{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:09:12Z","timestamp":1760058552005,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,4,6]],"date-time":"2025-04-06T00:00:00Z","timestamp":1743897600000},"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 Carlson\u2013Shaffer operator plays an important role in the geometric theory of analytic functions. It is associated with the hypergeometric function and the incomplete beta function. The Carlson\u2013Shaffer operator generalizes various other linear operators, such as the Ruscheweyh derivative operator, the Bernardi\u2013Libera\u2013Livingston operator, and the Srivastava\u2013Owa operator. Ideas in the theory of analytic functions are often symmetrically transferred to the theory of harmonic functions. By using the Carlson\u2013Shaffer operator, we introduce a class of harmonic functions defined by weak subordination. Next, we give some necessary and sufficient coefficient conditions for the class of functions. Furthermore, we determine coefficient estimates, distortion bounds, extreme points, and radii of starlikeness and convexity for the defined class.<\/jats:p>","DOI":"10.3390\/sym17040558","type":"journal-article","created":{"date-parts":[[2025,4,9]],"date-time":"2025-04-09T12:05:56Z","timestamp":1744200356000},"page":"558","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Classes of Harmonic Functions Defined by the Carlson\u2013Shaffer Operator"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1482-1080","authenticated-orcid":false,"given":"Jacek","family":"Dziok","sequence":"first","affiliation":[{"name":"Institute of Mathematics, University of Rzesz\u00f3w, 35-310 Rzesz\u00f3w, Poland"}]}],"member":"1968","published-online":{"date-parts":[[2025,4,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1315","DOI":"10.1007\/s13398-018-0542-8","article-title":"Classes of harmonic functions associated with Ruscheweyh derivatives","volume":"113","author":"Dziok","year":"2019","journal-title":"RACSAM"},{"key":"ref_2","first-page":"191","article-title":"Starlikeness of harmonic functions defined by Ruscheweyh derivatives","volume":"26","author":"Jahangiri","year":"2004","journal-title":"J. Indian Acad. Math."},{"key":"ref_3","first-page":"769","article-title":"Some properties for a new generalized subclass of close-to-convex harmonic functions","volume":"44","author":"Li","year":"2024","journal-title":"J. Math. Res. Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"33","DOI":"10.24193\/subbmath.2025.1.03","article-title":"Harmonic close-to-convex mappings associated with S\u0103l\u0103gean q-differential operator","volume":"70","author":"Mishra","year":"2025","journal-title":"Stud. Univ. Babe\u015f-Bolyai Math."},{"key":"ref_5","first-page":"99","article-title":"On certain classes of harmonic functions involving Ruscheweyh derivatives","volume":"96","author":"Murugusundaramoorthy","year":"2004","journal-title":"Bull. Calcutta Math. Soc."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1007\/s13370-023-01129-6","article-title":"Classes of p-valent harmonic functions defined by q-derivative operator","volume":"34","author":"Seoudy","year":"2023","journal-title":"Afr. Mat."},{"key":"ref_7","first-page":"648","article-title":"Properties and characteristics of certain subclass of close-to-convex harmonic mappings","volume":"44","author":"Shi","year":"2024","journal-title":"J. Math. Res. Appl."},{"key":"ref_8","first-page":"327","article-title":"On certain subclass of harmonic univalent functions defined by a generalized Ruscheweyh derivatives operator","volume":"4","author":"Joshi","year":"2010","journal-title":"Appl. Math. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"737","DOI":"10.1137\/0515057","article-title":"Starlike and prestarlike hypergeometric functions","volume":"15","author":"Carlson","year":"1984","journal-title":"SIAM J. Math. Anal."},{"key":"ref_10","first-page":"41","article-title":"On a class of functions with negative coefficients","volume":"28","author":"Dziok","year":"2006","journal-title":"J. Math. Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1090\/S0002-9939-1975-0367176-1","article-title":"New criteria for univalent functions","volume":"49","author":"Ruscheweyh","year":"1975","journal-title":"Proc. Am. Math Soc."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"429","DOI":"10.1090\/S0002-9947-1969-0232920-2","article-title":"Convex and starlike univalent functions","volume":"135","author":"Bernardi","year":"1969","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_13","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. Am. Math. Soc."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"352","DOI":"10.1090\/S0002-9939-1966-0188423-X","article-title":"On the radius of univalence of certain analytic functions","volume":"17","author":"Livingston","year":"1966","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_15","first-page":"53","article-title":"On the distortion theorems","volume":"18","author":"Owa","year":"1978","journal-title":"I Kyungpook Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1017\/S0027763000000854","article-title":"Some characterization and distortion theorems involving fractional calculus, generalized hypergeometric functions, Hadamard products, linear operators, and certain subclasses of analytic functions","volume":"106","author":"Srivastava","year":"1987","journal-title":"Nagoya Math. J."},{"key":"ref_17","first-page":"156","article-title":"Sur un type de transformation analytique g\u00e9n\u00e9ralisant la repr\u00e9sentation conforme et d\u00e9finie au moyen de fonctions harmoniques","volume":"89","author":"Choquet","year":"1945","journal-title":"Bull. Sci. Math."},{"key":"ref_18","first-page":"123","article-title":"Losung der Aufgabe 41","volume":"36","author":"Kneser","year":"1926","journal-title":"Jahresber. Dtsch. Math.-Ver."},{"key":"ref_19","first-page":"35","article-title":"Aufgabe 41","volume":"49","author":"Rado","year":"1926","journal-title":"Jahresber. Dtsch. Math.-Ver."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"3","DOI":"10.5186\/aasfm.1984.0905","article-title":"Harmonic univalent functions","volume":"9","author":"Clunie","year":"1984","journal-title":"Ann. Acad. Sci. Fenn. Ser. A I Math."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"862","DOI":"10.1016\/j.jmaa.2008.08.015","article-title":"Weak subordination for convex univalent harmonic functions","volume":"348","author":"Mauir","year":"2008","journal-title":"J. Math. Anal. Appl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1016\/j.crma.2015.08.001","article-title":"Generalizations of starlike harmonic functions","volume":"354","author":"Dziok","year":"2016","journal-title":"C. R. Math. Acad. Sci. Paris"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"470","DOI":"10.1006\/jmaa.1999.6377","article-title":"Harmonic functions starlike in the unit disk","volume":"235","author":"Jahangiri","year":"1999","journal-title":"J. Math. Anal. Appl."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1006\/jmaa.1997.5882","article-title":"Harmonic univalent functions with negative coefficients","volume":"220","author":"Silverman","year":"1998","journal-title":"Math. Anal. Appl."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1319","DOI":"10.1016\/j.aml.2005.02.003","article-title":"Certain multipliers of univalent harmonic functions","volume":"18","author":"Ahuja","year":"2005","journal-title":"Appl. Math. Lett."},{"key":"ref_26","first-page":"59","article-title":"Univalent harmonic functions","volume":"8","year":"2007","journal-title":"J. Inequal. Pure Appl. Math."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"762","DOI":"10.1006\/jmaa.1996.0411","article-title":"Some families of starlike functions with negative coefficients","volume":"203","author":"Aouf","year":"1996","journal-title":"J. Math. Anal. Appl."},{"key":"ref_28","first-page":"17","article-title":"On certain subclass of harmonic univalent functions","volume":"6","author":"Darus","year":"2008","journal-title":"J. Anal. Appl."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"469","DOI":"10.1016\/S0096-3003(02)00314-4","article-title":"A subclass of harmonic univalent functions with negative coefficients","volume":"142","author":"Karpuzoullari","year":"2003","journal-title":"Appl. Math. Comput."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"673","DOI":"10.1006\/jmaa.1995.1197","article-title":"A certain fractional derivative operator and its applications to a new class of analytic and multivalent functions with negative coefficients II","volume":"192","author":"Srivastava","year":"1995","journal-title":"J. Math. Anal. Appl."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"755","DOI":"10.1090\/S0002-9939-1965-0178131-2","article-title":"Some classes of regular univalent functions","volume":"16","author":"Libera","year":"1965","journal-title":"Proc. Am. Math. Soc."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/4\/558\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:11:18Z","timestamp":1760029878000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/4\/558"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,6]]},"references-count":31,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2025,4]]}},"alternative-id":["sym17040558"],"URL":"https:\/\/doi.org\/10.3390\/sym17040558","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2025,4,6]]}}}