{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,13]],"date-time":"2026-06-13T15:14:47Z","timestamp":1781363687794,"version":"3.54.1"},"reference-count":36,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,5,22]],"date-time":"2025-05-22T00:00:00Z","timestamp":1747872000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"University of Oradea, Romania"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Harmonic functions are renowned for their application in the analysis of minimal surfaces. These functions are also very important in applied mathematics. Any harmonic function in the open unit disk U=z\u2208C:z&lt;1 can be written as a sum f=h+g\u00af, where h and g are analytic functions in U and are called the analytic part and the co-analytic part of f, respectively. In this paper, the harmonic shear f=h+g\u00af\u2208SH and its rotation f\u03bc by \u03bc\u03bc\u2208C,\u03bc=1 are considered. Bounds are established for this rotation f\u03bc, specific inequalities that define the Jacobian of f\u03bc are obtained, and the integral representation is determined.<\/jats:p>","DOI":"10.3390\/axioms14060393","type":"journal-article","created":{"date-parts":[[2025,5,22]],"date-time":"2025-05-22T04:32:37Z","timestamp":1747888357000},"page":"393","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["On Certain Bounds of Harmonic Univalent Functions"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3884-3957","authenticated-orcid":false,"given":"Fethiye M\u00fcge","family":"Sakar","sequence":"first","affiliation":[{"name":"Department of Management, Dicle University, Diyarbakir 21280, Turkey"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9614-8656","authenticated-orcid":false,"given":"Omendra","family":"Mishra","sequence":"additional","affiliation":[{"name":"Department of Mathematical and Statistical Sciences, Institute of Natural Sciences and Humanities, Shri Ramswaroop Memorial University, Lucknow 225003, India"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2902-4455","authenticated-orcid":false,"given":"Georgia Irina","family":"Oros","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, University of Oradea, 410087 Oradea, Romania"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8608-8063","authenticated-orcid":false,"given":"Basem Aref","family":"Frasin","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Al al-Bayt University, Mafraq 25113, Jordan"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,5,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"689","DOI":"10.1090\/S0002-9904-1936-06397-4","article-title":"On the non vanishing of the Jacobian in certain one-to-one mappings","volume":"42","author":"Lewy","year":"1936","journal-title":"Bull. 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