{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,3]],"date-time":"2026-04-03T18:52:31Z","timestamp":1775242351564,"version":"3.50.1"},"reference-count":19,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2023,10,31]],"date-time":"2023-10-31T00:00:00Z","timestamp":1698710400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"United Arab Emirates University","award":["UPAR12S127"],"award-info":[{"award-number":["UPAR12S127"]}]},{"name":"United Arab Emirates University","award":["RSP2023R440"],"award-info":[{"award-number":["RSP2023R440"]}]},{"name":"King Saud University","award":["UPAR12S127"],"award-info":[{"award-number":["UPAR12S127"]}]},{"name":"King Saud University","award":["RSP2023R440"],"award-info":[{"award-number":["RSP2023R440"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper presents a geometric approach to the problems in differential subordination theory. The necessary conditions for a function to be in various subfamilies of the class of starlike functions and the class of Carath\u00e9odory functions are studied, respectively. Further, several consequences of the findings are derived.<\/jats:p>","DOI":"10.3390\/sym15112004","type":"journal-article","created":{"date-parts":[[2023,10,31]],"date-time":"2023-10-31T12:44:03Z","timestamp":1698756243000},"page":"2004","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Applications of First-Order Differential Subordination for Subfamilies of Analytic Functions Related to Symmetric Image Domains"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2239-6416","authenticated-orcid":false,"given":"Muhammad Ghaffar","family":"Khan","sequence":"first","affiliation":[{"name":"Institute of Numerical Sciences, Kohat University of Science and Technology, Kohat 26000, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2427-2003","authenticated-orcid":false,"given":"Bilal","family":"Khan","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, East China Normal University, Shanghai 200241, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2909-0970","authenticated-orcid":false,"given":"Jianhua","family":"Gong","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, United Arab Emirates University, Al Ain 15551, United Arab Emirates"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7855-508X","authenticated-orcid":false,"given":"Fairouz","family":"Tchier","sequence":"additional","affiliation":[{"name":"Mathematics Department, College of Science, King Saud University, Riyadh 22452, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1655-5185","authenticated-orcid":false,"given":"Ferdous M. O.","family":"Tawfiq","sequence":"additional","affiliation":[{"name":"Mathematics Department, College of Science, King Saud University, Riyadh 22452, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2023,10,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Khan, M.F., Al-shbeil, I., Khan, S., Khan, N., Haq, W.U., and Gong, J. (2022). Applications of a q-Differential Operator to a Class of Harmonic Mappings Defined by q-Mittag\u2013Leffler Functions. Symmetry, 14.","DOI":"10.3390\/sym14091905"},{"key":"ref_2","unstructured":"Li, Z., Ren, F., Yang, L., and Zhang, S. (1994). Proceeding of Conference on Complex Analytic, International Press."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"159","DOI":"10.4064\/ap-23-2-159-177","article-title":"Extremal problems for a family of functions with a positive real part and for some related families","volume":"23","author":"Janowski","year":"1971","journal-title":"Ann. Polonici Math."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Ur Rehman, M.S., Ahmad, Q.Z., Al-Shbeil, I., Ahmad, S., Khan, A., Khan, B., and Gong, J. (2022). Coefficient Inequalities for Multivalent Janowski Type q-Starlike Functions Involving Certain Conic Domains. Axioms, 11.","DOI":"10.3390\/axioms11100494"},{"key":"ref_5","first-page":"101","article-title":"Radius of convexity of some subclasses of strongly starlike functions","volume":"19","author":"Kanas","year":"1996","journal-title":"Zesz. Nauk. Politech. Rzeszowskiej Mat."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1007\/s41980-018-0127-5","article-title":"Radius problems for starlike functions associated with the sine function","volume":"45","author":"Cho","year":"2019","journal-title":"Bull. Iran. Math. 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Differential Subordinations Theory and Its Applications, Marcel Dekker Inc.","DOI":"10.1201\/9781482289817"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/11\/2004\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:15:00Z","timestamp":1760130900000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/11\/2004"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,10,31]]},"references-count":19,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2023,11]]}},"alternative-id":["sym15112004"],"URL":"https:\/\/doi.org\/10.3390\/sym15112004","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,10,31]]}}}