{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:02:48Z","timestamp":1760234568679,"version":"build-2065373602"},"reference-count":13,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2021,5,19]],"date-time":"2021-05-19T00:00:00Z","timestamp":1621382400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In their paper published in 1990, Miller and Mocanu have investigated the special function Gaussian hypergeometric function in view of its relation to the theory of analytic functions, stating conditions for this function to be univalent using a,b,c\u2208\u211d,\u00a0c\u22600,\u22121,\u22122,\u2026. The study done in this paper extends the results on the univalence of the considered function taking a,b,c\u2208\u2102, with c\u22600,\u22121,\u22122,\u2026 two criteria being stated in the corollaries of the proved theorems. An interpretation of the univalence results from the sets inclusion view is also given, underlining the geometrical properties of the outcomes. Examples showing how the univalence results can be applied are also included.<\/jats:p>","DOI":"10.3390\/sym13050904","type":"journal-article","created":{"date-parts":[[2021,5,19]],"date-time":"2021-05-19T21:49:21Z","timestamp":1621460961000},"page":"904","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Univalence Conditions for Gaussian Hypergeometric Function Involving Differential Inequalities"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2902-4455","authenticated-orcid":false,"given":"Georgia Irina","family":"Oros","sequence":"first","affiliation":[{"name":"Department of Mathematics and Computer Science, University of Oradea, 410087 Oradea, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,5,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1007\/BF02392821","article-title":"A proof of the Bieberbach conjecture","volume":"154","year":"1985","journal-title":"Acta Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1090\/S0002-9939-1990-1017006-8","article-title":"Univalence of Gaussian and confluent hypergeometric functions","volume":"110","author":"Miller","year":"1990","journal-title":"Proc. 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Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1080\/17476930903276134","article-title":"Starlikeness of the Gaussian hypergeometric functions","volume":"55","author":"Ponnusamy","year":"2010","journal-title":"Complex Var. Elliptic Equ."},{"key":"ref_7","first-page":"83","article-title":"Certain sufficiency conditions on Gaussian hypergeometric functions","volume":"5","author":"Swaminathan","year":"2004","journal-title":"J. Inequal. Pure App. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1016\/0022-247X(78)90181-6","article-title":"Second order differential inequalities in the complex plane","volume":"65","author":"Miller","year":"1978","journal-title":"J. Math. Anal. Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1307\/mmj\/1029002507","article-title":"Differential subordinations and univalent functions","volume":"28","author":"Miller","year":"1981","journal-title":"Mich. Math. J."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Miller, S.S., and Mocanu, P.T. (2000). Differential Subordinations. Theory and Applications, Marcel Dekker, Inc.","DOI":"10.1201\/9781482289817"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Oros, G.I. (2021). New Conditions for Univalence of Confluent Hypergeometric Function. Symmetry, 13.","DOI":"10.3390\/sym13010082"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Oros, G.I. (2021). Applications of Inequalities in the Complex Plane Associated with Confluent Hypergeometric Function. Symmetry, 13.","DOI":"10.3390\/sym13020259"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1007\/s13324-020-00372-5","article-title":"Systems of simultaneous differential inequalities, inclusions and subordinations in the complex plane","volume":"10","author":"Antonino","year":"2020","journal-title":"Anal. Math. Phys."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/5\/904\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:04:06Z","timestamp":1760162646000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/5\/904"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,5,19]]},"references-count":13,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2021,5]]}},"alternative-id":["sym13050904"],"URL":"https:\/\/doi.org\/10.3390\/sym13050904","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,5,19]]}}}