{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,9]],"date-time":"2026-04-09T20:14:47Z","timestamp":1775765687144,"version":"3.50.1"},"reference-count":31,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2021,12,4]],"date-time":"2021-12-04T00:00:00Z","timestamp":1638576000000},"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>The geometry of the image domain plays an important role in the characterization of analytic functions. Therefore, for a comprehensive and detailed study of these functions, a thorough analysis of the geometrical properties of their domains is of prime interest. In this regard, new geometrical structures are introduced and studied as an image domain and then their subsequent analytic functions are defined. Inspired and motivated by ongoing research, Malik et al. introduced a very innovative domain named the cardioid domain, which is symmetric about a real axis. Extending the same work on this symmetric cardioid domain, in this article, we provide a deeper analysis and define and study the convex functions associated with the symmetric cardioid domain, named cardio-convex functions.<\/jats:p>","DOI":"10.3390\/sym13122321","type":"journal-article","created":{"date-parts":[[2021,12,7]],"date-time":"2021-12-07T02:48:13Z","timestamp":1638845293000},"page":"2321","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["On Convex Functions Associated with Symmetric Cardioid Domain"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8940-0569","authenticated-orcid":false,"given":"Sarfraz Nawaz","family":"Malik","sequence":"first","affiliation":[{"name":"Department of Mathematics, Wah Campus, COMSATS University Islamabad, Wah Cantt 47040, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7466-7930","authenticated-orcid":false,"given":"Mohsan","family":"Raza","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Government College University Faisalabad, Faisalabad 38000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6178-8538","authenticated-orcid":false,"given":"Qin","family":"Xin","sequence":"additional","affiliation":[{"name":"Faculty of Science and Technology, University of the Faroe Islands, Vestarabryggja 15, FO 100 Torshavn, Faroe Islands"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1204-2286","authenticated-orcid":false,"given":"Janusz","family":"Sok\u00f3\u0142","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Rzesz\u00f3w University of Technology, Al. Powsta\u00f1c\u00f3w Warszawy 12, 35-959 Rzesz\u00f3w, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rabbiya","family":"Manzoor","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Wah Campus, COMSATS University Islamabad, Wah Cantt 47040, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6522-1451","authenticated-orcid":false,"given":"Saira","family":"Zainab","sequence":"additional","affiliation":[{"name":"School of Electrical Engineering and Computer Science (SEECS), National University of Sciences & Technology (NUST), Sector H-12, Islamabad 44000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,12,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Hitzer, E., and Sangwine, S. (2013). Generalized Analytic Signals in Image Processing: Comparison, Theory and Applications. 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