{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,2]],"date-time":"2026-03-02T22:29:09Z","timestamp":1772490549347,"version":"3.50.1"},"reference-count":37,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2023,12,4]],"date-time":"2023-12-04T00:00:00Z","timestamp":1701648000000},"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":["Symmetry"],"abstract":"<jats:p>In this study, we apply q-symmetric calculus operator theory and investigate a generalized symmetric S\u0103l\u0103gean q-differential operator for harmonic functions in an open unit disk. We consider a newly defined operator and establish new subclasses of harmonic functions in complex order. We determine the sharp results, such as the sufficient necessary coefficient bounds, the extreme of closed convex hulls, and the distortion theorems for a new family of harmonic functions. Further, we illustrate how we connect the findings of previous studies and the results of this article.<\/jats:p>","DOI":"10.3390\/sym15122156","type":"journal-article","created":{"date-parts":[[2023,12,4]],"date-time":"2023-12-04T05:28:21Z","timestamp":1701667701000},"page":"2156","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Sharp Estimates Involving a Generalized Symmetric S\u0103l\u0103gean q-Differential Operator for Harmonic Functions via Quantum Calculus"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5287-4656","authenticated-orcid":false,"given":"Isra","family":"Al-Shbeil","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, The University of Jordan, Amman 11942, Jordan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0361-4887","authenticated-orcid":false,"given":"Shahid","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Abbottabad University of Science and Technology, Abbottabad 22500, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7855-508X","authenticated-orcid":false,"given":"Fairouz","family":"Tchier","sequence":"additional","affiliation":[{"name":"Mathematics Department, College of Science, King Saud University, Riyadh 11495, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1655-5185","authenticated-orcid":false,"given":"Ferdous M. O.","family":"Tawfiq","sequence":"additional","affiliation":[{"name":"Mathematics Department, College of Science, King Saud University, Riyadh 11495, Saudi Arabia"}]},{"given":"Amani","family":"Shatarah","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science and Arts, Taibah University, Al-Madinah 42353, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1000-7375","authenticated-orcid":false,"given":"Adriana","family":"C\u0103ta\u015f","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, University of Oradea, 1 University Street, 410087 Oradea, Romania"}]}],"member":"1968","published-online":{"date-parts":[[2023,12,4]]},"reference":[{"key":"ref_1","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_2","doi-asserted-by":"crossref","unstructured":"Al-Shbeil, I., Gong, J., Ray, S., Khan, S., Khan, N., and Alaqad, H. (2023). The Properties of Meromorphic Multivalent q-Starlike Functions in the Janowski Domain. Fractal Fract., 7.","DOI":"10.3390\/fractalfract7060438"},{"key":"ref_3","first-page":"1","article-title":"On harmonic univalent mappings","volume":"44","author":"Avci","year":"1990","journal-title":"Ann. Univ. Mariae Curie-Sklodowska Sect. A"},{"key":"ref_4","unstructured":"Halim, S.A., and Janteng, A. (2008). Harmonic functions starlike of complex order. Proc. Int. Symp., 132\u2013140."},{"key":"ref_5","first-page":"57","article-title":"Coefficient bounds and univalence criteria for harmonic functions with negative coefficients","volume":"5","author":"Jahangiri","year":"1998","journal-title":"Ann. Univ. Mariae Curie-Sklodowska Sect. A"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"470","DOI":"10.1006\/jmaa.1999.6377","article-title":"Harmonic functions starlike in the unit disc","volume":"235","author":"Jahangiri","year":"1999","journal-title":"J. Math. Anal. Appl."},{"key":"ref_7","first-page":"77","article-title":"S\u0103l\u0103gean-type harmonic univalent functions","volume":"2","author":"Jahangiri","year":"2002","journal-title":"Southwest J. Pure Appl. Math."},{"key":"ref_8","first-page":"191","article-title":"Starlikeness of Rucheweyh type harmonic univalent functions","volume":"26","author":"Jahangiri","year":"2004","journal-title":"J. Indian Acad. Math."},{"key":"ref_9","first-page":"45","article-title":"Goodman-R\u00f8nning type harmonic univalent functions","volume":"41","author":"Rosy","year":"2001","journal-title":"Kyungpook Math. J."},{"key":"ref_10","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":"J. Math. Anal. Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1016\/j.aml.2004.05.003","article-title":"A new class of Salagean-type harmonic univalent functions","volume":"18","year":"2005","journal-title":"Appl. Math. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Masih, V.S., and Kanas, S. (2020). Subclasses of starlike and convex functions associated with the Lima\u00e7on domain. Symmetry, 12.","DOI":"10.3390\/sym12060942"},{"key":"ref_13","first-page":"23","article-title":"Hermite-Hadamard type inequalities for a new class of harmonically convex functions","volume":"38","author":"Olanipekuni","year":"2018","journal-title":"Note Mat."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"94","DOI":"10.1016\/j.amc.2016.03.025","article-title":"Some subclasses of close-to-convex mappings associated with conic regions","volume":"285","author":"Srivastava","year":"2016","journal-title":"Appl. Math. Comput."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1017\/S0080456800002751","article-title":"On q-functions and a certain difference operator","volume":"46","author":"Jackson","year":"1908","journal-title":"Trans. R. Soc. Edinb."},{"key":"ref_16","first-page":"193","article-title":"On q-definite integrals","volume":"41","author":"Jackson","year":"1910","journal-title":"Q. J. Pure Appl. Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"095009","DOI":"10.1103\/PhysRevD.62.095009","article-title":"q-Deformed conformal quantum mechanics","volume":"62","author":"Youm","year":"2000","journal-title":"Phys. Rev. D"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"3743","DOI":"10.1103\/PhysRevLett.71.3743","article-title":"Information in black hole radiation","volume":"71","author":"Strominger","year":"1993","journal-title":"Phys. Rev. Lett."},{"key":"ref_19","first-page":"107","article-title":"On some symmetric q-special functions","volume":"68","author":"Kamel","year":"2013","journal-title":"Matematiche"},{"key":"ref_20","unstructured":"Ernst, T. (2023, October 22). The History of q-Calculus and New Method. Licentiate Thesis, U.U.D.M. Report. Available online: http:\/\/math.uu.se\/thomas\/Lics.pdf."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2241","DOI":"10.1016\/j.camwa.2012.01.076","article-title":"The q-symmetric variational calculus","volume":"64","author":"Martins","year":"2012","journal-title":"Comput. Math. Appl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"222","DOI":"10.1186\/s13662-016-0947-7","article-title":"Certain fractional q-symmetric integrals and q-symmetric derivatives and their application","volume":"2016","author":"Sun","year":"2016","journal-title":"Adv. Differ. Equations"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1727","DOI":"10.1007\/s11253-019-01602-1","article-title":"Subclass of k uniformly starlike functions defined by symmetric q-derivative operator","volume":"70","author":"Kanas","year":"2019","journal-title":"Ukr. Math."},{"key":"ref_24","first-page":"8264693","article-title":"Radius and Differential Subordination Results for Starlikeness Associated with Lima\u00e7on Class","volume":"2022","author":"Saliu","year":"2022","journal-title":"J. Funct. Spaces"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"4250878","DOI":"10.1155\/2022\/4250878","article-title":"Applications of q-symmetric derivative operator to the subclass of analytic and bi-univalent functions involving the faber polynomial coefficients","volume":"2022","author":"Khan","year":"2022","journal-title":"Math. Probl. Eng."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Al-shbeil, I., Khan, S., AlAqad, H., Alnabulsi, S., and Khan, M.F. (2023). Applications of the symmetric quantum-difference operator for new subclasses of meromorphic functions. Symmetry, 15.","DOI":"10.3390\/sym15071439"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"667","DOI":"10.3934\/math.2022042","article-title":"Applications of q-difference symmetric operator in harmonic univalent functions","volume":"7","author":"Zhang","year":"2021","journal-title":"AIMS Math."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1016\/j.jmaa.2008.05.099","article-title":"Extreme points of the closed convex hull of composition operators","volume":"347","author":"Hosokawa","year":"2008","journal-title":"J. Math. Anal. Appl."},{"key":"ref_29","first-page":"39","article-title":"Harmonic univalent functions defined by q-calculus operators","volume":"5","author":"Jahangiri","year":"2018","journal-title":"Int. J. Math. Anal. Appl."},{"key":"ref_30","first-page":"81","article-title":"An application of q-calculus to harmonic univalent functions","volume":"14","author":"Porwal","year":"2018","journal-title":"J. Qual. Measure. Anal."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"362","DOI":"10.1007\/BFb0066543","article-title":"Subclasses of univalent functions","volume":"1013","year":"1983","journal-title":"Lect. Notes Math."},{"key":"ref_32","first-page":"172525","article-title":"On Univalent functions defined by a generalized Salagean operator","volume":"27","year":"2004","journal-title":"Int. J. Math. Math. Sci."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Al-shbeil, I., Khan, N., Tchier, f., Xin, Q., and Malik, S.N. (2023). Khan, S. Coefficient bounds for a family of s-fold symmetric bi-univalent functions. Axioms, 12.","DOI":"10.3390\/axioms12040317"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Rehman, M.U., Ahmad, Q.Z., Al-shbeil, I., Ahmad, S., Khan, A., and Khan, B. (2022). Coefficient inequalities for multivalent Janowski type q-starlike functions involving certain conic domains. Axioms, 11.","DOI":"10.3390\/axioms11100494"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Al-Shbeil, I., Shaba, T.G., and C\u0103ta\u015f, A. (2022). Second Hankel Determinant for the Subclass of Bi-Univalent Functions Using q-Chebyshev Polynomial and Hohlov Operator. Fractals Fract., 6.","DOI":"10.3390\/fractalfract6040186"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"310","DOI":"10.1016\/S0378-4371(01)00680-X","article-title":"q-Deformed structures and nonextensive statistics: A comparative study","volume":"305","author":"Lavagno","year":"2002","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1016\/S0034-4877(09)90021-0","article-title":"Basic-deformed quantum mechanics","volume":"64","author":"Lavagno","year":"2009","journal-title":"Rep. Math. Phys."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/12\/2156\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:37:29Z","timestamp":1760132249000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/12\/2156"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,12,4]]},"references-count":37,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2023,12]]}},"alternative-id":["sym15122156"],"URL":"https:\/\/doi.org\/10.3390\/sym15122156","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,12,4]]}}}