article-title":"New subclasses of analytic and bi-univalent functions endowed with coefficient estimate problems","volume":"11","author":"Yousef","year":"2021","journal-title":"Anal. Math. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Yousef, F., Frasin, B.A., and Al-Hawary, T. (2018). Fekete-Szeg\u00f6 inequality for analytic and bi-univalent functions subordinate to Chebyshev polynomials. arXiv.","DOI":"10.2298\/FIL1809229Y"},{"key":"ref_15","first-page":"59","article-title":"Coefficient estimates for a class of analytic and bi-univalent functions","volume":"43","author":"Bulut","year":"2013","journal-title":"Novi Sad J. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1007\/s13370-017-0535-3","article-title":"Chebyshev polynomial coefficient estimates for a class of analytic bi-univalent functions related to pseudo-starlike functions","volume":"29","author":"Magesh","year":"2018","journal-title":"Afr. Mat."},{"key":"ref_17","first-page":"32","article-title":"A comprehensive class of analytic bi-univalent functions by means of Chebyshev polynomials","volume":"8","author":"Bulut","year":"2017","journal-title":"J. Fract. Calc. Appl."},{"key":"ref_18","first-page":"83","article-title":"Initial bounds for analytic and bi-univalent functions by means of chebyshev polynomials","volume":"11","author":"Bulut","year":"2017","journal-title":"Analysis"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1569","DOI":"10.1016\/j.aml.2011.03.048","article-title":"New subclasses of bi-univalent functions","volume":"24","author":"Frasin","year":"2011","journal-title":"Appl. Math. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"631","DOI":"10.1007\/s13370-020-00850-w","article-title":"Some special families of holomorphic and Al-Oboudi type bi-univalent functions related to k-Fibonacci numbers involving modified Sigmoid activation function","volume":"32","author":"Frasin","year":"2020","journal-title":"Afr. Mat."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"573017","DOI":"10.1155\/2013\/573017","article-title":"Coefficient bounds for certain subclasses of bi-univalent function","volume":"2013","author":"Murugusundaramoorthy","year":"2013","journal-title":"Abstr. Appl. Anal."},{"key":"ref_22","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_23","first-page":"693908","article-title":"Coefficient estimate of bi-univalent functions of complex order associated with the Hohlov operator","volume":"2014","author":"Peng","year":"2014","journal-title":"J. Complex Anal."},{"key":"ref_24","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_25","first-page":"90","article-title":"Estimates on coefficients of a general subclass of bi-univalent functions associated with symmetric q\u2013derivative operator by means of the Chebyshev polynomials","volume":"4","author":"Altinkaya","year":"2017","journal-title":"Asia Pac. J. Math."},{"key":"ref_26","first-page":"559","article-title":"Coefficicent estimates for bi-univalent functions","volume":"5","author":"Tan","year":"1984","journal-title":"Chin. Ann. Math. Ser. A"},{"key":"ref_27","first-page":"2229960","article-title":"The Sharp Upper Bounds of the Hankel Determinant on Logarithmic Coefficients for Certain Analytic Functions Connected with Eight-Shaped Domains","volume":"2022","author":"Sunthrayuth","year":"2022","journal-title":"J. Funct. Spaces"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"6354994","DOI":"10.1155\/2022\/6354994","article-title":"Consolidation of a Certain Discrete Probability Distribution with a Subclass of Bi-Univalent Functions Involving Gegenbauer Polynomials","volume":"2022","author":"Amourah","year":"2022","journal-title":"Math. Probl. Eng."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/2\/128\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:17:43Z","timestamp":1760120263000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/2\/128"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,28]]},"references-count":28,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2023,2]]}},"alternative-id":["axioms12020128"],"URL":"https:\/\/doi.org\/10.3390\/axioms12020128","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,1,28]]}}}