{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,11]],"date-time":"2026-04-11T07:32:06Z","timestamp":1775892726551,"version":"3.50.1"},"reference-count":33,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2023,11,29]],"date-time":"2023-11-29T00:00:00Z","timestamp":1701216000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Scientific Research Deanship at University of Ha\u2019il\u2014Saudi Arabia","award":["RG-23 033"],"award-info":[{"award-number":["RG-23 033"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The aim of this work is to discuss some conditions for Touchard polynomials to be in the classes\u00a0TBb(\u03c1,\u03c3)\u00a0and\u00a0TKb(\u03c1,\u03c3). Also, we obtain some connection between\u00a0R\u03b7(D,E)\u00a0and\u00a0TKb(\u03c1,\u03c3). Also, we investigate several mapping properties involving these subclasses. Further, we discuss the geometric properties of an integral operator related to the Touchard polynomial. Additionally, briefly mentioned are specific instances of our primary results. Also, several particular examples are presented.<\/jats:p>","DOI":"10.3390\/sym15122125","type":"journal-article","created":{"date-parts":[[2023,11,29]],"date-time":"2023-11-29T09:00:18Z","timestamp":1701248418000},"page":"2125","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["An Application of Touchard Polynomials on Subclasses of Analytic Functions"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5477-0065","authenticated-orcid":false,"given":"Ekram E.","family":"Ali","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, University of Ha\u2019il, Ha\u2019il 81451, Saudi Arabia"},{"name":"Department of Mathematics and Computer Science, Faculty of Science, Port Said University, Port Said 42521, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Waffa Y.","family":"Kota","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Damietta University, New Damietta 34517, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5490-3745","authenticated-orcid":false,"given":"Rabha M.","family":"El-Ashwah","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Damietta University, New Damietta 34517, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7619-6869","authenticated-orcid":false,"given":"Abeer M.","family":"Albalahi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, University of Ha\u2019il, Ha\u2019il 81451, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fatma E.","family":"Mansour","sequence":"additional","affiliation":[{"name":"Department of Physics, College of Science and Arts in Methneb, Qassim University, Methneb 51931, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"R. A.","family":"Tahira","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, Jazan University, Jazan 45142, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,11,29]]},"reference":[{"key":"ref_1","unstructured":"Duren, P.L. (1983). Univalent Functions, Springer."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1007\/BF02863012","article-title":"Bounded starlike functions of complex order","volume":"92","author":"Nasr","year":"1983","journal-title":"Proc. Indian Acad. Sci. (Math. Sci.)"},{"key":"ref_3","first-page":"1","article-title":"Starlike function of complex order","volume":"25","author":"Nasr","year":"1985","journal-title":"J. Natar Sci. Math."},{"key":"ref_4","first-page":"565","article-title":"On convex functions of complex order","volume":"9","author":"Nasr","year":"1982","journal-title":"Bull. Fac. Sci. 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