{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,15]],"date-time":"2026-06-15T23:17:05Z","timestamp":1781565425565,"version":"3.54.5"},"reference-count":28,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2022,9,2]],"date-time":"2022-09-02T00:00:00Z","timestamp":1662076800000},"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>In the present paper, due to beta negative binomial distribution series and Laguerre polynomials, we investigate a new family F\u03a3(\u03b4,\u03b7,\u03bb,\u03b8;h) of normalized holomorphic and bi-univalent functions associated with Ozaki close-to-convex functions. We provide estimates on the initial Taylor\u2013Maclaurin coefficients and discuss Fekete\u2013Szeg\u0151 type inequality for functions in this family.<\/jats:p>","DOI":"10.3390\/axioms11090451","type":"journal-article","created":{"date-parts":[[2022,9,5]],"date-time":"2022-09-05T20:48:25Z","timestamp":1662410905000},"page":"451","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":20,"title":["Applications of Beta Negative Binomial Distribution and Laguerre Polynomials on Ozaki Bi-Close-to-Convex Functions"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5287-4656","authenticated-orcid":false,"given":"Isra","family":"Al-Shbeil","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, The University of Jordan, Amman 11942, Jordan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5838-7365","authenticated-orcid":false,"given":"Abbas Kareem","family":"Wanas","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, University of Al-Qadisiyah, Al Diwaniyah 58002, Iraq"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0647-3824","authenticated-orcid":false,"given":"Afis","family":"Saliu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of the Gambia, Birkama Campus, MDI Road, Kanifing Serrekunda P.O. Box 3530, The Gambia"},{"name":"Department of Mathematics, Gombe State University, P.M.B 127, Gombe 760253, Nigeria"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1000-7375","authenticated-orcid":false,"given":"Adriana","family":"C\u0103ta\u015f","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, University of Oradea, 1 University Street, 410087 Oradea, Romania"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,2]]},"reference":[{"key":"ref_1","unstructured":"Duren, P.L. (1983). Univalent Functions, Springer. Grundlehren der Mathematischen Wissenschaften, Band 259."},{"key":"ref_2","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. 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