{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,25]],"date-time":"2026-06-25T08:57:47Z","timestamp":1782377867679,"version":"3.54.5"},"reference-count":15,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2021,5,29]],"date-time":"2021-05-29T00:00:00Z","timestamp":1622246400000},"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 this paper we study a certain differential subordination related to the harmonic mean and its symmetry properties, in the case where a dominant is a linear function. In addition to the known general results for the differential subordinations of the harmonic mean in which the dominant was any convex function, one can study such differential subordinations for the selected convex function. In this case, a reasonable and difficult issue is to look for the best dominant or one that is close to it. This paper is devoted to this issue, in which the dominant is a linear function, and the differential subordination of the harmonic mean is a generalization of the Briot\u2013Bouquet differential subordination.<\/jats:p>","DOI":"10.3390\/sym13060966","type":"journal-article","created":{"date-parts":[[2021,5,31]],"date-time":"2021-05-31T03:45:29Z","timestamp":1622432729000},"page":"966","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["On Certain Differential Subordination of Harmonic Mean Related to a Linear Function"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2289-1986","authenticated-orcid":false,"given":"Anna","family":"Dobosz","sequence":"first","affiliation":[{"name":"Department of Complex Analysis, Faculty of Mathematics and Computer Science, University of Warmia and Mazury in Olsztyn, ul. S\u0142oneczna 54, 10-710 Olsztyn, Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1974-8305","authenticated-orcid":false,"given":"Piotr","family":"Jastrz\u0119bski","sequence":"additional","affiliation":[{"name":"Department of Complex Analysis, Faculty of Mathematics and Computer Science, University of Warmia and Mazury in Olsztyn, ul. S\u0142oneczna 54, 10-710 Olsztyn, Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0185-9402","authenticated-orcid":false,"given":"Adam","family":"Lecko","sequence":"additional","affiliation":[{"name":"Department of Complex Analysis, Faculty of Mathematics and Computer Science, University of Warmia and Mazury in Olsztyn, ul. S\u0142oneczna 54, 10-710 Olsztyn, Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,5,29]]},"reference":[{"key":"ref_1","first-page":"29","article-title":"On some differential subordination of harmonic mean","volume":"LXIV","author":"Chojnacka","year":"2014","journal-title":"Bull. Soc. Sci. Lett. \u0141\u00f3d\u017a"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"427","DOI":"10.1007\/s40315-015-0108-0","article-title":"Differential subordination of harmonic mean","volume":"15","author":"Cho","year":"2015","journal-title":"Comput. Methods Funct. Theory"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1475","DOI":"10.1216\/RMJ-2018-48-5-1475","article-title":"Differential subordination of a harmonic mean to a linear function","volume":"48","author":"Chojnacka","year":"2018","journal-title":"Rocky Mountain J. Math."},{"key":"ref_4","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_5","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1090\/S0002-9939-1976-0399438-7","article-title":"Generating functions for some classes of univalent functions","volume":"56","author":"Lewandowski","year":"1976","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_6","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_7","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1307\/mmj\/1029002507","article-title":"Differential subordination and univalent functions","volume":"28","author":"Miller","year":"1981","journal-title":"Mich. Math. 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