{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:37:17Z","timestamp":1760236637060,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2021,12,8]],"date-time":"2021-12-08T00:00:00Z","timestamp":1638921600000},"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 goal of the present investigation is to introduce a new class of analytic functions (Kt,q), defined in the open unit disk, by means of the q-difference operator, which may have symmetric or assymetric properties, and to establish the relationship between the new defined class and appropriate subordination. We derived relationships of this class and obtained sufficient conditions for an analytic function to be Kt,q. Finally, in the concluding section, we have taken the decision to restate the clearly-proved fact that any attempt to create the rather simple (p,q)-variations of the results, which we have provided in this paper, will be a rather inconsequential and trivial work, simply because the added parameter p is obviously redundant.<\/jats:p>","DOI":"10.3390\/sym13122361","type":"journal-article","created":{"date-parts":[[2021,12,8]],"date-time":"2021-12-08T23:30:00Z","timestamp":1639006200000},"page":"2361","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["A Generalized Class of Functions Defined by the q-Difference Operator"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6974-3430","authenticated-orcid":false,"given":"Loriana","family":"Andrei","sequence":"first","affiliation":[{"name":"Department of Mathematics and Computer Science, University of Oradea, 1 Universitatii Street, 410087 Oradea, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Vasile-Aurel","family":"Caus","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, University of Oradea, 1 Universitatii Street, 410087 Oradea, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,12,8]]},"reference":[{"key":"ref_1","first-page":"16","article-title":"Problems for Normalized Hyper-Bessel Function","volume":"5","author":"Radii","year":"2020","journal-title":"Turk. J. Sci."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1007\/s11139-018-0105-9","article-title":"Geometric and monotonic properties of hyper-Bessel functions","volume":"51","author":"Aktas","year":"2020","journal-title":"Ramanujan J."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Aral, A., Gupta, V., and Agarwal, R.P. (2013). Applications of q-Calculus in Operator Theory, Springer.","DOI":"10.1007\/978-1-4614-6946-9"},{"key":"ref_4","first-page":"77","article-title":"A generalization of starlike functions","volume":"14","author":"Ismail","year":"1990","journal-title":"Complex Var. Theory Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"55","DOI":"10.7146\/math.scand.a-15177","article-title":"Purohit and Ravinder Krishna Raina, Certain subclasses of analytic functions associated with fractional q-calculus operators","volume":"109","author":"Sunil","year":"2011","journal-title":"Math. Scand."},{"key":"ref_6","first-page":"454","article-title":"A generalized operator involving the q hypergeometric function","volume":"65","author":"Aabed","year":"2013","journal-title":"Mat. Vesnik"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1183","DOI":"10.2478\/s12175-014-0268-9","article-title":"Some class of analytic functions related to conic domains","volume":"64","author":"Kanas","year":"2014","journal-title":"Math. Slovaca"},{"key":"ref_8","first-page":"205682","article-title":"Certain bound for q-starlike and q-convex functions with respect to symmetric points","volume":"7","author":"Ramachandran","year":"2015","journal-title":"Int. J. Math. Math. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1260","DOI":"10.37863\/umzh.v73i9.814","article-title":"Certain subclasses of meromorphically -starlike functions associated with the q-derivative operators","volume":"73","author":"Srivastava","year":"2021","journal-title":"Ukr. Kyi Mat. Zhurnal"},{"key":"ref_10","first-page":"43","article-title":"Studies on Generalized Differential-Difference Operator of Normalized Analytic Functions","volume":"45","author":"Suzan","year":"2021","journal-title":"S. Asian Bull. Math."},{"key":"ref_11","first-page":"24","article-title":"Analytic Functions Expressed with q-Poisson Distribution Series","volume":"6","author":"Nezir","year":"2021","journal-title":"Turk. J. Sci."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1007\/s10587-010-0064-9","article-title":"Some properties of certain subclasses of analytic functions with negative coefficients by using generalized Ruscheweyh derivative operator","volume":"60","author":"Deniz","year":"2010","journal-title":"Czechoslovak Math. J."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"165","DOI":"10.4064\/ap-57-2-165-175","article-title":"Uniformly convex functions","volume":"57","author":"Ma","year":"1992","journal-title":"Annal. Polon. Math."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Miller, S.S., and Mocanu, P.T. (2000). Differential Subordinations: Theory and Applications, Marcel Dekker.","DOI":"10.1201\/9781482289817"},{"key":"ref_15","first-page":"127","article-title":"Properties of a new integral operator","volume":"24","author":"Bucur","year":"2016","journal-title":"Analele St. Univ. Ovidius"},{"key":"ref_16","first-page":"123","article-title":"On a class of analytic functions related to the class of the starlike funcctions","volume":"45","author":"Gao","year":"2005","journal-title":"Kyungpook Math. J."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"364","DOI":"10.1016\/0022-247X(91)90006-L","article-title":"On uniformly starlike functions","volume":"155","author":"Goodman","year":"1991","journal-title":"J. Math. Anal. Appl."},{"key":"ref_18","first-page":"33","article-title":"Certain results on a subclass of close-to-convex functions","volume":"3","author":"Goyal","year":"2014","journal-title":"Palestine J. Math."},{"key":"ref_19","first-page":"213","article-title":"Univalence conditions for a general integral operator","volume":"23","author":"Oprea","year":"2015","journal-title":"Analele St. Univ. Ovidius"},{"key":"ref_20","first-page":"11","article-title":"A new subclass of close-to-convex functions","volume":"11","author":"Prajapat","year":"2016","journal-title":"Surveys Math. Its Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1307\/mmj\/1031710535","article-title":"On close-to-convex univalent functions","volume":"3","author":"Reade","year":"1955","journal-title":"Michigan Math. J."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"93","DOI":"10.4064\/ap113-1-6","article-title":"On a generalization of close-to-convex functions","volume":"113","author":"Sahoo","year":"2014","journal-title":"Ann. Pol. Math."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"12","DOI":"10.2307\/2007212","article-title":"Functions which map the interior of the unit circle upon simple regions","volume":"17","author":"Alexander","year":"1915","journal-title":"Ann. Math."},{"key":"ref_24","unstructured":"Ernst, T. (2001). The History of q-Calculus and a New Method. [Licentiate Dissertation, Uppsala University]."},{"key":"ref_25","first-page":"111","article-title":"Close-to-convexity of basic hypergeometric functions using their Taylor coefficients","volume":"35","author":"Raghavendar","year":"2012","journal-title":"J. Math. Appl."},{"key":"ref_26","first-page":"7","article-title":"Growth and distortion theorems for generalized q-starlike functions","volume":"5","author":"Polatoglu","year":"2016","journal-title":"Adv. Math. Sci. J."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1007\/s40995-019-00815-0","article-title":"Operators of Basic (or q-) Calculus and Fractional q-Calculus and Their Applications in Geometric Function Theory of Complex Analysis","volume":"44","author":"Srivastava","year":"2020","journal-title":"Iran. J. Sci. Technol. Trans. A Sci."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Khan, B., Srivastava, H.M., Khan, N., Darus, M., Ahmad, Q.Z., and Tahir, M. (2021). Applications of Certain Conic Domains to a Subclass of q-Starlike Functions Associated with the Janowski Functions. Symmetry, 13.","DOI":"10.3390\/sym13040574"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Khan, B., Liu, Z.G., Srivastava, H.M., Khan, N., Darus, M., and Tahir, M. (2020). A Study of Some Families of Multivalent q-Starlike Functions Involving Higher-Order q-Derivatives. Mathematics, 8.","DOI":"10.3390\/math8091470"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"417","DOI":"10.18514\/MMN.2020.3102","article-title":"A certain class of analytic functions of complex order with a q-analogue of integral operators","volume":"21","author":"Srivastava","year":"2020","journal-title":"Miskolc Math. Notes"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/12\/2361\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:43:02Z","timestamp":1760168582000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/12\/2361"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,12,8]]},"references-count":30,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2021,12]]}},"alternative-id":["sym13122361"],"URL":"https:\/\/doi.org\/10.3390\/sym13122361","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,12,8]]}}}