{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,10]],"date-time":"2026-02-10T11:45:40Z","timestamp":1770723940220,"version":"3.49.0"},"reference-count":56,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2023,5,24]],"date-time":"2023-05-24T00:00:00Z","timestamp":1684886400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Umm Al-Qura University","award":["23UQU4320576DSR003"],"award-info":[{"award-number":["23UQU4320576DSR003"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The paper introduces a new family of analytic bi-univalent functions that are injective and possess analytic inverses, by employing a q-analogue of the derivative operator. Moreover, the article establishes the upper bounds of the Taylor\u2013Maclaurin coefficients of these functions, which can aid in approximating the accuracy of approximations using a finite number of terms. The upper bounds are obtained by approximating analytic functions using Faber polynomial expansions. These bounds apply to both the initial few coefficients and all coefficients in the series, making them general and early, respectively.<\/jats:p>","DOI":"10.3390\/axioms12060512","type":"journal-article","created":{"date-parts":[[2023,5,25]],"date-time":"2023-05-25T02:30:06Z","timestamp":1684981806000},"page":"512","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Coefficient Estimation Utilizing the Faber Polynomial for a Subfamily of Bi-Univalent Functions"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5183-2654","authenticated-orcid":false,"given":"Abdullah","family":"Alsoboh","sequence":"first","affiliation":[{"name":"Department of Mathematics, Al-Leith University College, Umm Al-Qura University, Mecca 24382, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9287-7704","authenticated-orcid":false,"given":"Ala","family":"Amourah","sequence":"additional","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\/0000-0002-3884-3957","authenticated-orcid":false,"given":"Fethiye M\u00fcge","family":"Sakar","sequence":"additional","affiliation":[{"name":"Department of Management, Faculty of Economics and Administrative Sciences, Dicle University, Diyarbakir 21280, Turkey"}],"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"}]},{"given":"Nasser","family":"Zomot","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Zarqa University, Zarqa 13110, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,5,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1007\/BF01444293","article-title":"\u00dcber polynomische entwickelungen","volume":"57","author":"Faber","year":"1903","journal-title":"Math. 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